Toán Học & Tuổi Trẻ

Báo Toán Học và Tuổi Trẻ (TH&TT) ra số đầu tiên từ tháng 10 năm 1964. Từ đó cho tới nay, báo TH&TT với biết bao bài toàn với cách giải thông minh luôn được các bạn học sinh giỏi toán THCS THPT, các thầy cô giáo chào đón nồng nhiệt.

MOlympiad tổng hợp và chia sẻ đến các bạn toàn bộ các số báo TH&TT từ năm 1978 cho đến nay, và sẽ tiếp tục cập nhật các số báo TH&TT mới nhất để bạn đọc tham khảo!. Bạn không cần phải download các số báo TH&TT (tổng dung lượng các tập tin lên đến hàng chục GB), bạn chỉ cần đăng nhập vào tài khoản Google để lưu (save) vào tài khoản của bạn là bạn đã có thể đọc và tra cứu lúc cần thiết.



Tại đây bạn cũng tìm thấy 15 số báo đặc san đặc biệt của báo Toán Học Tuổi Trẻ từ năm 2011 đến năm 2014, hai cuốn Tuyển tập 5 nămTuyển tập 30 năm,  Tuyển chọn theo chuyên đề Toán học và Tuổi trẻ, và Các bài toán chọn lọc 45 năm Tạp chí Toán học và Tuổi trẻ các cuốn sách này tuyển chọn các bài viết chuyên đề thú vị, các bài toán hay trên tạp chí.
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Đề Thi Tuyển Sinh Lớp 10 THPT TP Đà Nẵng 2014-2015

  1. a) Tính giá trị của biểu thức $A=\sqrt{9}-\sqrt{4}$.
    b) Rút gọn biểu thức sau với $x > 0$, $x \ne 2$ $$P=\frac{x\sqrt{2}}{2\sqrt{x}+x\sqrt{2}}+\frac{\sqrt{2x}-2}{x-2}$$
  2. Giải hệ phương trình $$\begin{cases}3x+4y&=5\\6x+7y&=8\end{cases}.$$
  3. Cho hàm số $y = x^2$ có đồ thị $(P)$ và hàm số $y = 4x + m$ có đồ thị $(d_m)$.
    a) Vẽ đồ thị $(P)$.
    b) Tìm tất cả các giá trị của $m$ sao cho $(d_m)$ và $(P)$ cắt nhau tại hai điểm phân biệt, trong đó tung độ của một trong hai giao điểm đó bằng $1$.
  4. Cho phương trình tham số $m$ sau $$x^2 + 2(m - 2)x - m^2 = 0$$ a) Giải phương trình khi $m = 0$.
    b) Giả sử phương trình có hai nghiệm phân biệt $x_1$ và $x_2$ với $x_1 < x_2$. Tìm tất cả các giá trị của $m$ sao $|x_1| - |x_2| = 6$.
  5. Cho tam giác $ABC$ vuông tại $A$ có đường cao $AH$ ($H$ thuộc $BC$). Vẽ đường tròn $(C)$ có tâm $C$, bán kính $CA$. Đường thẳng $AH$ cắt đường tròn $(C)$ tại điểm thứ hai là $D$.
    a) Chứng minh $BD$ là tiếp tuyến của đường tròn $(C)$.
    b) Trên cung nhỏ $\overset\frown{AD}$ của đường tròn $(C)$ lấy điểm $E$ sao cho $HE$ song song với $AB$. Đường thẳng $BE$ cắt đường tròn $(C)$ tại điểm thứ hai là $F$. Gọi $K$ là trung điểm của $EF$. Chứng minh rằng $BA^2 = BE\cdot BF$, $\widehat{BHE}=\widehat{BFC}$ và ba đường thẳng $AF$, $ED$, $HK$ song song với nhau từng đôi một.
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Đề Thi Tuyển Sinh Lớp 10 THPT Chuyên Tỉnh Vĩnh Phúc 2016-2017 (Vòng 1)

  1. Cho phương trình tham số $m$ sau $$x^2-2mx+m+2=0.$$ a) Giải phương trình khi $m=2$.
    b) Tìm tất cả các giá trị của $m$ để phương trình có nghiệm duy nhất.
  2. Cho biểu thức sau với $x>0$, $x \neq 1$ $$A=\left(\dfrac{\sqrt{x}-1}{\sqrt{x}+1}-\dfrac{\sqrt{x}+1}{\sqrt{x}-1}\right)\left(\dfrac{1}{2\sqrt{x}}-\dfrac{\sqrt{x}}{2}\right)^2.$$ a) Rút gọn $A$.
    b) Tìm tất cả các giá trị của $x$ để $\dfrac{A}{\sqrt{x}} >3$.
  3. Cho hệ phương trình tham số $m$ sau $$\begin{cases} \left(m+1\right)x-2y&=-1 \\ x+my&=5\end{cases}$$ a) Giải hệ phương trình khi $m=2$.
    b) Tìm tất cả các giá trị của $m$ để hệ có nghiệm duy nhất $(x,y)$ sao cho $5x+y$ lớn nhất
  4. Cho nửa đường tròn $(O)$ có tâm là $(O)$ và đường kính $AB=2R$ ($R$ là một số dương cho trước). Gọi $M$, $N$ là hai điểm di động trên đường tròn $(O)$. sao cho $M$ thuộc cung $\stackrel\frown{AN}$ và tổng khoảng cách từ $A$ và $B$ đến đường thẳng $MN$ bằng $R\sqrt{3}$. Gọi $I$ là giao điểm của các đường thẳng $AN$ và $BM$; $K$ là giao điểm của các đường thẳng $AM$ và $BN$.
    a) Chứng minh rằng bốn điểm $K$, $I$, $M$, $N$ cùng nằm trên một đường tròn $(C)$.
    b) Tính độ dài đoạn thẳng $MN$ và bán kính đường tròn $(C)$ theo $R$.
    c) Xác định ví trí của $M$, $N$ sao cho tam giác $KAB$ có diện tích lớn nhất. Tính giá trị lớn nhất đó theo $R$
  5. Cho $x,y,z$ là các số thực không âm thỏa mãn $x^2+y^2+z^2+xyz=4$. Tìm giá trị nhỏ nhất và giá trị lớn nhất của biểu thức $$P=x+y+z.$$
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Đề Thi Tuyển Sinh Lớp 10 THPT Chuyên Tỉnh Hải Dương 2013-2014 (Vòng 2)

  1. a) Phân tích đa thức sau thành nhân tử $$a^2(b-2c)+b^2(c-a)+2c^2(a-b)+abc.$$ b) Cho $x$, $y$ thỏa $$x=\sqrt[3]{y-\sqrt{y^2+1}}+\sqrt[3]{y+\sqrt{y^2+1}}.$$ Tính giá trị biểu thức sau $$A=x^4+x^3y+3x^2+xy-2y^2+1.$$
  2. a) Giải phương trình $$(x^2-4x+11)(x^4-8x^2+21)=35.$$ b) Giải hệ phương trình $$\begin{cases} (x+\sqrt{x^2+2012})(y+\sqrt{y^2+2012})&=2012\\ x^2+z^2-4(y+z)+8&=0 \end{cases}.$$
  3. a) Chứng minh rằng với mọi số nguyên $n$ thì $n^2+n+1$ không chia hết cho $9$.
    b) Xét phương trình $$x^2-m^2x+2m+2=0.$$ Tìm $m$ nguyên dương để phương trình có nghiệm nguyên.
  4. Cho tam giác $ABC$ vuông tại $A$ có $AB< AC$ ngoại tiếp đường tròn tâm $O$. Gọi $D$, $E$, $F$ lần lượt là tiếp điểm của $(O)$ với các cạnh $AB$, $AC$, $BC$. $BO$ cắt $EF$ tại $I$. $M$ là điểm di chuyển trên đoạn $CE$.
    a) Tính số đo góc $BIF$.
    b) Gọi $H$ là giao điểm của $BM$ và $EF$. Chứng minh rằng nếu $AM=AB$ thì tứ giác $ABHI$ nội tiếp.
    c) Gọi $N$ là giao điểm của $BM$ với cung nhỏ $EF$ của đường tròn $(O)$, $P$ và $Q$ lần lượt là hình chiếu của $N$ trên các đường thẳng $DE$, $DF$. Xác định vị trí của $M$ để độ dài đoạn $PQ$ lớn nhất.
  5. Cho ba số $a$, $b$, $c$ thỏa mãn $0\leq a\leq b\leq c\leq 1$. Tìm giá trị lớn nhất của biểu thức  $$B=(a+b+c+3)\left(\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\right)$$
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[Đáp Án] Đề Thi Chọn Học Sinh Giỏi Quốc Gia THPT 2018-2019

  1. Cho hàm số $f:\;\mathbb R\to\mathbb R^+$ liên tục và thỏa mãn\[\mathop {\lim }\limits_{x \to - \infty } f\left( x \right) = \mathop {\lim }\limits_{x \to + \infty } f\left( x \right) = 0.\] a) Chứng minh rằng tồn tại giá trị lớn nhất của $f(x)$ trên $\mathbb R$.
    b) Chứng minh rằng tồn tại hai dãy số $\left(x_n\right)$ và $\left(y_n\right)$ sao cho $\displaystyle\lim_{n\to\infty} x_n = \lim_{n\to\infty} y_n$ và \[{x_n} < {y_n},\, f\left( {{x_n}} \right) = f\left( {{y_n}} \right),\, \forall n\in\mathbb N.\]
  2. Xét dãy số nguyên $\left( {{x_n}} \right)$ thỏa $0\le x_0<x_1\le 100$ và \[{x_{n + 2}} = 7{x_{n+1}} - {x_n} + 280,\,\forall n \in \mathbb N.\] a) Chứng minh rằng nếu $x_0=2$, $x_1=3$ thì tổng các ước số dương của $$x_{n}x_{n+1}+x_{n+1}x_{n+2}+x_{n+2}x_{n+3}+2018$$ là bội số của $24$.
    b) Tìm các cặp $\left(x_0,x_1\right)$ sao cho $x_nx_{n+1}+2019$ là số chính phương với vô số số tự nhiên $n$. 
  3. Với mỗi đa thức $f(x)=a_0+a_1x+a_2x^2+....+a_nx^n$, đặt $$\Gamma( f(x))=a_0^2+a_1^2+...+a_n^2.$$ Cho $P(x)=(x+1)(x+2)...(x+2020)$. Chứng minh rằng tồn tại ít nhất $2^{2019}$ đa thức đôi một khác nhau $Q_k(x)$ $(1\leq k \leq 2^{2019})$ với hệ số là các số thực dương sao cho $\deg Q_k(x)=2020$ và $\Gamma(Q_k(x)^n)=\Gamma(P(x)^n)$ với mọi số nguyên dương $n$. 
  4. Cho tam giác $ABC$ có trực tâm $H$ và tâm đường tròn nội tiếp $I$, trên các tia $AB$, $AC$, $BC$, $BA$, $CA$, $CB$ lần lượt lấy các điểm $A_1$, $A_2$, $B_1$, $B_2$, $C_1$, $C_2$ sao cho $AA_1=AA_2=BC$, $BB_1=BB_2=AC$, $CC_1=CC_2=AB$. Các cặp đường thẳng $\left(B_1B_2, C_1C_2 \right)$, $\left( C_1C_2, A_1A_2 \right)$, $\left( B_1B_2, A_1A_2 \right)$ lần lượt có giao điểm là $A'$, $B'$, $C'$.
    a) Chứng minh rằng diện tích tam giác $A'B'C'$ không vượt quá diện tích tam giác $ABC$.
    b) Gọi $J$ là tâm đường tròn ngoại tiếp tam giác $A'B'C'$. Các đường thẳng $AJ$, $BJ$, $CJ$ lần lượt cắt các đường thẳng $BC$, $CA$, $AB$ tại $R$, $S$, $T$ tương ứng. Các đường tròn ngoại tiếp các tam giác $AST$, $BTR$, $CRS$ cùng đi qua một điểm $K$. Chứng minh rằng nếu tam giác $ABC$ không cân thì $IHJK$ là hình bình hành. 
  5. Xét đa thức $f(x)=x^2-\alpha x+1$ $(\alpha\in\mathbb{R})$.
    a) Khi $\alpha = \dfrac{\sqrt{15}}{2}$, hãy viết $f(x)$ thành thương của hai đa thức với các hệ số không âm.
    b) Tìm tất cả các giá trị của $\alpha$ để $f(x)$ viết được thành thương của hai đa thức với các hệ số không âm.
  6. Cho tam giác $ABC$ nhọn và không cân, nội tiếp đường tròn $(O)$ và có trực tâm $(H)$. Gọi $M$, $N$, $P$ lần lượt là trung điểm các cạnh $BC$, $CA$, $AB$ và $D$, $E$, $F$ lần lượt là chân các đường cao tương ứng với các đỉnh $A$, $B$, $C$ của tam giác $ABC$. Gọi $K$ là điểm đối xứng của $H$ qua $BC$. Hai đường thẳng $DE$ và $MP$ cắt nhau tại $X$; hai đường thẳng $DF$ và $MN$ cắt nhau tại $Y$.
    a) Đường thẳng $XY$ cắt cung nhỏ $BC$ của $(O)$ tại $Z$. Chứng minh rằng bốn điểm $K$, $Z$, $E$, $F$ đồng viên.
    b) Hai đường thẳng $KE$, $KF$ lần lượt cắt $(O)$ tại các điểm thứ hai là $S$ và $T$ (khác $K$). Chứng minh rằng các đường thẳng $BS$, $CT$ và $XY$ đồng quy.
  7. Có một số mảnh giấy hình vuông có cùng kích thước, mỗi mảnh được chia caro thành $5\times 5$ ô vuông ở cả hai mặt. Ta dùng $n$ màu đẻ tô cho các mảnh giấy sao cho mỗi ô của mỗi mảnh giấy được tô cả hai mặt bởi cùng một màu. Hai mảnh giấy được coi là giống nhau nếu có thể xếp chúng khít lên nhau sao cho các cặp ô vuông ở cùng vị trí có cùng màu. Chứng minhh rằng không có quá $\dfrac{1}{8}\left( {{n^{25}} + 4{n^{15}} + {n^{13}} + 2{n^7}} \right)$ mảnh giấy đôi một không giống nhau.
  1. Do $f(0)>0$ và giả thiết về giới hạn ở hai đầu vô cực, nên tồn tại $m>0$ sao cho $f(x)\le f(0)$ với mỗi $x$ không ở trong đoạn $D=[-m, \,m]$. Vì hàm liên tục nên trên đoạn đóng $D$, nên hàm đạt gía trị lớn nhất là $M$ nào đó.
    • Nếu $M\ge f(0)$, thì $M\ge f(x)$ với mọi $x$, và có $M$ là giá trị lớn nhất trên $\mathbb R$ của $f(x)$.
    • Nếu $M<f(0)$, thì $f(0)$ là giá trị lớn nhất của $f(x)$ trên $\mathbb R$.
    Bây giờ, giả sử $M=f(a)$, để ý là theo định lý Bolzano-Cauchy thì với mỗi số nguyên dương $n$ sẽ tồn tại $x_n\in (-\infty;\,a)$ và $y_n\in (a,\,+\infty)$ sao cho\[f\left( {{x_n}} \right) = f\left( {{y_n}} \right) = M\left( {1 – \frac{1}{{n + 1}}} \right).\] Ta có thể xây dựng $\left(x_n\right)$ tăng, còn $\left(y_n\right)$ giảm và cùng tụ về $a$.
  2. Ta có các đồng dư sau với mỗi số tự nhiên $n$\[\begin{array}{l}{x_{n + 2}} \equiv – \left( {{x_{n + 1}} + {x_n}} \right) \pmod 8,\\{x_{n + 2}} \equiv {x_{n + 1}} – {x_n} + 1 \pmod 3.\end{array}\] Từ đây, khảo sát số dư của dãy khi chia $8$ và chia $3$, qua phép truy toán ta được $x_n$ chia cho $3$ dư $2$ nếu $n$ chẵn và $x_n$ chia $3$ dư $0$ nếu $n$ lẻ. Đồng thời, $x_n$ chia cho $8$ dư $2$ nếu $3\mid n$, và khi $3\nmid n$ thì $x_n$ chia $8$ dư $2$. Cho nên có\[\begin{array}{l}{M_n} &= {x_n}{x_{n + 1}} + {x_{n + 1}}{x_{n + 2}} + {x_{n + 2}}{x_{n + 3}} + 2018\\&\equiv {x_n}{x_{n + 1}} + {x_{n + 1}}{x_n} + {x_n}{x_{n + 1}} + 2018\\&\equiv – 1 \pmod 3.\end{array}\] Trong ba số nguyên liên tiếp, sẽ có duy nhất một số chia hết cho $3$, nên\[\begin{array}{l}{M_n} &= {x_n}{x_{n + 1}} + {x_{n + 1}}{x_{n + 2}} + {x_{n + 2}}{x_{n + 3}} + 2018\\& \equiv {x_n}{x_{n + 1}} + {x_{n + 1}}{x_n} + {x_{n + 2}}{x_n} + 2018\\&\equiv 2\times 3+3\times 3+3\times 2+2018\\& \equiv – 1 \pmod 8.\end{array}\] Tổng hợp lại ta sẽ có\[{M_n} = {x_n}{x_{n + 1}} + {x_{n + 1}}{x_{n + 2}} + {x_{n + 2}}{x_{n + 3}} + 2018 \equiv – 1 \pmod{24}.\] Do $M_n\equiv -1\pmod 3$, nên $M_n$ không thể là số chính phương, và ta viết tổng các ước số dương của $M_n$ thành \[\sigma = \sum\limits_{d} {\left( {d + \frac{{{M_n}}}{d}} \right)} .\] Trong đó, $d$ chạy khắp các ước dương của $M_n$ nhỏ hơn $\sqrt{M_n}$. Để ý là nếu $d\mid M_n$ thì $d$ lẻ và $3\nmid d$, cho nên từ $M_n\equiv -1\pmod{24}$ có \[\begin{array}{l} d + \dfrac{{{M_n}}}{d} = \dfrac{{{d^2} + {M_n}}}{d} \equiv 0 \pmod 3,\\d + \dfrac{{{M_n}}}{d} = \dfrac{{{d^2} + {M_n}}}{d} \equiv 0 \pmod 8.\end{array}\] Từ đó lấy tổng lại là có $$24\mid\sigma .$$ b) Giả sử $x_0$, $x_1$ là các số thỏa yêu cầu, ta có đẳng thức sau\[{x_{n + 3}} – 7{x_{n + 2}} + {x_{n + 1}} = {x_{n + 2}} – 7{x_{n + 1}} + {x_n},\;\;\;{\kern 1pt} \forall {\mkern 1mu} n \in \mathbb N.\]Từ đó, nếu với mỗi $n\in\mathbb N$ đặt $x_{n+2}-9x_{x_n+1}-x_n=a_n$, thế thì có\[{a_{n + 1}} = – {a_n} – 18{x_{n + 1}},\;\;\;{\kern 1pt} \forall {\mkern 1mu} n \in \mathbb N.\]Bình phương hai vế ta có\[\begin{array}{l}a_{n + 1}^2 – 36{x_{n + 2}}{x_{n + 1}} &= a_n^2 + 36{a_n}{x_{n + 1}} + {18^2}x_{n + 1}^2 – 36{x_{n + 2}}{x_{n + 1}}\\&= a_n^2 + 36\left( {{x_{n + 2}} – 9{x_{n + 1}} – {x_n}} \right)x_{n+1} + {18^2}x_{n + 1}^2 – 36{x_{n + 2}}{x_{n + 1}}\\&= a_n^2 – 36{x_{n + 1}}{x_n}.\end{array}\]Như vậy, $a_n^2-36x_{n+1}x_n$ là dãy hằng, và ta có\[\begin{array}{l}a_n^2 – 36{x_{n + 1}}{x_n} &= {a_0}^2 – 36{x_1}{x_0}\\&= {\left( {{x_2} – 9{x_1} – {x_0}} \right)^2} – 36{x_1}{x_0}\\&= {\left( {7{x_1} – {x_0} – 9{x_1} – {x_0}} \right)^2} – 36{x_1}{x_0}\\&= {\left( {280 – 2{x_1} – 2{x_0}} \right)^2} – 36{x_1}{x_0}.\end{array}\] Từ đó mà có được\[36\left( {{x_n}{x_{n + 1}} + 2019} \right) = a_n^2 – D.\]Trong đó, $D= {\left( {280 – 2{x_1} – 2{x_0}} \right)^2} – 36{x_1}{x_0}-36\times 2019$, đó là một hằng số và nếu ta kết hợp với việc $\lim x_nx_{n+1}=+\infty$ kéo theo $\lim \left| {{a_n}} \right|=+\infty$ thế thì với $n$ đủ lớn, sẽ xảy ra bất đẳng thức $$2\left| {{a_n}} \right| + 1 > – D > – 2\left| {{a_n}} \right| + 1,$$ nó dẫn đến\[{\left( {\left| {{a_n}} \right| + 1} \right)^2} > a_n^2 – D > {\left( {\left| {{a_n}} \right| – 1} \right)^2}.\]Vì $a_n^2 – D$ là số chính phương với $n$ lớn thỏa thích, thế nên $D=0$, tức là có\[{\left( {140 – {x_1} – {x_0}} \right)^2} = 9\left( {2019 + {x_1}{x_0}} \right).\]Giờ để ý $0\le x_0<x_1\le 100$, để có đánh giá sau\[2019 + {x_0}{x_1} = \frac{{{{\left( {140 – {x_1} – {x_0}} \right)}^2}}}{9} < 2178.\]Do $2019 + {x_0}{x_1}$ là số chính phương nằm giữa $2019$ và $2178$, nên có hai khả năng sau.
    • Nếu $2019 + {x_0}{x_1}=46^2$, khi đó $x_0x_1=97$ và $x_0+x_1=2$, nên không xảy ra tình huống này.
    • Nếu $2019 + {x_0}{x_1}=45^2$, khi đó $x_0x_1=6$ và $x_0+x_1=5$. Từ đây, $x_0=2$ và $x_1=3$.
    Vậy, tất cả các cặp cần tìm là \[\left( {{x_0},{\mkern 1mu} {x_1}} \right) = \left( {2,\,3} \right).\]
  3. Trước hết ta có các tính chất sau
    • Cho $f(x)=a_0+a_1x+a_2x^2+....+a_nx^n$ và $a,b$ là các số thực dương. Khi đó $$\Gamma((ax+b)f(x))=(a^2+b^2)(a_0^2+a_1^2+...+a_n ^2)+2ab(a_0.a_1+a_1a_2+...+a_{n-1}.a_n).$$
    • Từ tính chất trên suy ra $$\Gamma((ax+b)f(x))=\Gamma((bx+a)f(x)).$$
    • Với mọi số nguyên dương $n$ ta có $$\Gamma((ax+b)^nf^n(x))=\Gamma((ax+b)^{n-1}(bx+a)f^n(x))=...=\Gamma((bx+a)^nf^n(x)).$$
    Trở lại bài toán. Đặt $X=\{2;3;...;2020\}$, Gọi $A$ là tập con bất kỳ của $X$, đặt $$Q_{A}(x)=(x+1)\prod\limits_{k \notin A} {\left( {x + k} \right)}\prod\limits_{k \in A} {\left( {kx + 1} \right)}.$$ Áp dụng tính chất $(ii)$, ta được $\Gamma(Q_A(x))=\Gamma(P(x))$ và cũng từ tính chất $(iii)$ suy ra $\Gamma(Q_A^n(x))=\Gamma (P^n(x))$ với mọi số nguyên dương $n$. Hay $Q_A(x)$ là một đa thức thỏa mãn điều kiện bài toán. Do tập $X$ có $2^{2019}$ tập con. Nên có ít nhất $2^{2019}$ đa thức thỏa mãn yêu cầu bài toán. 
  4. http://analgeomatica.blogspot.com/2019/01/bai-hinh-vmo-bai-4-ngay-1-nam-2019.html hoặc https://nguyenvanlinh.files.wordpress.com/2019/01/bai-4-vmo-2019.pdf
  5. Trước hết ta có một số kết quả sau
    • $\displaystyle\lim_{n\to\infty}\dfrac{n}{q^n}=0$ với $q>1$ cho trước.
    • $\displaystyle\lim_{n\to\infty}\dfrac{a^n}{C_{2n}^n}=0$ với mọi số thực dương $a<4$. Thật vậy, ta có $$0<\dfrac{a^n}{C_{2n}^n}<\dfrac{a^n}{\dfrac{1}{2n}. 4^n}=\dfrac{2n}{q^n}$$ với $q=\dfrac{4}{a}>1$. Suy ra $\displaystyle\lim_{n\to\infty}\dfrac{a^n}{C_{2n}^n}=0$.
    Trở lại bài toán, ta sẽ chứng minh phần $(b)$ trước. Giả sử rằng $x^2-\alpha x+1=\dfrac{P(x)}{Q(x)}$ với $P(x),Q(x)$ là hai đa thức với hệ số không âm. Từ đây suy ra $2-\alpha=\dfrac{P(1)}{Q(1)}>0$. Hay $\alpha <2$. Và đây là tất cả những số $\alpha$ thỏa mãn yêu cầu bài toán. Khi $\alpha \leq 0$ là trường hợp tầm thường (vì $f(x)=\dfrac{f(x)}{1}$). Ta chỉ xét $\alpha >0$. Do $\alpha<2$ nên $a=2+\alpha <4$. Từ tính chất $(ii)$, suy ra tồn tại $n$ sao $\dfrac{(2+\alpha)^{2^{n-1}}}{C_{2^n}^{2^{n-1}}}<1$ hay $(2+\alpha)^{2^{n-1}}<C_{2^n}^{2^{n-1}}$. Xét các biểu thức $$\begin{align}P(x)&=x^{2^{n-1}}[(\sqrt{x}+\dfrac{1}{\sqrt{x}})^{2^n}-(2+\alpha)^{2^{n-1}}] \\ Q(x)&=x^{2^{n-1}-1}[(\sqrt{x}+\dfrac{1}{\sqrt{x}})^{2}+(2+\alpha)]...[(\sqrt{x}+\dfrac{1}{\sqrt{x}})^{2^{n-1}}+(2+\alpha)^{2^{n-2}}]\end{align}.$$ Rõ ràng $P(x)$ và $Q(x)$ là các đa thức với hệ số không âm, đồng thời $$\dfrac{P(x)}{Q(x)}=x[(\sqrt{x}+\dfrac{1}{\sqrt{x}})^{2}-(2+\alpha)]=f(x).$$ Hay ta có điều phải chứng minh.
  6. http://analgeomatica.blogspot.com/2019/01/bai-hinh-vmo-bai-6-ngay-2-nam-2019.html hoặc https://nguyenvanlinh.files.wordpress.com/2019/01/bai-6-vmo-2019.pdf
  7. Trước hết ta có một số nhận xét sau
    • Có tất cả là $n^{25}$ cách tô màu tất cả các ô của mảnh giấy (kể cả giống nhau do quay và lật).
    • Phép lật ngược mảnh giấy được xem như là phép đối đối xứng của hình vuông (bao gồm phép đối xứng ngang dọc hoặc phép đối xứng chéo).
    • Một số mảnh giấy có thể tự xoay hoặc tự đối xứng thành chính nó và tất cả các mảnh giấy được chia thành $8$ trường hợp sau
      • Quay được $90^\circ$ và đối xứng được: Trường hợp này có tất cả là $x_1=n^6$ cách tô màu và mỗi chúng xuất hiện đúng $1$ lần. 
      • Quay được $90^\circ$ và không đối xứng được: Trường hợp này có tất cả là $x_2=n^7-n^6$ cách tô màu và mỗi chúng xuất hiện đúng $2$ lần. 
      • Quay $180^\circ$ và đối xứng ngang hoặc dọc (không quay $90^\circ$): Trường hợp này có tất cả là $x_3=n^8-n^6$ cách tô màu và mỗi chúng xuất hiện đúng $2$ lần. 
      • Quay $180^\circ$ và đối xứng chéo (không quay $90^\circ$): Trường hợp này có tất cả là $x_4=n^9-n^6$ cách tô màu và mỗi chúng xuất hiện đúng $2$ lần. 
      • Quay $180^\circ$ và không đối xứng (không quay $90^\circ$): Trường hợp này có $x_5\leq n^{13}-n^7$ cách tô màu và mỗi chúng xuất hiện đúng $4$ lần. 
      • Không quay được nhưng đối xưng ngang hoặc dọc: Trường hợp này có tất cả là $x_6\leq n^{14}-n^8$ cách tô màu và mỗi chúng xuất hiện đúng $4$ lần. 
      • Không quay được nhưng đối xứng chéo: Trường hợp này có tất cả là $x_7\leq n^{15}-n^9$ cách tô màu và mỗi chúng xuất hiện đúng $4$ lần. 
      • Không quay và không đối xứng: Đây là những cách tô màu còn lại, có tất cả là $x_8=n^{25}-x_1-x_2-x_3-x_4-x_5-x_6-x_7$ cách tô màu và mỗi chúng xuất hiện đúng $8$ lần.
    Vậy số cách tô màu thỏa mãn yêu cầu bài toán là $$\begin{align}S&=x_1+\frac{x_2+x_3+x_4}{2}+\frac{x_5+x_6+x_7}{4}+\frac{x_8}{8}\\&=\frac{n^{25}+x_7+x_6+x_5+3x_4+3x_3+3x_2+7x_1}{8}\\&\leq \frac{n^{25}+n^{15}+n^{14}+n^{13}+2n^9+2n^8+2n^7-2n^6}{8}\\&\leq\frac{n^{25}+4.n^{15}+n^{13}+2n^7}{8}\end{align}$$
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Đề Thi Olympic Chuyên Khoa Học Tự Nhiên Hà Nội 2019

  1. Tìm tất cả các số nguyên dương $n$ sao cho $n^3$ là ước của $3^n-1$.
  2. Với $k$ là số nguyên dương, cho dãy số $(u_n)$ xác định bởi $$u_1=k,\quad u_{n+1}=\dfrac{(n+2)u_n-2k+4}{n},\,\forall n \in \mathbb{Z}^+.$$ Chứng minh rằng tồn tại số nguyên dương $k$ sao cho trong dãy số $(u_n)$ có đúng $2019$ số hạng là số chính phương.
  3. Cho tam giác $ABC$. Giả sử có điểm $P$ nằm trong tam giác $ABC$ sao cho $\angle BPC=\angle CPA=\angle APB$. $PB$, $PC$ theo thứ tự cắt $CA$, $AB$ tại $E$, $F$. $D$ là điểm di chuyển trên cạnh $BC$. Đường thẳng $DF$ cắt đường thẳng $AC$ tại $M$. Đường thẳng $DE$ cắt đường thẳng $AB$ tại $N$.
    a) Chứng minh rằng số đo góc $\angle MPN$ không đổi khi $D$ thay đổi.
    b) Gọi giao của đường thẳng $EF$ với đường thẳng $MN$ là $Q$. Chứng minh rằng $PQ$ là phân giác của góc $\angle MPN$.
  4. Chứng minh rằng với mọi số thực dương $a$, $b$, $c$ ta luôn có $$\dfrac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{3}{2} \cdot \frac{ab+bc+ca}{a^2+b^2+c^2} \geq \frac{9}{2}$$
  5. Tìm tất cả các đa thức hệ số thực $P(x)$ sao cho $$P(x^3+x^2+1)=P(x+2)P(x^2+1)$$
  6. Cho ngũ giác lồi $ABCDE$ nội tiếp trong đường tròn $(O)$ sao cho $AD$ là đường kính, đồng thời $EA=ED$. Dựng ra ngoài ngũ giác $ABCDE$, tam giác $BCF$ vuông cân tại $F$, và hai hình vuông $ABMN$, $CDPQ$. Giả sử $MQ$ cắt $NP$ tại $R$. Gọi $S$, $T$ lần lượt là trung điểm $MQ$ và $OS$. Chứng minh rằng $RT \perp EF$.
  7. Một khu vực quốc tế có $512$ sân bay. Mỗi sân bay đều có thể bay trực tiếp tới ít nhất $5$ sân bay khác. Biết rằng ta có thể đi từ bất kì sân bay nào đến bất kì sân bay khác thông qua một hoặc nhiều chuyến bay trực tiếp. Với mỗi cặp sân bay ta xét tuyến đường ngắn nhất nối giữa chúng, tức là tuyến đường mà nó gồm số lượng ít nhất các đường bay trực tiếp nối giữa hai sân bay này. Hỏi số lượng đường bay trực tiếp lớn nhất có thể có trong một tuyến đường ngắn nhất giữa hai sân bay nào đó là bao nhiêu?.
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    [Solutions] Czech-Polish-Slovak Mathematics Competition 2018

    1. Determine all functions $f : \mathbb R \to \mathbb R$ such that for all real numbers $x$ and $y$, $$f(x^2 + xy) = f(x)f(y) + yf(x) + xf(x+y).$$
    2. Let $ABC$ be an acute scalene triangle. Let $D$ and $E$ be points on the sides $AB$ and $AC$, respectively, such that $BD=CE$. Denote by $O_1$ and $O_2$ the circumcentres of the triangles $ABE$ and $ACD$, respectively. Prove that the circumcircles of the triangles $ABC$, $ADE$, and $AO_1O_2$ have a common point different from $A$.
    3. There are $2018$ players sitting around a round table. At the beginning of the game we arbitrarily deal all the cards from a deck of $K$ cards to the players (some players may receive no cards). In each turn we choose a player who draws one card from each of the two neighbors. It is only allowed to choose a player whose each neighbor holds a nonzero number of cards. The game terminates when there is no such player. Determine the largest possible value of $K$ such that, no matter how we deal the cards and how we choose the players, the game always terminates after a finite number of turns.
    4. Let $ABC$ be an acute triangle with the perimeter of $2s$. We are given three pairwise disjoint circles with pairwise disjoint interiors with the centers $A, B$, and $C$, respectively. Prove that there exists a circle with the radius of $s$ which contains all the three circles.
    5. In a $2 \times 3$ rectangle there is a polyline of length $36$, which can have self-intersections. Show that there exists a line parallel to two sides of the rectangle, which intersects the other two sides in their interior points and intersects the polyline in fewer than $10$ points.
    6. We say that a positive integer $n$ is fantastic if there exist positive rational numbers $a$ and $b$ such that $$ n = a + \frac 1a + b + \frac 1b.$$ a) Prove that there exist infinitely many prime numbers $p$ such that no multiple of $p$ is fantastic.
      b) Prove that there exist infinitely many prime numbers $p$ such that some multiple of $p$ is fantastic.
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    Czech-Polish-Slovak Mathematics Competition 2017

    1. Find all positive real numbers $c$ such that there are infinitely many pairs of positive integers $(n,m)$ satisfying the following conditions $$n \ge m+c\sqrt{m - 1}+1$$ and among numbers $n, n+1,...., 2n-m$ there is no square of an integer.
    2. Let ${\omega}$ be the circumcircle of an acute-angled triangle ${ABC}$. Point ${D}$ lies on the arc ${BC}$ of ${\omega}$ not containing point ${A}$. Point ${E}$ lies in the interior of the triangle ${ABC}$, does not lie on the line ${AD}$, and satisfies ${\angle DBE =\angle ACB}$ and ${\angle DCE = \angle ABC}$. Let ${F}$ be a point on the line ${AD}$ such that lines ${EF}$ and ${BC}$ are parallel, and let ${G}$ be a point on ${\omega}$ different from ${A}$ such that ${AF = FG}$. Prove that points $D$, $E$, $F$, $G$ lie on one circle.
    3. Let ${k}$ be a fixed positive integer. A finite sequence of integers ${x_1,x_2, ..., x_n}$ is written on a blackboard. Pepa and Geoff are playing a game that proceeds in rounds as follows.
      • In each round, Pepa first partitions the sequence that is currently on the blackboard into two or more contiguous subsequences (that is, consisting of numbers appearing consecutively). However, if the number of these subsequences is larger than ${2}$, then the sum of numbers in each of them has to be divisible by ${k}$.
      • Then Geoff selects one of the subsequences that Pepa has formed and wipes all the other subsequences from the blackboard.
      The game finishes once there is only one number left on the board. Prove that Pepa may choose his moves so that independently of the moves of Geoff, the game finishes after at most ${3k}$ rounds.
    4. Let ${ABC}$ be a triangle. Line $l$ is parallel to ${BC}$ and it respectively intersects side ${AB}$ at point ${D}$, side ${AC}$ at point ${E}$, and the circumcircle of the triangle ${ABC}$ at points ${F}$ and ${G}$, where points ${F,D,E,G}$ lie in this order on $l$. The circumcircles of triangles ${FEB}$ and ${DGC}$ intersect at points ${P}$ and ${Q}$. Prove that points $A$, $P$, $Q$ are collinear.
    5. Each of the ${4n^2}$ unit squares of a ${2n \times 2n}$ board ${(n \ge 1) }$ has been colored blue or red. A set of four different unit squares of the board is called pretty if these squares can be labeled ${A,B,C,D}$ in such a way that ${A}$ and ${B}$ lie in the same row, ${C}$ and ${D}$ lie in the same row, ${A}$ and ${C}$ lie in the same column, ${B}$ and ${D}$ lie in the same column, ${A}$ and ${D}$ are blue, and ${B}$ and ${C}$ are red. Determine the largest possible number of different pretty sets on such a board.
    6. Find all functions ${f : (0, +\infty) \to \mathbb R}$ satisfying $$f(x) - f(x+ y) = f \left( \frac{x}{y}\right) f(x + y),\,\forall x, y > 0.$$
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    [Nguyễn Quản Bá Hồng] Một Cách Đổi Biến Và Ứng Dụng Trong Chứng Minh Bất Đẳng Thức

    Chuyên đề trình bày về một cách đổi biến đặc biệt, được giới thiệu trong quyển Problems from the book. Đồng thời, khai thác và mở rộng để tăng phạm vi ứng dụng cho phương pháp này.

    Đề tài nghiên cứu này đề cập đến một phương pháp nhỏ, rất cơ bản trong chứng minh bất đẳng thức. Đó là phương pháp đổi biến (hay thường được gọi là đặt ẩn phụ). Nhưng trong chuyên đề này, chúng ta sẽ không thảo luận tất cả các cách, các phương pháp đổi biến, mà chỉ thảo luận riêng về một cách đổi biến trong số các cách đó mà thôi. Tuy nhiên, để mở ra cái nhìn rộng hơn cho đề tài, chúng tôi sẽ dành phần cuối trong chuyên đề này để tổng hợp các cách đổi biến hay trong giải toán đại số nói chung và trong chứng minh bất đẳng thức nói riêng. Và thêm một phần nữa để nêu ra một sốbài toán hay có thể sử dụng phương pháp này để giải. Cũng để phân biệt rõ ràng với các phương pháp khác, chúng tôi sẽ gọi đối tượng nghiên cứu trong chuyên đề này là cách đổi biến $X$. Ký hiệu này được sử dụng chung cho toàn bộ chuyên đề.

    Vấn đề về phạm vi hay khả năng ứng dụng của một phương pháp, một kỹ thuật chứng minh thường được quan tâm đến rất nhiều. Chẳng hạn đối với một phương pháp, kỹ thuật nào đó, ta thường quan tâm để phạm vi ứng dụng của nó. Nó có thể giải được những bài toán như thế nào?. Giải được nhiều bài toán hay không? Có thể đối phó được các bài toán khó haykhông? Đối với bài toán như thể nào thì phương pháp đó trở nên vô dụng? Rất nhiều câu hỏi! Rõ ràng chúng ta đang quan tâm đến sự hiệu quả và phạm vi ứng dụngcủa một phương pháp. Trong chuyên đề này, chúng tôi cũng hết sức quan tâm vấn đề đó. Đối với mảng đặc biệt như bất đẳng thức. Phương pháp nào cũng mang tính tương đối cả. Tùy vào bài toán cụ thể, mỗi phương pháp sẽ có hiệu quả nhất định. Có nhiều bài toán khó, phải sử dụng đến phương pháp hiện đại mới có thể giải quyết được, trong khi các phương pháp truyền thống thì chẳng giúp được gì. Nhưng lại có những bài toán khác, chỉ có thể sử dụng phương pháp truyền thống mớicó thể cho một lời giải ngắn và đẹp được, trong khi các phương pháp hiện đại trở nên vụng về và nặng nề về tính toán. Tất cả đều mang tính tuyệt đối! Quan trọng là chúng ta sử dụng các phương pháp đó như thể nào?. Một tư tưởng nhỏ của chuyên đề này mà chúng tôi muốn thể hiện, đó làcải tiếnnhững phương pháp đã có, để tăng hiệu quả, và phạm vi ứng dụng. Mục đích của chúng tôi là
    1. Cải tiến những công cụ đơn giản.
    2. Xây dựng công cụ mới từ những công cụ đã có.
    Với hai mục đích này, chúng tôi sẽ cải tiến, tức là mở rộng và tổng quát mộtcách đổi biến đã được giới thiệu trong Problems from the book để mở rộng phạm vi ứng dụng cho nó. Tiếp theo, từ cách đổi biến này, chúng tôi sẽ xây dựng một số phương pháp, kỹ thuật chứng minh mới. Suốt chuyên đề, các nhận xét sẽ giúp mọi người theo dõi hiệu quả và phạm vi ứng dụng của từng đối tượng được trình bày. 
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    [Solutions] Czech-Polish-Slovak Mathematics Competition 2016

    1. Let $P$ be a non-degenerate polygon with $n$ sides, where $n > 4$. Prove that there exist three distinct vertices $A, B, C$ of $P$ with the following property:If $\ell_1,\ell_2,\ell_3$ are the lengths of the three polygonal chains into which $A, B, C$ break the perimeter of $P$, then there is a triangle with side lengths $\ell_1,\ell_2$ and $\ell_3$.
      Remark: By a non-degenerate polygon we mean a polygon in which every two sides are disjoint, apart from consecutive ones, which share only the common endpoint.
    2. Let $m,n > 2$ be even integers. Consider a board of size $m \times n$ whose every cell is colored either black or white. The Guesser does not see the coloring of the board but may ask the Oracle some questions about it. In particular, she may inquire about two adjacent cells (sharing an edge) and the Oracle discloses whether the two adjacent cells have the same color or not. The Guesser eventually wants to find the parity of the number of adjacent cell-pairs whose colors are different. What is the minimum number of inquiries the Guesser needs to make so that she is guaranteed to find her answer?.
    3. Let $n$ be a positive integer. For a finite set $M$ of positive integers and each $i \in \{0,1,..., n-1\}$, we denote $s_i$ the number of non-empty subsets of $M$ whose sum of elements gives remainder $i$ after division by $n$. We say that $M$ is "$n$-balanced" if $s_0 = s_1 =....= s_{n-1}$. Prove that for every odd number $n$ there exists a non-empty $n$-balanced subset of $\{0,1,..., n\}$. (For example if $n = 5$ and $M = \{1,3,4\}$, we have $s_0 = s_1 = s_2 = 1$, $s_3 = s_4 = 2$ so $M$ is not $5$-balanced.)
    4. Find all quadruplets $(a, b, c, d)$ of real numbers satisfying the system $$(a + b)(a^2 + b^2) = (c + d)(c^2 + d^2)$$ $$(a + c)(a^2 + c^2) = (b + d)(b^2 + d^2)$$ $$(a + d)(a^2 + d^2) = (b + c)(b^2 + c^2)$$
    5. Prove that for every non-negative integer $n$ there exist integers $x, y, z$ with $\gcd(x, y, z) = 1$, such that $x^2 + y^2 + z^2 = 3^{2^n}$
    6. Let $ABC$ be an acute-angled triangle with $AB < AC$. Tangent to its circumcircle $\Omega$ at $A$ intersects the line $BC$ at $D$. Let $G$ be the centroid of $\triangle ABC$ and let $AG$ meet $\Omega$ again at $H \neq A$. Suppose the line $DG$ intersects the lines $AB$ and $AC$ at $E$ and $F$, respectively. Prove that $\angle EHG = \angle GHF$.
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    [Đáp Án] Đề Thi Olympic Toán Duyên Hải Bắc Bộ 2018-2019 (Khối 11)


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    [Đáp Án] Đề Thi Olympic Toán Duyên Hải Bắc Bộ 2018-2019 (Khối 10)


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    [Solutions] Romanian Team Selection Tests For Junior Balkan Mathematical Olympiad 2017

    1. Let $n$ and $k$ be two positive integers such that $1\leq n \leq k$. Prove that, if $d^k+k$ is a prime number for each positive divisor $d$ of $n$, then $n+k$ is a prime number.
    2. Given $x_1,x_2,...,x_n$ real numbers, prove that there exists a real number $y$, such that $$\{y-x_1\}+\{y-x_2\}+...+\{y-x_n\} \leq \frac{n-1}{2}$$
    3. Let $I$ be the incenter of the scalene $\Delta ABC$ such that $AB<AC$, and let $I'$ be the reflection of point $I$ in line $BC$. The angle bisector $AI$ meets $BC$ at $D$ and circumcircle of $\Delta ABC$ at $E$. The line $EI'$ meets the circumcircle at $F$. Prove that
      a) $\dfrac{AI}{IE}=\dfrac{ID}{DE}$.
      b) $IA=IF$.
    4. The sides of an equilateral triangle are divided into n equal parts by $n-1$ points on each side. Through these points one draws parallel lines to the sides of the triangle. Thus, the initial triangle is divides into $n^2$ equal equilateral triangles. In every vertex of such a triangle there is a beetle. The beetles start crawling simultaneously, with equal speed, along the sides of the small triangles. When they reach a vertex, the beetles change the direction of their movement by $60^{\circ}$ or by $120^{\circ}$.
      a) Prove that if $n \geq 7$ then the beetles can move indefinitely on the sides of the small triangles without two beetles ever meeting in a vertex of a small triangle.
      b) Determine all the values of $n \geq 1$ for which the beetles can move along the sides of the small triangles without meeting in their vertices.

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    [Solutions] Romanian Team Selection Tests For Junior Balkan Mathematical Olympiad 2015

    1. Let $ABC$ be an acute triangle with $AB \neq AC$. Also let $M$ be the midpoint of the side $BC$, $H$ the orthocenter of the triangle $ABC$, $O_1$ the midpoint of the segment $AH$ and $O_2$ the center of the circumscribed circle of the triangle $BCH$. Prove that $O_1AMO_2$ is a parallelogram.
    2. Find the smallest positive integer $n$ such that if we color in red $n$ arbitrary vertices of the cube, there will be a vertex of the cube which has the three vertices adjacent to it colored in red.
    3. Let $x,y,z>0$. Show that $$\frac{x^3}{z^3+x^2y}+\frac{y^3}{x^3+y^2z}+\frac{z^3}{y^3+z^2x} \geq \frac{3}{2}$$
    4. Solve in nonnegative integers the following equation $$21^x+4^y=z^2$$
    5. Let $ABCD$ be a convex quadrilateral with non perpendicular diagonals and with the sides $AB$ and $CD$ non parallel. Denote by $O$ the intersection of the diagonals, $H_1$ the orthocenter of the triangle $AOB$ and $H_2$ the orthocenter of the triangle $COD$. Also denote with $M$ the midpoint of the side $AB$ and with $N$ the midpoint of the side $CD$. Prove that $H_1H_2$ and $MN$ are parallel if and only if $AC=BD$.
    6. Find all the positive integers $N$ with an even number of digits with the property that if we multiply the two numbers formed by cutting the number in the middle we get a number that is a divisor of $N$ (for example $12$ works because $1 \cdot 2$ divides $12$).
    7. Let $a,b,c>0$ such that $a \geq bc^2$, $b \geq ca^2$ and $c \geq ab^2$. Find the maximum value that the expression $$E=abc(a-bc^2)(b-ca^2)(c-ab^2)$$ can acheive.
    8. Can we partition the positive integers in two sets such that none of the sets contains an infinite arithmetic progression of nonzero ratio?.
    9. Let $ABC$ be a triangle with $AB \neq BC$ and let $BD$ the interior bisectrix of $ \angle ABC$ with $D \in AC$. Let $M$ be the midpoint of the arc $AC$ that contains the point $B$ in the circumcircle of the triangle $ABC$. The circumcircle of the triangle $BDM$ intersects the segment $AB$ in $K \neq B$. Denote by $J$ the symmetric of $A$ with respect to $K$. Prove that if $DJ$ intersects $AM$ in $O$ the $J$, $B$, $M$, $O$ are concyclic.
    10. Define the set $M_q=\{x \in \mathbb{Q} \mid x^3-2015x=q \}$, where $q$ is an arbitrary rational number.
      a) Show that there exists values for $q$ such that the set is null as well as values for which it has exactly one element.
      b) Determine all the possible values for the cardinality of $M_q$.
    11. Find all the triplets of real numbers $(x , y , z)$ such that $$\begin{cases}y&=\dfrac{x^3+12x}{3x^2+4},\\ z&=\dfrac{y^3+12y}{3y^2+4},\\ x&=\dfrac{z^3+12z}{3z^2+4}\end{cases}$$
    12. Let $ABC$ be an acute triangle with $AB \neq AC$ and denote its orthocenter by $H$. The point $D$ is located on the side $BC$ and the circumcircles of the triangles $ABD$ and $ACD$ intersects for the second time the lines $AC$, respectively $AB$ in the points $E$ respectively $F$. If we denote by $P$ the intersection point of $BE$ and $CF$ then show that $HP \parallel BC$ if and only if $AD$ passes through the circumcenter of the triangle $ABC$.
    13. The vertices of a regular $n$-gon are initially marked with one of the signs $+$ or $-$. A move consists in choosing three consecutive vertices and changing the signs from the vertices, from $+$ to $-$ and from $-$ to $+$.
      a) Prove that if $n=2015$ then for any initial configuration of signs, there exists a sequence of moves such that we'll arrive at a configuration with only $+$ signs.
      b) Prove that if $n=2016$, then there exists an initial configuration of signs such that no matter how we make the moves we'll never arrive at a configuration with only $+$ signs.
    14. Let $n\in \mathbb{N}$, $n \geq 4$. Determine all sets $$A = \{a_1, a_2, . . . , a_n\} \subset \mathbb{N}$$ containing $2015$ and having the property that $ |a_i - a_j|$ is prime, for all distinct $i, j\in \{1, 2, . . . , n\}.$
    15. Solve in $\mathbb{N}^*$ the equation $$ 4^a \cdot 5^b - 3^c \cdot 11^d = 1.$$
    16. Let $ABC$ be a triangle with $AB \ne AC$ and $ I$ its incenter. Let $M$ be the midpoint of the side $BC$ and $D$ the projection of $I$ on $BC.$ The line $AI$ intersects the circle with center $M$ and radius $MD$ at $P$ and $Q.$ Prove that $\angle BAC + \angle PMQ = 180^{\circ}.$
    17. We have $n$ integers $a_1, a_2,. . . , a_n$, not necessarily distinct, with sum $2S.$ An integer $k$ is called separator if $k$ of the numbers can be chosen with sum equal to $S.$ What is the maximum possible number of separators?
    18. Prove that number $1$ can be represented as a sum of a finite number $n$ of real numbers, less than $1,$ not necessarily distinct, which contain in their decimal representation only the digits $0$ and/or $7.$ Which is the least possible number $n$?
    19. Two players, $A$ and $B,$ alternatively take stones from a pile of $n \geq 2$ stones. $A$ plays first and in his first move he must take at least one stone and at most $n-1$ stones. Then each player must take at least one stone and at most as many stones as his opponent took in the previous move. The player who takes the last stone wins. Which player has a winning strategy?
    20. Prove that if $a,b,c>0$ and $a+b+c=1,$ then $$\frac{bc+a+1}{a^2+1}+\frac{ca+b+1}{b^2+1}+\frac{ab+c+1}{c^2+1}\leq \frac{39}{10}$$
    21. Let $ABC$ be a triangle inscribed in circle $\omega$ and $P$ a point in its interior. The lines $AP$, $BP$ and $CP$ intersect circle $\omega$ for the second time at $D$, $E$ and $F$, respectively. If $A'$, $B'$, $C'$ are the reflections of $A$, $B$, $C$ with respect to the lines $EF$, $FD$, $DE$, respectively, prove that the triangles $ABC$ and $A'B'C'$ are similar.

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    [Solutions] Romanian Team Selection Tests For Junior Balkan Mathematical Olympiad 2001

    1. Let $ABC$ be an arbitrary triangle. A circle passes through $B$ and $C$ and intersects the lines $AB$ and $AC$ at $D$ and $E$, respectively. The projections of the points $B$ and $E$ on $CD$ are denoted by $B'$ and $E'$, respectively. The projections of the points $D$ and $C$ on $BE$ are denoted by $D'$ and $C'$, respectively. Prove that the points $B'$, $D'$, $E'$ and $C'$ lie on the same circle.
    2. Find all $n\in\mathbb{Z}$ such that the number $\sqrt{\dfrac{4n-2}{n+5}}$ is rational.
    3. In the interior of a circle centred at $O$ consider the $1200$ points $A_1,A_2,\ldots ,A_{1200}$, where for every $i,j$ with $1\le i\le j\le 1200$, the points $O,A_i$ and $A_j$ are not collinear. Prove that there exist the points $M$ and $N$ on the circle, with $\angle MON=30^{\circ}$, such that in the interior of the angle $\angle MON$ lie exactly $100$ points.
    4. Three students write on the blackboard next to each other three two-digit squares. In the end, they observe that the 6-digit number thus obtained is also a square. Find this number!
    5. Let $ABCD$ be a rectangle. We consider the points $E\in CA$, $F\in AB$, $G\in BC$ such that $DC\perp CA$, $EF\perp AB$ and $EG\perp BC$. Solve in the set of rational numbers the equation $$AC^x=EF^x+EG^x$$
    6. Let $A$ be a non-empty subset of $\mathbb{R}$ with the property that for every real numbers $x,y$, if $x+y\in A$ then $xy\in A$. Prove that $A=\mathbb{R}$.
    7. Let $ABCD$ be a quadrilateral inscribed in the circle $O$. For a point $E\in O$, its projections $K$, $L$, $M$, $N$ on the lines $DA$, $AB$, $BC$, $CD$, respectively, are considered. Prove that if $N$ is the orthocentre of the triangle $KLM$ for some point $E$, different from $A,B,C,D$, then this holds for every point $E$ of the circle.
    8. Determine all positive integers in the form $a<b<c<d$ with the property that each of them divides the sum of the other three.
    9. Let $n$ be a non-negative integer. Find all non-negative integers $a,b,c,d$ such that \[a^2+b^2+c^2+d^2=7\cdot 4^n\]
    10. Let $ABCDEF$ be a hexagon with $AB||DE$, $BC||EF$, $CD||FA$ and in which the diagonals $AD$, $BE$ and $CF$ are congruent. Prove that the hexagon can be inscribed in a circle.
    11. Let $n\ge 2$ be a positive integer. Find the positive integers $x$ \[\sqrt{x+\sqrt{x+\ldots +\sqrt{x}}}<n \] for any number of radicals.
    12. Determine a right parallelepiped with minimal area, if its volume is strictly greater than $1000$, and the lengths of it sides are integer numbers.
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    [Shortlists & Solutions] Junior Balkan Mathematical Olympiad 2017

    Algebra

    1. Let $a, b, c$ be positive real numbers such that $$a + b + c + ab + bc + ca + abc = 7.$$ Prove that $$\sqrt{a^2 + b^2 + 2 }+\sqrt{b^2 + c^2 + 2 }+\sqrt{c^2 + a^2 + 2 } \ge 6.$$
    2. Let $a$ and $b$ be positive real numbers such that $$3a^2 + 2b^2 = 3a + 2b.$$ Find the minimum value of $$A =\sqrt{\frac{a}{b(3a+2)}} + \sqrt{\frac{b}{a(2b+3)}}$$
    3. Let $a\le b\le c \le d$. Show that $$ab^3+bc^3+cd^3+da^3\ge a^2b^2+b^2c^2+c^2d^2+d^2a^2$$
    4. Let $x,y,z$ be positive integers such that $x\neq y\neq z \neq x$. Prove that $$(x+y+z)(xy+yz+zx-2)\geq 9xyz.$$ When does the equality hold?

    Combinatorics

    1. Consider a regular $2n + 1$-gon $P$ in the plane, where n is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$, if the line segment $SE$ contains no other points that lie on the sides of $P$ except $S$. We want to color the sides of $P$ in $3$ colors, such that every side is colored in exactly one color, and each color must be used at least once. Moreover, from every point in the plane external to $P$, at most $2$ different colors on $P$ can be seen (ignore the vertices of $P$, we consider them colorless). Find the largest positive integer for which such a coloring is possible.
    2. Consider a regular $2n$-gon $ P$, $A_1,A_2,\cdots ,A_{2n}$ in the plane where $n$ is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$ if the line segment $SE$ contains no other points that lie on the sides of $P$ except $S$. We color the sides of $P$ in $3$ different colors (ignore the vertices of $P$, we consider them colorless), such that every side is colored in exactly one color, and each color is used at least once. Moreover ,from every point in the plane external to $P$, points of most $2$ different colors on $P$ can be seen .Find the number of distinct such colorings of $P$ (two colorings are considered distinct if at least one of sides is colored differently).
    3. We have two piles with $2000$ and $2017$ coins respectively. Ann and Bob take alternate turns making the following moves: The player whose turn is to move picks a pile with at least two coins, removes from that pile $t$ coins for some $2\le t \le 4$, and adds to the other pile $1$ coin. The players can choose a different $t$ at each turn, and the player who cannot make a move loses. If Ann plays first determine which player has a winning strategy.

    Geometry

    1. Given a parallelogram $ABCD$. The line perpendicular to $AC$ passing through $C$ and the line perpendicular to $BD$ passing through $A$ intersect at point $P$. The circle centered at point $P$ and radius $PC$ intersects the line $BC$ at point $X$, ($X \ne C$) and the line $DC$ at point $Y$ ($Y \ne C$). Prove that the line $AX$ passes through the point $Y$ .
    2. Let $ABC$ be an acute triangle such that $AB$ is the shortest side of the triangle. Let $D$ be the midpoint of the side $AB$ and $P$ be an interior point of the triangle such that $\angle CAP = \angle CBP = \angle ACB$. Denote by M and $N$ the feet of the perpendiculars from $P$ to $BC$ and $AC$, respectively. Let $p$ be the line through $ M$ parallel to $AC$ and $q$ be the line through $N$ parallel to $BC$. If $p$ and $q$ intersect at $K$ prove that $D$ is the circumcenter of triangle $MNK$.
    3. Consider triangle $ABC$ such that $AB \le AC$. Point $D$ on the arc $BC$ of thecircumcirle of $ABC$ not containing point $A$ and point $E$ on side $BC$ are such that $\angle BAD = \angle CAE < \frac12 \angle BAC$. Let $S$ be the midpoint of segment $AD$. Prove that if $\angle ADE = \angle ABC - \angle ACB$ then $\angle BSC = 2 \angle BAC$.
    4. Let $ABC $ be an acute triangle such that $AB\neq AC$ with circumcircle $ \Gamma$ and circumcenter $O$. Let $M$ be the midpoint of $BC$ and $D$ be a point on $ \Gamma$ such that $AD \perp BC$. Let $T$ be a point such that $BDCT$ is a parallelogram and $Q$ a point on the same side of $BC$ as $A$ such that $\angle{BQM}=\angle{BCA}$ and $\angle{CQM}=\angle{CBA}$. Let the line $AO$ intersect $ \Gamma$ at $E$ $(E\neq A)$ and let the circumcircle of $\triangle ETQ$ intersect $ \Gamma$ at point $X\neq E$. Prove that the point $A$, $M$ and $X$ are collinear.
    5. A point $P$ lies in the interior of the triangle $ABC$. The lines $AP, BP$, and $CP$ intersect $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$, respectively. Prove that if two of the quadrilaterals $ABDE$, $BCEF$, $CAFD$, $AEPF$, $BFPD$, and $CDPE$ are concyclic, then all six are concyclic.

    Number Theory

    1. Determine all the sets of six consecutive positive integers such that the product of some two of them added to the product of some other two of them is equal to the product of the remaining two numbers.
    2. Determine all positive integers $n$ such that $$n^2 \mid (n - 1)!$$
    3. Find all pairs of positive integers $(x,y)$ such that $2^x + 3^y$ is a perfect square.
    4. Solve in nonnegative integers the equation $$5^t + 3^x4^y = z^2.$$
    5. Find all positive integers $n$ such that there exists a prime number $p$, such that $p^n-(p-1)^n$ is a power of $3$. (A power of $3$ is a number of the form $3^a$ where $a$ is a positive integer.)
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    Junior Balkan Mathematical Olympiad Shortlist 2016

    Algebra

    1. Let $a, b, c$ be positive real numbers such that $abc = 8$. Prove that $$\frac{ab + 4}{a + 2}+\frac{bc + 4}{b + 2}+\frac{ca + 4}{c + 2}\ge 6.$$
    2. Let $a,b,c$ be positive real numbers. Prove that $$\frac{8}{(a+b)^2 + 4abc} + \frac{8}{(b+c)^2 + 4abc} + \frac{8}{(a+c)^2 + 4abc} + a^2 + b^2 + c ^2 \\ \ge \frac{8}{a+3} + \frac{8}{b+3} + \frac{8}{c+3}$$
    3. Find all the pairs of integers $ (m, n)$ such that $$\sqrt {n +\sqrt {2016}} +\sqrt {m-\sqrt {2016}} \in \mathbb {Q}.$$
    4. If the non-negative reals $x,y,z$ satisfy $x^2+y^2+z^2=x+y+z$. Prove that $$\displaystyle\frac{x+1}{\sqrt{x^5+x+1}}+\frac{y+1}{\sqrt{y^5+y+1}}+\frac{z+1}{\sqrt{z^5+z+1}}\geq 3.$$ When does the equality occur?
    5. Let $x,y,z$ be positive real numbers such that $$x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$$ Prove that \[x+y+z\geq \sqrt{\frac{xy+1}{2}}+\sqrt{\frac{yz+1}{2}}+\sqrt{\frac{zx+1}{2}} \ .\]

    Combinatorics

    1. Let $S_n$ be the sum of reciprocal values of non-zero digits of all positive integers up to (and including) $n$. For instance, $$S_{13} = \frac{1}{1}+ \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}+ \frac{1}{6}+ \frac{1}{7}+ \frac{1}{8}+ \frac{1}{9}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{2}+ \frac{1}{1}+ \frac{1}{3}.$$ Find the least positive integer $k$ making the number $k!\cdot S_{2016}$ an integer.
    2. The natural numbers from $1$ to $50$ are written down on the blackboard. At least how many of them should be deleted, in order that the sum of any two of the remaining numbers is not a prime?
    3. A $5 \times 5$ table is called regular if each of its cells contains one of four pairwise distinct real numbers, such that each of them occurs exactly one in every $2 \times 2$ subtable. The sum of all numbers of a regular table is called the total sum of the table. With any four numbers, one constructs all possible regular tables, computes their total sums and counts the distinct outcomes.Determine the maximum possible count.
    4. A splitting of a planar polygon is a finite set of triangles whose interiors are pairwise disjoint, and whose union is the polygon in question. Given an integer $n \ge 3$, determine the largest integer $m$ such that no planar $n$-gon splits into less than $m$ triangles.

    Geometry

    1. Let ${ABC}$ be an acute angled triangle, let ${O}$ be its circumcentre, and let $D$, $E$, $F$ be points on the sides $BC$, $CA$, $AB$, respectively. The circle ${(c_1)}$ of radius ${FA}$, centered at ${F}$, crosses the segment ${OA}$ at ${A'}$ and the circumcircle ${(c)}$ of the triangle ${ABC}$again at ${K}$. Similarly, the circle ${(c_2)}$ of radius $DB$, centered at $D$, crosses the segment $\left( OB \right)$ at ${B}'$ and the circle ${(c)}$ again at ${L}$. Finally, the circle ${(c_3)}$ of radius $EC$, centered at $E$, crosses the segment $\left( OC \right)$at ${C}'$ and the circle ${(c)}$ again at ${M}$. Prove that the quadrilaterals $BKF{A}'$, $CLD{B}'$ and $AME{C}'$ are all cyclic, and their circumcircles share a common point.
    2. Let ${ABC}$ be a triangle with $\angle BAC={{60}^{{}^\circ }}$. Let $D$ and $E$ be the feet of the perpendiculars from ${A}$ to the external angle bisectors of $\angle ABC$ and $\angle ACB$, respectively. Let ${O}$ be the circumcenter of the triangle ${ABC}$. Prove that the circumcircles of the triangles ${ADE}$and ${BOC}$ are tangent to each other.
    3. A trapezoid $ABCD$ ($AB || CF$, $AB > CD$) is circumscribed. The incircle of the triangle $ABC$ touches the lines $AB$ and $AC$ at the points $M$ and $N$, respectively. Prove that the incenter of the trapezoid $ABCD$ lies on the line $MN$.
    4. Let ${ABC}$ be an acute angled triangle whose shortest side is ${BC}$. Consider a variable point ${P}$ on the side ${BC}$, and let ${D}$ and ${E}$ be points on ${AB}$ and ${AC}$, respectively, such that ${BD=BP}$ and ${CP=CE}$. Prove that, as ${P}$ traces ${BC}$, the circumcircle of the triangle ${ADE}$ passes through a fixed point.
    5. Let $ABC$ be an acute angled triangle with orthocenter ${H}$ and circumcenter ${O}$. Assume the circumcenter ${X}$ of ${BHC}$lies on the circumcircle of ${ABC}$. Reflect O across ${X}$ to obtain ${O'}$, and let the lines ${XH}$ and ${O'A}$ meet at ${K}$. Let $L,M$ and $N$ be the midpoints of $\left[ XB \right]$, $\left[ XC \right]$ and $\left[ BC \right]$, respectively. Prove that the points $K$, $L$, $M$ and $K$, $L$, $M$, $N$ are cocyclic.
    6. Given an acute triangle ${ABC}$, erect triangles ${ABD}$ and ${ACE}$ externally, so that $\angle ADB= \angle AEC=90^\circ$ and $\angle BAD= \angle CAE$. Let ${{A}_{1}}\in BC$, ${{B}_{1}}\in AC$ and ${{C}_{1}}\in AB$ be the feet of the altitudes of the triangle ${ABC}$, and let $K$ and $K$, $L$ be the midpoints of $[ B{{C}_{1}} ]$ and ${BC_1, CB_1}$, respectively. Prove that the circumcenters of the triangles $AKL$, ${{A}_{1}}{{B}_{1}}{{C}_{1}}$ and ${DEA_1}$ are collinear.
    7. Let ${AB}$ be a chord of a circle ${(c)}$ centered at ${O}$, and let ${K}$ be a point on the segment ${AB}$ such that ${AK<BK}$. Two circles through ${K}$, internally tangent to ${(c)}$ at ${A}$ and ${B}$, respectively, meet again at ${L}$. Let ${P}$ be one of the points of intersection of the line ${KL}$and the circle ${(c)}$, and let the lines ${AB}$and ${LO}$meet at ${M}$. Prove that the line ${MP}$is tangent to the circle ${(c)}$.

    Number Theory

    1. Determine the largest positive integer $n$ that divides $p^6 - 1$ for all primes $p > 7$.
    2. Find the maximum number of natural numbers $x_1,x_2, ... , x_m$ satisfying the conditions
      • No $x_i - x_j , 1 \le i < j \le m$ is divisible by $11$, and
      • The sum $x_2x_3 ...x_m + x_1x_3 ... x_m + \cdot \cdot \cdot + x_1x_2... x_{m-1}$ is divisible by $11$.
    3. Find all positive integers $n$ such that the number $A_n =\frac{ 2^{4n+2}+1}{65}$ is
      a) an integer,
      b) a prime.
    4. Find all triplets of integers $(a,b,c)$ such that the number $$N = \frac{(a-b)(b-c)(c-a)}{2} + 2$$ is a power of $2016$. (A power of $2016$ is an integer of form $2016^n$ where $n$ is a non-negative integer.)
    5. Determine all four-digit numbers $\overline{abcd} $ such that $$(a + b)(a + c)(a + d)(b + c)(b + d)(c + d) =\overline{abcd}$$
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    [Solutions] Junior Balkan Mathematical Olympiad Shortlist 2015

    Algebra

    1. Let $x$, $y$, $z$ be real numbers, satisfying the relations $x \ge 20$, $y \ge 40$, $z \ge 1675$ and $x + y + z = 2015$. Find the greatest value of the product $P = xy z$
    2. Assume that $x$ satisfies $$x^3-3\sqrt3 x^2 +9x - 3\sqrt3 -64=0.$$ Find the value of $$x^6-8x^5+13x^4-5x^3+49x^2-137x+2015.$$
    3. Let $a,b,c$ be positive real numbers. Prove that $$\frac{a}{b}+\sqrt{\frac{b}{c}}+\sqrt[3]{\frac{c}{a}}>2.$$
    4. Let $a,b,c$ be positive real numbers such that $a+b+c = 3$. Find the minimum value of the expression \[A=\dfrac{2-a^3}a+\dfrac{2-b^3}b+\dfrac{2-c^3}c.\]
    5. The positive real $x, y, z$ are such that $x^2+y^2+z^2 = 3$. Prove that $$\frac{x^2+yz}{x^2+yz +1}+\frac{y^2+zx}{y^2+zx+1}+\frac{z^2+xy}{z^2+xy+1}\leq 2$$

    Geometry

    1. Around the triangle $ABC$ the circle is circumscribed, and at the vertex ${C}$ tangent ${t}$ to this circle is drawn. The line ${p}$, which is parallel to this tangent intersects the lines ${BC}$ and ${AC}$ at the points ${D}$ and ${E}$, respectively. Prove that the points $A$, $B$, $D$, $E$ belong to the same circle.
    2. The point ${P}$ is outside the circle ${\Omega}$. Two tangent lines, passing from the point ${P}$ touch the circle ${\Omega}$ at the points ${A}$ and ${B}$. The median${AM \left(M\in BP\right)}$ intersects the circle ${\Omega}$ at the point ${C}$ and the line ${PC}$ intersects again the circle ${\Omega}$ at the point ${D}$. Prove that the lines ${AD}$ and ${BP}$ are parallel.
    3. Let ${c\equiv c\left(O, R\right)}$ be a circle with center ${O}$ and radius ${R}$ and ${A, B}$ be two points on it, not belonging to the same diameter. The bisector of angle${\angle{ABO}}$ intersects the circle ${c}$ at point ${C}$, the circumcircle of the triangle $AOB$, say ${c_1}$ at point ${K}$ and the circumcircle of the triangle $AOC$, say ${{c}_{2}}$ at point ${L}$. Prove that point ${K}$ is the circumcircle of the triangle $AOC$ and that point ${L}$ is the incenter of the triangle $AOB$.
    4. Let $ABC$ be an acute triangle.The lines $l_1$ and $l_2$ are perpendicular to $AB$ at the points $A$ and $B$, respectively.The perpendicular lines from the midpoint $M$ of $AB$ to the lines $AC$ and $BC$ intersect $l_1$ and $l_2$ at the points $E$ and $F$, respectively. Prove that if $D$ is the intersection point of the lines $EF$ and $MC$ then \[\angle ADB = \angle EMF.\]
    5. Let $ABC$ be an acute triangle with ${AB\neq AC}$. The incircle ${\omega}$ of the triangle κύκλος touches the sides ${BC, CA}$ and ${AB}$ at ${D, E}$ and ${F}$, respectively. The perpendicular line erected at ${C}$onto ${BC}$ meets ${EF}$at ${M}$, and similarly the perpendicular line erected at ${B}$onto ${BC}$ meets ${EF}$at${N}$. The line ${DM}$ meets ${\omega}$ again in ${P}$, and the line ${DN}$ meets ${\omega}$ again at ${Q}$. Prove that ${DP=DQ}$.

    Number Theory

    1. What is the greatest number of integers that can be selected from a set of $2015$ consecutive numbers so that no sum of any two selected numbers is divisible by their difference?.
    2. A positive integer is called a repunit, if it is written only by ones. The repunit with $n$ digits will be denoted as $\underbrace{{11\cdots1}}_{n}$. Prove that a) the repunit $\underbrace{{11\cdots1}}_{n}$is divisible by $37$ if and only if $n$ is divisible by $3$. b) there exists a positive integer $k$ such that the repunit $\underbrace{{11\cdots1}}_{n}$ is divisible by $41$ if $n$ is divisible by $k$.
    3. a) Show that the product of all differences of possible couples of six given positive integers is divisible by $960$. b) Show that the product of all differences of possible couples of six given positive integers is divisible by $34560$.
    4. Find all prime numbers $a,b,c$ and positive integers $k$ satisfying the equation \[a^2+b^2+16c^2 = 9k^2 + 1.\]
    5. Check if there exists positive integers $ a, b$ and prime number $p$ such that $$a^3-b^3=4p^2$$

    Combinatorics

    1. A board $n \times n$ ($n \ge 3$) is divided into $n^2$ unit squares. Integers from $O$ to $n$ included, are written down: one integer in each unit square, in such a way that the sums of integers in each $2\times 2$ square of the board are different. Find all $n$ for which such boards exist.
    2. $2015$ points are given in a plane such that from any five points we can choose two points with distance less than $1$ unit. Prove that $504$ of the given points lie on a unit disc.
    3. Positive integers are put into the following table $$\begin{array}{|l|l|l|l|l|l|l|l|l|l|} \hline 1 & 3 & 6 & 10 & 15 & 21 & 28 & 36 & & \\ \hline 2 & 5 & 9 & 14 & 20 & 27 & 35 & 44 & & \\ \hline 4 & 8 & 13 & 19 & 26 & 34 & 43 & 53 & & \\ \hline 7 & 12 & 18 & 25 & 33 & 42 & & & & \\ \hline 11 & 17 & 24 & 32 & 41 & & & & & \\ \hline 16 & 23 & & & & & & & & \\ \hline ... & & & & & & & & & \\ \hline ... & & & & & & & & & \\ \hline \end{array}$$
      Find the number of the line and column where the number $2015$ stays.
    4. Let $n\ge 1$ be a positive integer. A square of side length $n$ is divided by lines parallel to each side into $n^2$ squares of side length $1$. Find the number of parallelograms which have vertices among the vertices of the $n^2$ squares of side length $1$, with both sides smaller or equal to $2$, and which have tha area equal to $2$.
    5. An $L$-shape is one of the following four pieces, each consisting of three unit squares. A $5\times 5$ board, consisting of $25$ unit squares, a positive integer $k\leq 25$ and an unlimited supply of L-shapes are given. Two players $A$ and $B$, play the following game: starting with $A$ they play alternatively mark a previously unmarked unit square until they marked a total of $k$ unit squares. e say that a placement of $L$-shapes on unmarked unit squares is called good if the L-shapes do not overlap and each of them covers exactly three unmarked unit squares of the board. $B$ wins if every good placement of $L$-shapes leaves uncovered at least three unmarked unit squares. Determine the minimum value of $k$ for which $B$ has a winning strategy.
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    Junior Balkan Mathematical Olympiad Shortlist 2014

    Geometry

    1. Let ${ABC}$ be a triangle with $m\left( \angle B \right)=m\left( \angle C \right)={{40}^{{}^\circ }}$ Line bisector of ${\angle{B}}$ intersects ${AC}$ at point ${D}$. Prove that $BD+DA=BC$.
    2. Acute-angled triangle ${ABC}$ with ${AB<AC<BC}$ and let be ${c(O,R)}$ it’s circumicircle. Diametes ${BD}$ and ${CE}$ are drawn. Circle ${c_1(A,AE)}$ interescts ${AC}$ at ${K}$ Circle ${{c}_{2}(A,AD)}$ intersects ${BA}$ at ${L}$ (${A}$ lies between ${B}$ and ${L}$). Prove that lines $EK$ and $DL$ intersect at circle.
    3. Consider an acute triangle $ABC$ of area $S$. Let $CD \perp AB$ ($D \in AB$), $DM \perp AC$ ($M \in AC$) and $DN \perp BC$ ($N \in BC$). Denote by $H_1$ and $H_2$ the orthocentres of the triangles $MNC$, respectively $MND$. Find the area of the quadrilateral $AH_1BH_2$ in terms of $S$.
    4. Let $ABC$ be an acute triangle such that $AB\not=AC$. Let $M$ be the midpoint $BC$, $H$ the orthocenter of $\triangle ABC$, $O_1$ the midpoint of $AH$ and $O_2$ the circumcenter of $\triangle BCH$. Prove that $O_1AMO_2$ is a parallelogram.
    5. Let $ABC$ be a triangle with ${AB\ne BC}$ and let ${BD}$ be the internal bisector of $\angle ABC$ $\left( D\in AC \right)$. Denote by ${M}$ the midpoint of the arc ${AC}$ which contains point ${B}$. The circumscribed circle of the triangle $\triangle BDM$ intersects the segment ${AB}$ at point ${K\neq B}$. Let ${J}$ be the reflection of ${A}$ with respect to ${K}$. Prove that if ${DJ\cap AM=\left\{O\right\}}$ then the points $J$, $B$, $M$, $O$ belong to the same circle.
    6. Let $ABCD$ be a quadrilateral whose diagonals are not perpendicular and whose sides $AB$ and $CD$ are not parallel.Let $O$ be the intersection of its diagonals. Denote with $H_1$ and $H_2$ the orthocenters of triangles $AOB$ and $COD$, respectively. If $M$ and $N$ are the midpoints of the segment lines $AB$ and $CD$, respectively. Prove that the lines $H_1H_2$ and $MN$ are parallel if and only if $AC=BD$.
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    Junior Balkan Mathematical Olympiad Shortlist 2013

    Geometry

    1. Let ${AB}$ be a diameter of a circle ${\omega}$ and center ${O}$, ${OC}$ a radius of ${\omega}$ perpendicular to $AB$, ${M}$ be a point of the segment $\left( OC \right)$. Let ${N}$ be the second intersection point of line ${AM}$ with ${\omega}$ and ${P}$ the intersection point of the tangents of ${\omega}$ at points ${N}$ and ${B.}$ Prove that points $M$, $O$, $P$, $N$ are cocyclic.
    2. Circles ${\omega_1}$, ${\omega_2}$ are externally tangent at point M and tangent internally with circle ${\omega_3}$ at points ${K}$ and $L$ respectively. Let ${A}$ and ${B}$be the points that their common tangent at point ${M}$ of circles ${\omega_1}$ and ${\omega_2}$ intersect with circle ${\omega_3.}$ Prove that if ${\angle KAB=\angle LAB}$ then the segment ${AB}$ is diameter of circle ${\omega_3.}$
    3. Let $ABC$ be an acute-angled triangle with $AB<AC$ and let $O$ be the centre of its circumcircle $\omega$. Let $D$ be a point on the line segment $BC$ such that $\angle BAD = \angle CAO$. Let $E$ be the second point of intersection of $\omega$ and the line $AD$. If $M$, $N$ and $P$ are the midpoints of the line segments $BE$, $OD$ and $AC$, respectively, show that the points $M$, $N$ and $P$ are collinear.
    4. Let $I$ be the incenter and $AB$ the shortest side of the triangle $ABC$. The circle centered at $I$ passing through $C$ intersects the ray $AB$ in $P$ and the ray $BA$ in $Q$. Let $D$ be the point of tangency of the $A$-excircle of the triangle $ABC$ with the side $BC$. Let $E$ be the reflection of $C$ with respect to the point $D$. Prove that $PE\perp CQ$.
    5. A circle passing through the midpoint $M$ of the side $BC$ and the vertex $A$ of the triangle $ABC$ intersects the segments $AB$ and $AC$ for the second time in the points $P$ and $Q$, respectively. Prove that if $\angle BAC=60^{\circ}$ then $$AP+AQ+PQ<AB+AC+\frac{1}{2} BC.$$
    6. Let $P$ and $Q$ be the midpoints of the sides $BC$ and $CD$, respectively in a rectangle $ABCD$. Let $K$ and $M$ be the intersections of the line $PD$ with the lines $QB$ and $QA$, respectively, and let $N$ be the intersection of the lines $PA$ and $QB$. Let $X$, $Y$ and $Z$ be the midpoints of the segments $AN$, $KN$ and $AM$, respectively. Let $\ell_1$ be the line passing through $X$ and perpendicular to $MK$, $\ell_2$ be the line passing through $Y$ and perpendicular to $AM$ and $\ell_3$ the line passing through $Z$ and perpendicular to $KN$. Prove that the lines $\ell_1$, $\ell_2$ and $\ell_3$ are concurrent.
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    [Solutions] Junior Balkan Mathematical Olympiad Shortlist 2012

    Algebra

    1. Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that \[\frac {a}{b} + \frac {a}{c} + \frac {c}{b} + \frac {c}{a} + \frac {b}{c} + \frac {b}{a} + 6 \geq 2\sqrt{2}\left (\sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\right ).\] When does equality hold?
    2. Let $a$, $b$, $c$ be positive real numbers such that $abc=1$ . Show that \[\frac{1}{a^3+bc}+\frac{1}{b^3+ca}+\frac{1}{c^3+ab} \leq \frac{ \left (ab+bc+ca \right )^2 }{6}\]
    3. Let $a$, $b$, $c$ be positive real numbers such that $$a+b+c=a^2+b^2+c^2.$$ Prove that \[\frac{a^2}{a^2+ab}+\frac{b^2}{b^2+bc}+\frac{c^2}{c^2+ca} \geq \frac{a+b+c}{2}\]
    4. Solve the following equation for $x , y , z \in \mathbb{N}$ \[\left (1+ \frac{x}{y+z} \right )^2+\left (1+ \frac{y}{z+x} \right )^2+\left (1+ \frac{z}{x+y} \right )^2=\frac{27}{4}\]
    5. Find the largest positive integer $n$ for which the inequality \[ \frac{a+b+c}{abc+1}+\sqrt[n]{abc} \leq \frac{5}{2}\] holds true for all $a, b, c \in [0,1]$. Here we make the convention $\sqrt[1]{abc}=abc$.

    Geometry

    1. Let $ABC$ be an equilateral triangle and $P$ be a point on the circumcircle of the triangle but distinct from $A$, $B$ and $C$. The lines through $P$ and parallel to $BC$, $CA$, $AB$ intersect the lines $CA$, $AB$, $BC$ at $M$, $N$ and $Q$ respectively. Prove that $M$, $N$ and $Q$ are collinear .
    2. Let $ABC$ be an isosceles triangle with $AB=AC$. Let also $\omega$ be a circle of center $K$ tangent to the line $AC$ at $C$ which intersects the segment $BC$ again at $H$. Prove that $HK \bot AB $.
    3. Let $AB$ and $CD$ be chords in a circle of center $O$ with $A$, $B$, $C$, $D$ distinct, and with the lines $AB$ and $CD$ meeting at a right angle at point $E$. Let also $M$ and $N$ be the midpoints of $AC$ and $BD$ respectively. Prove that if $MN \bot OE$ then $AD \parallel BC$.
    4. Let $ABC$ be an acute-angled triangle with circumcircle $\omega$, and let $O$, $H$ be the triangle's circumcenter and orthocenter respectively. Let also $A'$ be the point where the angle bisector of the angle $BAC$ meets $\omega$. Find the measure of the angle $BAC$ if $A'H=AH$.
    5. Let the circles $k_1$ and $k_2$ intersect at two points $A$ and $B$, and let $t$ be a common tangent of $k_1$ and $k_2$ that touches $k_1$ and $k_2$ at $M$ and $N$ respectively. If $t\perp AM$ and $MN=2AM$, evaluate the angle $NMB$.
    6. Let $O_1$ be a point in the exterior of the circle $\omega$ of center $O$ and radius $R$, and let $O_1N$, $O_1D$ be the tangent segments from $O_1$ to the circle. On the segment $O_1N$ consider the point $B$ such that $BN=R$. Let the line from $B$ parallel to $ON$ intersect the segment $O_1D$ at $C$. If $A$ is a point on the segment $O_1D$ other than $C$ so that $BC=BA=a$, and if the incircle of the triangle $ABC$ has radius $r$, then find the area of $\triangle ABC$ in terms of $a$, $R$, $r$.
    7. Let $MNPQ$ be a square of side length $1$, and $A$, $B$, $C$, $D$ points on the sides $MN$, $NP$, $PQ$ and $QM$ respectively such that $AC \cdot BD=\dfrac{5}{4}$. Can the set $ \{AB , BC , CD , DA \}$ be partitioned into two subsets $S_1$ and $S_2$ of two elements each so that each one has the sum of his elements a positive integer?.

    Combinatorics

    1. Along a round table are arranged $11$ cards with the names ( all distinct ) of the $11$ members of the $16^{th}$ JBMO Problem Selection Committee. The cards are arranged in a regular polygon manner. Assume that in the first meeting of the Committee none of its $11$ members sits in front of the card with his name. Is it possible to rotate the table by some angle so that at the end at least two members sit in front of the card with their names?.
    2. On a board there are $n$ nails, each two connected by a rope. Each rope is colored in one of $n$ given distinct colors. For each three distinct colors, there exist three nails connected with ropes of these three colors. a) Can $n$ be $6$?. b) Can $n$ be $7$?.
    3. In a circle of diameter $1$ consider $65$ points, no three of them collinear. Prove that there exist three among these points which are the vertices of a triangle with area less than or equal to $\dfrac{1}{72}$.

    Number Theory

    1. let $a$, $b$ be integers and $$s=a^3+b^3-60ab(a+b)\geq 2012.$$ Find the least possible value of $s$.
    2. Do there exist prime numbers $p$ and $q$ such that $$p^2(p^3-1)=q(q+1)$$
    3. Decipher the equality \[(\overline{VER}-\overline{IA})=G^{R^E} (\overline {GRE}+\overline{ECE}) \] assuming that the number $\overline {GREECE}$ has a maximum value. Each letter corresponds to a unique digit from $0$ to $9$ and different letters correspond to different digits. It's also supposed that all the letters $G$, $E$, $V$ and $I$ are different from $0$.
    4. Determine all triples $(m , n , p)$ satisfying \[n^{2p}=m^2+n^2+p+1\] where $m$ and $n$ are integers and $p$ is a prime number.
    5. Find all positive integers $x,y,z$ and $t$ such that $$2^x3^y+5^z=7^t.$$
    6. Let $a$, $b$, $c$, $d$ be integers and $$\begin{align*}A&=2(a-2b+c)^4+2(b-2c+a)^4+2(c-2a+b)^4,\\ B&=d(d+1)(d+2)(d+3)+1.\end{align*}$$ Prove that $\left (\sqrt{A}+1 \right )^2 +B$ cannot be a perfect square.
    7. Find all $a , b , c \in \mathbb{N}$ for which \[1997^a+15^b=2012^c\]
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    Junior Balkan Mathematical Olympiad Shortlist 2011

    Algebra

    1. Let $a,b,c$ be positive real numbers such that $abc = 1$. Prove that $$\prod(a^5+a^4+a^3+a^2+a+1)\geq 8(a^2+a+1)(b^2+b+1)(c^2+c+1)$$
    2. Let $x, y, z$ be positive real numbers. Prove that $$\frac{x + 2y}{z + 2x + 3y}+\frac{y + 2z}{x + 2y + 3z}+\frac{z + 2x}{y + 2z + 3x} \le \frac{3}{2}$$
    3. If $a,b$ be positive real numbers, show that $$\sqrt{\dfrac{a^2+ab+b^2}{3}}+\sqrt{ab}\leq a+b$$
    4. Let $x,y$ be positive reals satisfying the condition $x^3+y^3\leq x^2+y^2$. Find the maximum value of $xy$.
    5. Determine all positive integers $a,b$ such that $$a^{2}b^{2}+208=4(\text{gcl}(a,b)+\text{lcm}(a,b))^2$$
    6. Let $x_i> 1$ for all $i \in \left \{1, 2, 3, \ldots, 2011 \right \}$. Show that $$\frac{x^2_1}{x_2-1}+\frac{x^2_2}{x_3-1}+\frac{x^2_3}{x_4-1}+\ldots+\frac{x^2_{2010}}{x_{2011}-1}+\frac{x^2_{2011}}{x_1-1}\geq 8044.$$ When the equality holds?
    7. Let $a,b,c$ be positive reals such that $abc=1$. Prove the inequality $$\frac{2a^2+\frac{1}{a}}{b+\frac{1}{a}+1} + \frac{2b^2+\frac{1}{b}}{c+\frac{1}{b}+1} + \frac{2c^2+\frac{1}{c}}{a+\frac{1}{c}+1}\geq 3$$
    8. Decipher the equality $$(\overline{LARN} -\overline{ACA}) : (\overline{CYP} +\overline{RUS}) = C^{Y^P} \cdot R^{U^S}$$ where different symbols correspond to different digits and equal symbols correspond to equal digits. It is also supposed that all these digits are different from $0$.
    9. Let $x_1,x_2, ..., x_n$ be real numbers satisfying $$\sum_{k=1}^{n-1} \min(x_k; x_{k+1}) = \min(x_1; x_n).$$ Prove that $$\sum_{k=2}^{n-1} x_k \ge 0.$$

    Combinatorics

    1. Inside of a square whose side length is $1$ there are a few circles such that the sum of their circumferences is equal to $10$. Show that there exists a line that meets at least four of these circles.
    2. Can we divide an equilateral triangle $\triangle ABC$ into $2011$ small triangles using $122$ straight lines? (there should be $2011$ triangles that are not themselves divided into smaller parts and there should be no polygons which are not triangles)
    3. We can change a natural number $n$ in three ways 
        • If the number $n$ has at least two digits, we erase the last digit and we subtract that digit from the remaining number (for example, from $123$ we get $12 - 3 = 9$);
        • If the last digit is different from $0$, we can change the order of the digits in the opposite one (for example, from $123$ we get $321$);
        • We can multiply the number $n$ by a number from the set $ \{1, 2, 3,..., 2010\}$.
        1. Can we get the number $21062011$ from the number $1012011$?
        2. In a group of $n$ people, each one had a different ball. They performed a sequence of swaps, in each swap, two people swapped the ball they had at that moment. Each pair of people performed at least one swap. In the end each person had the ball he/she had at the start. Find the least possible number of swaps, if
          a) $n = 5$,
          b) $n = 6$.
        3. A set $S$ of natural numbers is called good, if for each element $x \in S, x$ does not divide the sum of the remaining numbers in $S$. Find the maximal possible number of elements of a good set which is a subset of the set $$A = \{1,2, 3, ...,63\}.$$
        4. Let $n>3$ be a positive integer. Equilateral triangle ABC is divided into $n^2$ smaller congruent equilateral triangles (with sides parallel to its sides). Let $m$ be the number of rhombuses that contain two small equilateral triangles and $d$ the number of rhombuses that contain eight small equilateral triangles. Find the difference $m-d$ in terms of $n$.
        5. Consider a rectangle whose lengths of sides are natural numbers. If someone places as many squares as possible, each with area $3$, inside of the given rectangle, such that the sides of the squares are parallel to the rectangle sides, then the maximal number of these squares fill exactly half of the area of the rectangle. Determine the dimensions of all rectangles with this property.
        6. Determine the polygons with $n$ sides $(n \ge 4)$, not necessarily convex, which satisfy the property that the reflection of every vertex of polygon with respect to every diagonal of the polygon does not fall outside the polygon. (Each segment joining two non-neighboring vertices of the polygon is a diagonal. The reflection is considered with respect to the support line of the diagonal.)
        7. Decide if it is possible to consider $2011$ points in a plane such that the distance between every two of these points is different from $1$ and each unit circle centered at one of these points leaves exactly $1005$ points outside the circle.

        Geometry

        1. Let $ABC$ be an isosceles triangle with $AB=AC$. On the extension of the side ${CA}$ we consider the point ${D}$ such that ${AD<AC}$. The perpendicular bisector of the segment ${BD}$ meets the internal and the external bisectors of the angle $\angle BAC$ at the points ${E}$and ${Z}$, respectively. Prove that the points ${A, E, D, Z}$ are concyclic.
        2. Let $AD$, $BF$ and ${CE}$ be the altitudes of $\triangle ABC$. A line passing through ${D}$ and parallel to ${AB}$ intersects the line ${EF}$ at the point ${G}$. If ${H}$ is the orthocenter of $\triangle ABC$, find the angle ${\angle{CGH}}$.
        3. Let $ABC$ be a triangle in which (${BL}$is the angle bisector of ${\angle{ABC}}$ $\left( L\in AC \right)$, ${AH}$ is an altitude of $\triangle ABC$ $\left( H\in BC \right)$ and ${M}$ is the midpoint of the side ${AB}$. It is known that the midpoints of the segments ${BL}$ and ${MH}$ coincides. Determine the internal angles of triangle $\triangle ABC$.
        4. Point ${D}$ lies on the side ${BC}$ of $\triangle ABC$. The circumcenters of $\triangle ADC$ and $\triangle BAD$ are ${O_1}$ and ${O_2}$, respectively and ${O_1O_2\parallel AB}$. The orthocenter of $\triangle ADC$ is ${H}$ and $AH=O_1O_2$. Find the angles of $\triangle ABC$ if $2\angle C=3\angle B.$
        5. Inside the square ${ABCD}$, the equilateral triangle $\triangle ABE$ is constructed. Let ${M}$ be an interior point of the triangle $\triangle ABE$ such that $MB=\sqrt{2}$, $MC=\sqrt{6}$, $MD=\sqrt{5}$ and $ME=\sqrt{3}$. Find the area of the square ${ABCD}$.
        6. Let $ABCD$ be a convex quadrilateral and points $E$ and $F$ on sides $AB$, $CD$ such that \[\frac{AB}{AE}=\frac{CD}{DF}=n.\] Show that iIf $S$ is the area of $AEFD$ then $${S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}$$

        Number Theory

        1. Solve in positive integers the equation $$1005^x + 2011^y = 1006^z.$$
        2. Find all primes $p$ such that there exist positive integers $x,y$ that satisfy $$x(y^2-p)+y(x^2-p)=5p$$
        3. Find all positive integers $n$ such that the equation $$y^2 + xy + 3x = n(x^2 + xy + 3y)$$ has at least a solution $(x, y)$ in positive integers. 
        4. Find all primes $p,q$ such that $$2p^3-q^2=2(p+q)^2.$$
        5. Find the least positive integer such that the sum of its digits is $2011$ and the product of its digits is a power of $6$.
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        Junior Balkan Mathematical Olympiad Shortlist 2010

        Algebra

        1. The real numbers $a$, $b$, $c$, $d$ satisfy simultaneously the equations $abc -d = 1$, $bcd - a = 2$, $cda- b = 3$ and $dab - c = -6$. Prove that $$a + b + c + d \not = 0.$$
        2. Determine all four digit numbers $\bar{a}\bar{b}\bar{c}\bar{d}$ such that $$a(a+b+c+d)(a^{2}+b^{2}+c^{2}+d^{2})(a^{6}+2b^{6}+3c^{6}+4d^{6})=\bar{a}\bar{b}\bar{c}\bar{d}$$
        3. Find all pairs $(x,y)$ of real numbers such that $$\begin{cases}|x|+ |y|&=1340,\\ x^{3}+y^{3}+2010xy &= 670^{3}.\end{cases}$$
        4. Let $a,b,c $ be real positive numbers such that $abc(a+b+c)=3$. Prove that $$(a+b)(b+c)(c+a) \geq 8$$
        5. Let $ x, y, z > 0 $ with $ x \leq 2$, $y \leq 3$ and $x+y+z = 11 $. Prove that $$xyz \leq 36$$

        Combinatorics

        1. There are two piles of coins, each containing $2010$ pieces. Two players $A$ and $B$ play a game taking turns ($A$ plays first). At each turn, the player on play has to take one or more coins from one pile or exactly one coin from each pile. Whoever takes the last coin is the winner. Which player will win if they both play in the best possible way?
        2. A $9\times 7$ rectangle is tiled with tiles of the two types: L-shaped tiles composed by three unit squares (can be rotated repeatedly with $90^\circ$) and square tiles composed by four unit squares. Let $n\ge 0$ be the number of the $2 \times 2 $ tiles which can be used in such a tiling. Find all the values of $n$.

        Geometry

        1. Consider a triangle $ABC$ with $\angle ACB=90^{\circ}$. Let $F$ be the foot of the altitude from $C$. Circle $\omega$ touches the line segment $FB$ at point $P$, the altitude $CF$ at point $Q$ and the circumcircle of $ABC$ at point $R$. Prove that points $A$, $Q$, $R$ are collinear and $AP = AC$.
        2. Let $ABC$ be acute-angled triangle. A circle $\omega_1(O_1,R_1)$ passes through points $B$ and $C$ and meets the sides $AB$ and $AC$ at points $D$ and $E$, respectively. Let $\omega_2(O_2,R_2)$ be the circumcircle of triangle $ADE$. Prove that $O_1O_2$ is equal to the circumradius of triangle $ABC$.
        3. Consider a triangle ${ABC}$ and let ${M}$ be the midpoint of the side $BC$. Suppose $\angle MAC=\angle ABC$ and $\angle BAM=105^\circ$. Find the measure of ${\angle ABC}$.
        4. Let $AL$ and $BK$ be angle bisectors in the non-isosceles triangle $ABC$ ($L$ lies on the side $BC$, $K$ lies on the side $AC$). The perpendicular bisector of $BK$ intersects the line $AL$ at point $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK$. Prove that $LN = NA$.

        Number Theory

        1. Find all integers $n$, $n \ge 1$, such that $n \cdot 2^{n+1}+1$ is a perfect square.
        2. Find $n$ such that $36^n-6$ is the product of three consecutive natural numbers
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        Junior Balkan Mathematical Olympiad Shortlist 2009

        Algebra

        1. Determine all integers $a, b, c$ satisfying identities $$\begin{cases}a + b + c &= 15 \\ (a - 3)^3 + (b - 5)^3 + (c -7)^3 &= 540\end{cases}$$
        2. Find the maximum value of $z+x$ if $x,y,z$ are satisfying the given conditions $$\begin{cases}x^2+y^2&=4,\\ z^2+t^2&=9, \\ xt+yz&\geq 6\end{cases}$$
        3. Find all values of the real parameter $a$, for which the system $$\begin{cases}(|x| + |y| - 2)^2 &= 1,\\ y &= ax + 5\end{cases}$$ has exactly three solutions.
        4. Let $ x$, $ y$, $ z$ be real numbers such that $ 0 < x,y,z < 1$ and $$xyz = (1 - x)(1 - y)(1 - z).$$ Show that at least one of the numbers $$(1 - x)y,\quad (1 - y)z,\quad (1 - z)x$$ is greater than or equal to $ \dfrac {1}{4}$.
        5. Let $x,y,z$ be positive reals. Prove that $$(x^2+y+1)(x^2+z+1)(y^2+x+1)(y^2+z+1)(z^2+x+1)(z^2+y+1) \\ \geq (x+y+z)^6$$

        Combinatorics

        1. Each one of 2009 distinct points in the plane is coloured in blue or red, so that on every blue-centered unit circle there are exactly two red points. Find the gratest possible number of blue points.
        2. Five players $(A,B,C,D,E)$ take part in a bridge tournament. Every two players must play (as partners) against every other two players. Any two given players can be partners not more than once per a day. What is the least number of days needed for this tournament?
        3. a) In how many ways can we read the word SARAJEVO from the table below, if it is allowed to jump from cell to an adjacent cell (by vertex or a side) cell?
        4. b) After the letter in one cell was deleted, only $525$ ways to read the word SARAJEVO remained. Find all possible positions of that cell.
        5. Determine all pairs of $(m, n)$ such that is possible to tile the table $ m \times n$ with figure ”corner” as in figure with condition that in that tilling does not exist rectangle (except $m \times n$) regularly covered with figures.

        Geometry

        1. Parallelogram ${ABCD}$ is given with ${AC>BD}$, and ${O}$ point of intersection of ${AC}$ and ${BD}$. Circle with center at ${O}$and radius ${OA}$ intersects extensions of ${AD}$and ${AB}$ at points ${G}$ and ${L}$, respectively. Let ${Z}$ be intersection point of lines ${BD}$and ${GL}$. Prove that $\angle ZCA=90^\circ$.
        2. In right trapezoid ${ABCD \left(AB\parallel CD\right)}$ the angle at vertex B measures $75^\circ$. Point ${H}$is the foot of the perpendicular from point ${A}$ to the line ${BC}$. If ${BH=DC}$ and${AD+AH=8}$, find the area of ${ABCD}$.
        3. Parallelogram ${ABCD}$with obtuse angle $\angle ABC$ is given. After rotation of the triangle ${ACD}$ around the vertex ${C}$, we get a triangle ${CD'A'}$, such that points $B,C$ and ${D'}$are collinear. Extensions of median of triangle ${CD'A'}$ that passes through ${D'}$intersects the straight line ${BD}$at point ${P}$. Prove that ${PC}$is the bisector of the angle $\angle BP{D}'$.
        4. Let $ ABCDE$ be a convex pentagon such that $ AB+CD=BC+DE$ and $ k$ a circle with center on side $ AE$ that touches the sides $ AB$, $ BC$, $ CD$ and $ DE$ at points $ P$, $ Q$, $ R$ and $ S$ (different from vertices of the pentagon) respectively. Prove that lines $ PS$ and $ AE$ are parallel.
        5. Let $A$, $B$, $C$ and ${O}$ be four points in plane, such that $\angle ABC>90^circ$ and ${OA=OB=OC}$. Define the point ${D\in AB}$ and the line ${l}$ such that $D\in l$, $AC\perp DC$ and ${l\perp AO}$. Line ${l}$ cuts ${AC}$at ${E}$ and circumcircle of ${ABC}$ at ${F}$. Prove that the circumcircles of triangles ${BEF}$and ${CFD}$are tangent at ${F}$.

        Number Theory

        1. Solve in non-negative integers the equation $$2^{a}3^{b} + 9 = c^{2}$$
        2. A group of $n > 1$ pirates of different age owned total of $2009$ coins. Initially each pirate (except the youngest one) had one coin more than the next younger.
          a) Find all possible values of $n$.
          b) Every day a pirate was chosen. The chosen pirate gave a coin to each of the other pirates. If $n = 7$, find the largest possible number of coins a pirate can have after several days.
        3. Find all pairs $(x,y)$ of integers which satisfy the equation $$(x + y)^2(x^2 + y^2) = 2009^2$$
        4. Determine all prime numbers $$p_1, p_2,..., p_{12}, p_{13}, p_1 \le p_2 \le ... \le p_{12} \le p_{13}$$ such that $$p_1^2+ p_2^2+ ... + p_{12}^2 = p_{13}^2$$ and one of them is equal to $2p_1 + p_9$.
        5. Show that there are infinitely many positive integers $c$, such that the following equations both have solutions in positive integers $$\begin{cases}(x^2 - c)(y^2 -c) &= z^2 -c,\\ (x^2 + c)(y^2 - c) &= z^2 - c. \end{cases}$$
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        Junior Balkan Mathematical Olympiad Shorlist 2008

        Algebra

        1. If for the real numbers $x, y,z, k$ the following conditions are valid, $x \ne y \ne z \ne x$ and $$x^3 +y^3 +k(x^2 +y^2) = y^3 +z^3 +k(y^2 +z^2) = z^3 +x^3 +k(z^2 +x^2) = 2008.$$ Find the product $xyz$.
        2. Find all real numbers $ a,b,c,d$ such that \[\begin{cases}a + b + c + d &= 20, \\ ab + ac + ad + bc + bd + cd &= 150. \end{cases}.\]
        3. Let the real parameter $p$ be such that the system $$\begin{cases} p(x^2 - y^2) &= (p^2- 1)xy \\ |x - 1|+ |y| &= 1 \end{cases}$$ has at least three different real solutions. Find $p$ and solve the system for that $p$.
        4. Find all triples $(x,y,z)$ of real numbers that satisfy the system $$\begin{cases} x + y + z &= 2008 \\ x^2 + y^2 + z^2 &= 6024^2 \\ \dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}&=\dfrac{1}{2008} \end{cases}$$
        5. Find all triples $(x, y, z)$ of real positive numbers, which satisfy the system $$\begin{cases} \dfrac{1}{x}+\dfrac{4}{y}+\dfrac{9}{z}=3 \\ x + y + z \le 12 \end{cases}.$$ If the real numbers $a, b, c, d$ are such that $0 < a,b,c,d < 1$, show that $$1 + ab + bc + cd + da + ac + bd > a + b + c + d.$$
        6. Let $a, b$ and $c$ be positive real numbers such that $abc = 1$. Prove the inequality $$\Big(ab + bc +\frac{1}{ca}\Big)\Big(bc + ca +\frac{1}{ab}\Big)\Big(ca + ab +\frac{1}{bc}\Big)\ge (1 + 2a)(1 + 2b)(1 + 2c)$$
        7. Show that, for all real positive numbers $x, y $ and $z$,  $$(x + y + z) \big(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\big) \ge 4 \Big(\frac{x}{xy+1}+\frac{y}{yz+1}+\frac{z}{zx+1}\Big)^2$$
        8. Consider an integer $n \ge 4 $ and a sequence of real numbers $x_1, x_2, x_3,..., x_n$. An operation consists in eliminating all numbers not having the rank of the form $4k + 3$, thus leaving only the numbers $x_3. x_7. x_{11}, ...$(for example, the sequence $4,5,9,3,6, 6,1, 8$ produces the sequence $9,1$). Upon the sequence $1, 2, 3, ..., 1024 $ the operation is performed successively for $5$ times. Show that at the end only one number remains and find this number.

        Combinatorics

        1. On a $5 \times 5$ board, $n$ white markers are positioned, each marker in a distinct $1 \times 1$ square. A smart child got an assignment to recolor in black as many markers as possible, in the following manner: a white marker is taken from the board, it is colored in black, and then put back on the board on an empty square such that none of the neighboring squares contains a white marker (two squares are called neighboring if they share a common side). If it is possible for the child to succeed in coloring all the markers black, we say that the initial positioning of the markers was good.
          a) Prove that if $n = 20$, then a good initial positioning exists.
          b) Prove that if $n = 21$, then a good initial positioning does not exist.
        2. Kostas and Helene have the following dialogue
          • Kostas: I have in my mind three positive real numbers with product $1$ and sum equal to the sum of all their pairwise products.
          • Helene: I think that I know the numbers you have in mind. They are all equal to $1$.
          • Kostas: In fact, the numbers you mentioned satisfy my conditions, but I did not think of these numbers. The numbers you mentioned have the minimal sum between all possible solutions of the problem.
          Can you decide if Kostas is right? (Explain your answer).
        3. Integers $1,2, ...,2n$ are arbitrarily assigned to boxes labeled with numbers $1, 2,..., 2n$. Now, we add the number assigned to the box to the number on the box label. Show that two such sums give the same remainder modulo $2n$.
        4. Every cell of table $4 \times 4$ is colored into white. It is permitted to place the cross (pictured below) on the table such that its center lies on the table (the whole figure does not need to lie on the table) and change colors of every cell which is covered into opposite (white and black). Find all $n$ such that after $n$ steps it is possible to get the table with every cell colored black.

        Geometry

        1. Two perpendicular chords of a circle, $AM$, $BN$ which intersect at point $K$, define on the circle four arcs with pairwise different length, with $AB$ being the smallest of them. We draw the chords $AD$, $BC$ with $AD || BC$ and $C$, $D$ different from $N$, $M$. If $L$ is the point of intersection of $DN$, $MC$ and $T$ the point of intersection of $DC$, $KL$. Prove that $\angle KTC = \angle KNL$.
        2. For a fixed triangle $ABC$ we choose a point $M$ on the ray $CA$ (after $A$), a point $N$ on the ray $AB$ (after $B$) and a point $P$ on the ray $BC$ (after $C$) in a way such that $$AM -BC = BN- AC = CP - AB.$$ Prove that the angles of triangle $MNP$ do not depend on the choice of $M$, $N$, $P$.
        3. The vertices $ A$ and $ B$ of an equilateral triangle $ ABC$ lie on a circle $k$ of radius $1$, and the vertex $ C$ is in the interior of the circle $ k$. A point $ D$, different from $ B$, lies on $ k$ so that $ AD=AB$. The line $ DC$ intersects $ k$ for the second time at point $ E$. Find the length of the line segment $ CE$.
        4. Let $ABC$ be a triangle $(BC < AB)$. The line $l$ passing trough the vertices $C$ and orthogonal to the angle bisector $BE$ of $\angle B$, meets $BE$ and the median $BD$ of the side $AC$ at points $F$ and $G$, respectively. Prove that segment $DF$ bisects the segment $EG$.
        5. Is it possible to cover a given square with a few congruent right-angled triangles with acute angle equal to ${{30}^{o}}$? (The triangles may not overlap and may not exceed the margins of the square.)
        6. Let $ABC$ be a triangle with $\angle A<{{90}^{o}} $. Outside of a triangle we consider isosceles triangles $ABE$ and $ACZ$ with bases $AB$ and $AC$, respectively. If the midpoint $D$ of the side $BC$ is such that $DE \perp DZ$ and $EZ = 2ED$, prove that $\angle AEB = 2 \angle AZC$ .
        7. Let $ABC$ be an isosceles triangle with $AC = BC$. The point $D$ lies on the side $AB$ such that the semicircle with diameter $BD$ and center $O$ is tangent to the side $AC$ in the point $P$ and intersects the side $BC$ at the point $Q$. The radius $OP$ intersects the chord $DQ$ at the point $E$ such that $5PE = 3DE$. Find the ratio $\dfrac{AB}{BC}$ .
        8. The side lengths of a parallelogram are $a, b$ and diagonals have lengths $x$ and $y$. Knowing that $ab = \dfrac{xy}{2}$, show that $$\left( a,b \right)=\left( \frac{x}{\sqrt{2}},\frac{y}{\sqrt{2}} \right) \quad \text{or} \quad \left( a,b \right)=\left( \frac{y}{\sqrt{2}},\frac{x}{\sqrt{2}} \right).$$
        9. Let $O$ be a point inside the parallelogram $ABCD$ such that $$\angle AOB + \angle COD = \angle BOC + \angle AOD.$$ Prove that there exists a circle $k$ tangent to the circumscribed circles of the triangles $\vartriangle AOB$, $\vartriangle BOC$, $\vartriangle COD$ and $\vartriangle DOA$.
        10. Let $\Gamma$ be a circle of center $O$, and $\delta$. be a line in the plane of $\Gamma$, not intersecting it. Denote by $A$ the foot of the perpendicular from $O$ onto $\delta$, and let $M$ be a (variable) point on $\Gamma$. Denote by $\gamma$ the circle of diameter $AM$, by $X$ the (other than $M$) intersection point of $\gamma$ and $\Gamma$, and by $Y$ the (other than $A$) intersection point of $\gamma$ and $\delta$. Prove that the line $XY$ passes through a fixed point.
        11. Consider $ABC$ an acute-angled triangle with $AB \ne AC$. Denote by $M$ the midpoint of $BC$, by $D, E$ the feet of the altitudes from $B, C$ respectively and let $P$ be the intersection point of the lines $DE$ and $BC$. The perpendicular from $M$ to $AC$ meets the perpendicular from $C$ to $BC$ at point $R$. Prove that lines $PR$ and $AM$ are perpendicular.

        Number Theory

        1. Find all the positive integers $x$ and $y$ that satisfy the equation $$x(x - y) = 8y - 7$$
        2. Let $n \ge 2$ be a fixed positive integer. An integer will be called "$n$-free" if it is not a multiple of an $n$-th power of a prime. Let $M$ be an infinite set of rational numbers, such that the product of every $n$ elements of $M$ is an $n$-free integer. Prove that $M$ contains only integers.
        3. Let $s(a)$ denote the sum of digits of a given positive integer a. The sequence $a_1, a_2,..., a_n, ...$ of positive integers is such that $a_{n+1} = a_n+s(a_n)$ for each positive integer $n$. Find the greatest possible n for which it is possible to have $a_n = 2008$.
        4. Find all integers $n$ such that $n^4 + 8n + 11$ is a product of two or more consecutive integers.
        5. Is it possible to arrange the numbers $1^1, 2^2,..., 2008^{2008}$ one after the other, in such a way that the obtained number is a perfect square? (Explain your answer.)
        6. Let $f : N \to R$ be a function, satisfying the following condition:
          for every integer $n > 1$, there exists a prime divisor $p$ of $n$ such that $$f(n) = f \Big(\frac{n}{p}\Big)-f(p).$$ If $f(2^{2007}) + f(3^{2008}) + f(5^{2009}) = 2006$, determine the value of $$f(2007^2) + f(2008^3) + f(2009^5)$$
        7. Determine the minimum value of prime $p> 3$ for which there is no natural number $n> 0$ such that $2^n+3^n\equiv 0\pmod{p} $.
        8. Let $a, b, c, d, e, f$ are nonzero digits such that the natural numbers $\overline{abc}, \overline{def}$ and $\overline{abcdef }$ are squares.
          a) Prove that $\overline{abcdef}$ can be represented in two different ways as a sum of three squares of natural numbers.
          b) Give an example of such a number.
        9. Let $p$ be a prime number. Find all positive integers $a$ and $b$ such that $$\frac{4a + p}{b}+\frac{4b + p}{a} \quad \text{and} \quad \frac{a^2}{b}+\frac{b^2}{a}$$ are integers.
        10. Prove that $2^n + 3^n$ is not a perfect cube for any positive integer $n$.
        11. Determine the greatest number with $n$ digits in the decimal representation which is divisible by $429$ and has the sum of all digits less than or equal to $11$.
        12. Find all prime numbers $p,q,r$ such that $$\frac{p}{q}-\frac{4}{r+1}=1$$
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        Statictis