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[Solutions] European Mathematical Cup 2018

Junior Division

  1. Let $a, b, c$ be non-zero real numbers such that $$a^2+b+c=\frac{1}{a}, \quad b^2+c+a=\frac{1}{b},\quad c^2+a+b=\frac{1}{c}.$$ Prove that at least two of $a, b, c$ are equal.
  2. Find all pairs $ (x; y) $ of positive integers such that $$xy | x^2 + 2y -1.$$
  3. Let $ABC$ be an acute triangle with $|AB | < |AC |$ and orthocenter $H$. The circle with center A and radius $|AC |$ intersects the circumcircle of $\triangle ABC$ at point $D$ and the circle with center $A$ and radius $ |AB |$ intersects the segment $\overline{AD}$ at point $K. $ The line through $K$ parallel to $CD $ intersects $BC$ at the point $ L.$ If $M$ is the midpoint of $\overline{BC}$ and $N$ is the foot of the perpendicular from $H$ to $AL, $ prove that the line $ MN $ bisects the segment $\overline{AH}$.
  4. Let $n$ be a positive integer. Ana and Banana are playing the following game: First, Ana arranges $2n$ cups in a row on a table, each facing upside-down. She then places a ball under a cup and makes a hole in the table under some other cup. Banana then gives a finite sequence of commands to Ana, where each command consists of swapping two adjacent cups in the row. Her goal is to achieve that the ball has fallen into the hole during the game. Assuming Banana has no information about the position of the hole and the position of the ball at any point, what is the smallest number of commands she has to give in order to achieve her goal?

Senior Division

  1. A partition of a positive integer is even if all its elements are even numbers. Similarly, a partition is odd if all its elements are odd. Determine all positive integers $n$ such that the number of even partitions of $n$ is equal to the number of odd partitions of $n$.
    Remark: A partition of a positive integer $n$ is a non-decreasing sequence of positive integers whose sum of elements equals $n$. For example, $(2; 3; 4), (1; 2; 2; 2; 2)$ and $(9) $ are partitions of $9.$
  2. Let ABC be a triangle with $|AB|< |AC|. $ Let $k$ be the circumcircle of $\triangle ABC$ and let $O$ be the center of $k$. Point $M$ is the midpoint of the arc $BC $ of $k$ not containing $A$. Let $D $ be the second intersection of the perpendicular line from $M$ to $AB$ with $ k$ and $E$ be the second intersection of the perpendicular line from $M$ to $AC $ with $k$. Points $X $and $Y $ are the intersections of $CD$ and $BE$ with $OM$ respectively. Denote by $k_b$ and $k_c$ circumcircles of triangles $BDX$ and $CEY$ respectively. Let $G$ and $H$ be the second intersections of $k_b$ and $k_c $ with $AB$ and $AC$ respectively. Denote by ka the circumcircle of triangle $AGH.$ Prove that $O$ is the circumcenter of $\triangle O_aO_bO_c$, where $O_a$, $O_b$, $O_c $ are the centers of $k_a$, $k_b$, $k_c$ respectively.
  3. For which real numbers $k > 1$ does there exist a bounded set of positive real numbers $S$ with at least $3$ elements such that $k(a - b)\in S$ for all $a,b\in S $ with $a > b$?.
    Remark: A set of positive real numbers $S$ is bounded if there exists a positive real number $M$ such that $x < M$ for all $x \in S.$
  4. Let $x; y; m; n$ be integers greater than $1$ such that $$
    \underset{m\text{ times}}{\underbrace{x^{x^{x^{\dots^{x}}}}}}=\underset{n\text{ times}}{\underbrace{y^{y^{y^{\dots^{y}}}}}} .$$ Does it follow that $m = n$?.
    Remark: This is a tetration operation, so we can also write $^mx = ^ny$ for the initial condition.

COMMENTS

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Name

Ả-rập Xê-út,1,Abel,2,Albania,2,American Mathematical Monthly,2,AMM,1,Amsterdam,8,Ấn Độ,1,An Giang,16,Andrew Wiles,1,Anh,2,Áo,1,APMO,16,Arabia,1,Ba Lan,1,Bà Rịa Vũng Tàu,44,Bắc Bộ,23,Bắc Giang,40,Bạc Liêu,7,Bắc Ninh,34,Bắc Trung Bộ,8,Bài Toán Hay,3,Balkan,29,Baltic Way,29,BAMO,1,Bất Đẳng Thức,77,BDHSG,14,Bến Tre,21,Benelux,11,Bình Định,36,Bình Dương,18,Bình Phước,20,Bình Thuận,25,Birch,1,Bosnia Herzegovina,2,BoxMath,3,Brazil,2,Bùi Đắc Hiên,1,Bùi Văn Tuyên,1,Bulgaria,5,BxMO,10,Cà Mau,12,Cần Thơ,12,Canada,63,Cao Bằng,5,Cao Quang Minh,1,Câu Chuyện Toán Học,30,Chọn Đội Tuyển,274,Chu Tuấn Anh,1,Chuyên Đề,104,Chuyên Sư Phạm,28,Collection,8,College Mathematics Journal,1,Concours,1,Cono Sur,1,Correspondence,1,Cosmin Poahata,1,CPS,4,Crux,2,Đà Nẵng,36,Đa Thức,2,Đại Số,31,Đắk Lắk,48,Đắk Nông,4,Đan Phượng,1,Đào Thái Hiệp,1,ĐBSCL,2,Đề Thi,1957,Đề Thi HSG,1108,Đề Thi JMO,1,Điện Biên,5,Định Lý,1,Định Lý Beaty,1,Đoàn Quỳnh,1,Đoàn Văn Trung,1,Đống Đa,3,Đồng Nai,42,Đồng Tháp,40,Đức,1,E-Book,19,EGMO,12,ELMO,17,EMC,7,Estonian,5,Evan Chen,1,Fermat,3,Finland,4,G. Polya,3,Gặp Gỡ Toán Học,21,GDTX,3,Geometry,5,Gia Lai,20,Giải Tích Hàm,1,Giảng Võ,1,Giới hạn,2,Goldbach,1,Hà Giang,2,Hà Lan,1,Hà Nam,21,Hà Nội,151,Hà Tĩnh,60,Hà Trung Kiên,1,Hải Dương,41,Hải Phòng,36,Hàn Quốc,4,Hậu Giang,3,Hilbert,1,Hình Học,49,HKUST,6,Hòa Bình,12,Hoàng Minh Quân,1,Hodge,1,Hojoo Lee,2,Hong Kong,1,HongKong,6,HSG 10,86,HSG 11,63,HSG 12,469,HSG 9,309,HSG Cấp Trường,64,HSG Quốc Gia,86,HSG Quốc Tế,13,Hứa Lâm Phong,1,Huế,30,Hùng Vương,25,Hưng Yên,24,Hy Lạp,1,IMC,23,IMO,40,India,37,Inequality,13,International,208,Iran,4,Jakob,1,JBMO,16,Journal,16,K2pi,1,Kazakhstan,1,Khánh Hòa,10,KHTN,46,Kiên Giang,26,Kon Tum,17,Kvant,2,Kỷ Yếu,37,Lai Châu,3,Lâm Đồng,20,Lạng Sơn,17,Langlands,1,Lào Cai,9,Lê Hoành Phò,4,Lê Khánh Sỹ,3,Lê Minh Cường,1,Lê Phúc Lữ,4,Lê Viết Hải,1,Lê Việt Hưng,1,Long An,33,Lớp 10,8,Lớp 10 Chuyên,342,Lớp 10 Không Chuyên,140,Lớp 11,1,Lượng giác,1,Lương Tài,1,Lưu Giang Nam,2,Macedonian,1,Malaysia,1,Mark Levi,1,Mathematical Excalibur,1,Mathematical Reflections,1,Mathematics Magazine,1,Mathematics Today Magazine,1,MathProblems Journal,1,Mathscope,8,MEMO,9,Metropolises,3,Mexico,1,Michael Guillen,1,Mochizuki,1,Moldova,1,Moscow,1,Mỹ,7,MYM,74,MYTS,1,Nam Định,26,Nam Phi,1,National,177,Nesbitt,1,Nghệ An,43,Ngô Bảo Châu,1,Ngô Việt Hải,1,Ngọc Huyền,2,Nguyễn Anh Tuyến,1,Nguyễn Bá Đang,1,Nguyễn Đình Thi,2,Nguyễn Đức Tấn,1,Nguyễn Duy Khương,1,Nguyễn Duy Tùng,1,Nguyễn Hữu Điển,3,Nguyễn Mình Hà,1,Nguyễn Minh Tuấn,4,Nguyễn Phan Tài Vương,1,Nguyễn Phú Khánh,1,Nguyễn Phúc Tăng,1,Nguyễn Quang Sơn,1,Nguyễn Tài Chung,4,Nguyễn Tăng Vũ,1,Nguyễn Tất Thu,1,Nguyễn Thúc Vũ Hoàng,1,Nguyễn Trung Tuấn,7,Nguyễn Tuấn Anh,2,Nguyễn Văn Huyện,3,Nguyễn Văn Mậu,23,Nguyễn Văn Nho,1,Nguyễn Văn Quý,1,Nguyễn Văn Thông,1,Nguyễn Việt Anh,1,Nguyễn Vũ Lương,2,Nhật Bản,2,Nhóm Toán,3,Ninh Bình,36,Ninh Thuận,13,Nội Suy Lagrange,1,Nội Suy Newton,1,Nordic,18,Olympiad Corner,1,Olympiad Preliminary,2,Olympic 10,87,Olympic 10/3,3,Olympic 11,79,Olympic 12,27,Olympic 24/3,6,Olympic 27/4,19,Olympic 30/4,56,Olympic KHTN,5,Olympic Sinh Viên,63,Olympic Toán,258,PAMO,1,Phạm Đình Đồng,1,Phạm Đức Tài,1,Phạm Huy Hoàng,1,Pham Kim Hung,3,Phạm Quốc Sang,2,Phan Huy Khải,1,Phan Thành Nam,1,Pháp,2,Philippine,1,Philippines,4,Phú Thọ,24,Phú Yên,21,Phùng Hồ Hải,1,Phương Trình Hàm,26,Phương Trình Pythagoras,1,Pi,1,Problems,1,PT-HPT,32,PTNK,37,Putnam,24,Quảng Bình,37,Quảng Nam,26,Quảng Ngãi,29,Quảng Ninh,32,Quảng Trị,17,Riemann,1,RMM,11,Romania,8,Romanian Mathematical Magazine,1,Russia,1,Sách Thường Thức Toán,7,Sách Toán,79,Sách Toán Cao Học,1,Sách Toán THCS,7,Saudi,2,Scholze,1,Serbia,17,Sharygin,19,Shortlists,35,Simon Singh,1,Singapore,1,Số học,38,Sóc Trăng,7,Sơn La,10,Swinnerton-Dyer,1,Talent Search,1,Tăng Hải Tuân,2,Tạp Chí,15,Tây Ban Nha,1,Tây Ninh,24,Thái Bình,33,Thái Nguyên,31,Thanh Hóa,46,THCS,2,Thổ Nhĩ Kỳ,4,Thomas J. Mildorf,1,THPTQG,11,THTT,7,Tiền Giang,16,Titu Andreescu,2,Tổ hợp,7,Toán 12,7,Toán Cao Cấp,3,Toán Chuyên,2,Toán Rời Rạc,20,Toán Tuổi Thơ,2,TOT,1,TPHCM,99,Trà Vinh,5,Trắc Nghiệm,1,Trắc Nghiệm Toán,2,Trại Hè,32,Trại Hè Phương Nam,5,Trần Đăng Phúc,1,Trần Minh Hiền,2,Trần Nam Dũng,8,Trần Phương,1,Trần Quang Hùng,1,Trần Quốc Anh,1,Trần Quốc Luật,1,Trần Tiến Tự,1,Trịnh Đào Chiến,2,Trung Quốc,11,Trường Đông,16,Trường Hè,7,Trường Thu,1,Trường Xuân,2,TST,44,Tuyên Quang,6,Tuyển sinh,10,Tuyển Tập,33,Tuymaada,1,Undergraduate,61,USA,28,USAJMO,1,USATST,5,Uzbekistan,1,Vasile Cîrtoaje,3,Viện Toán Học,1,Vietnam,2,Viktor Prasolov,1,VIMF,1,Vinh,23,Vĩnh Long,17,Vĩnh Phúc,55,Virginia Tech,1,VLTT,1,VMEO,4,VMF,8,VMO,38,VNTST,18,Võ Quốc Bá Cẩn,18,Võ Thành Văn,1,Vojtěch Jarník,5,Vũ Hữu Bình,7,Vương Trung Dũng,1,WFNMC Journal,1,Wiles,1,Yên Bái,15,Yên Định,1,Zhautykov,10,Zhou Yuan Zhe,1,
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Mathematical Olympiad Contests Collection: [Solutions] European Mathematical Cup 2018
[Solutions] European Mathematical Cup 2018
Mathematical Olympiad Contests Collection
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