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[Solutions] International Mathematical Olympiad 2017

  1. For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as $$a_{n+1} = \begin{cases} \sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\ a_n + 3 & \text{otherwise.} \end{cases} $$Determine all values of $a_0$ such that there exists a number $A$ such that $a_n = A$ for infinitely many values of $n$.
  2. Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for any real numbers $x$ and $y$, \[ f(f(x)f(y)) + f(x+y) = f(xy). \]
  3. A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit's starting point, $A_0,$ and the hunter's starting point, $B_0$ are the same. After $n-1$ rounds of the game, the rabbit is at point $A_{n-1}$ and the hunter is at point $B_{n-1}.$ In the $n^{\text{th}}$ round of the game, three things occur in order
    • The rabbit moves invisibly to a point $A_n$ such that the distance between $A_{n-1}$ and $A_n$ is exactly $1.$
    • A tracking device reports a point $P_n$ to the hunter. The only guarantee provided by the tracking device to the hunter is that the distance between $P_n$ and $A_n$ is at most $1.$
    The hunter moves visibly to a point $B_n$ such that the distance between $B_{n-1}$ and $B_n$ is exactly $1.$ Is it always possible, no matter how the rabbit moves, and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after $10^9$ rounds, she can ensure that the distance between her and the rabbit is at most $100?$
  4. Let $R$ and $S$ be different points on a circle $\Omega$ such that $RS$ is not a diameter. Let $\ell$ be the tangent line to $\Omega$ at $R$. Point $T$ is such that $S$ is the midpoint of the line segment $RT$. Point $J$ is chosen on the shorter arc $RS$ of $\Omega$ so that the circumcircle $\Gamma$ of triangle $JST$ intersects $\ell$ at two distinct points. Let $A$ be the common point of $\Gamma$ and $\ell$ that is closer to $R$. Line $AJ$ meets $\Omega$ again at $K$. Prove that the line $KT$ is tangent to $\Gamma$.
  5. An integer $N \ge 2$ is given. A collection of $N(N + 1)$ soccer players, no two of whom are of the same height, stand in a row. Sir Alex wants to remove $N(N - 1)$ players from this row leaving a new row of $2N$ players in which the following $N$ conditions hold:
    • ($1$) no one stands between the two tallest players,
    • ($2$) no one stands between the third and fourth tallest players,
    • $\;\;\vdots$
    • ($N$) no one stands between the two shortest players.
    Show that this is always possible.
  6. An ordered pair $(x, y)$ of integers is a primitive point if the greatest common divisor of $x$ and $y$ is $1$. Given a finite set $S$ of primitive points, prove that there exist a positive integer $n$ and integers $a_0, a_1, \ldots , a_n$ such that, for each $(x, y)$ in $S$, we have $$a_0x^n + a_1x^{n-1} y + a_2x^{n-2}y^2 + \cdots + a_{n-1}xy^{n-1} + a_ny^n = 1.$$

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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] International Mathematical Olympiad 2017
[Solutions] International Mathematical Olympiad 2017
Mathematical Olympiad Contests Collection
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