[Shortlists] International Mathematical Olympiad 2008


  1. Find all functions $ f: (0, \infty) \mapsto (0, \infty)$ (so $ f$ is a function from the positive real numbers) such that \[ \frac {\left( f(w) \right)^2 + \left( f(x) \right)^2}{f(y^2) + f(z^2) } = \frac {w^2 + x^2}{y^2 + z^2} \] for all positive real numbers $ w,x,y,z,$ satisfying $ wx = yz.$
  2. a) Prove that \[\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1\] for all real numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz=1$.
    b) Prove that equality holds above for infinitely many triples of rational numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz=1$.
  3. Let $ S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $ (f, g)$ of functions from $ S$ into $ S$ is a Spanish Couple on $ S$, if they satisfy the following conditions
    • both functions are strictly increasing, i.e. $ f(x) < f(y)$ and $ g(x) < g(y)$ for all $ x$, $ y\in S$ with $ x < y$;
    • the inequality $ f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $ x\in S$.
    Decide whether there exists a Spanish Couple
    a) on the set $ S = \mathbb{N}$ of positive integers;
    b) on the set $ S = \{a - \frac {1}{b}: a, b\in\mathbb{N}\}$
  4. For an integer $ m$, denote by $ t(m)$ the unique number in $ \{1, 2, 3\}$ such that $ m + t(m)$ is a multiple of $ 3$. A function $ f: \mathbb{Z}\to\mathbb{Z}$ satisfies $ f( - 1) = 0$, $ f(0) = 1$, $ f(1) = - 1$ and $ f\left(2^{n} + m\right) = f\left(2^n - t(m)\right) - f(m)$ for all integers $ m$, $ n\ge 0$ with $ 2^n > m$. Prove that $ f(3p)\ge 0$ holds for all integers $ p\ge 0$.
  5. Let $ a$, $ b$, $ c$, $ d$ be positive real numbers such that $ abcd = 1$ and $ a + b + c + d > \dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{d} + \dfrac{d}{a}$. Prove that \[ a + b + c + d < \dfrac{b}{a} + \dfrac{c}{b} + \dfrac{d}{c} + \dfrac{a}{d}\]
  6. Let $ f: \mathbb{R}\to\mathbb{N}$ be a function which satisfies $$f\left(x + \dfrac{1}{f(y)}\right) = f\left(y + \dfrac{1}{f(x)}\right)$$ for all $ x$, $ y\in\mathbb{R}$. Prove that there is a positive integer which is not a value of $ f$.
  7. Prove that for any four positive real numbers $ a$, $ b$, $ c$, $ d$ the inequality \[ \frac {(a - b)(a - c)}{a + b + c} + \frac {(b - c)(b - d)}{b + c + d} + \frac {(c - d)(c - a)}{c + d + a} + \frac {(d - a)(d - b)}{d + a + b}\ge 0\] holds. Determine all cases of equality.


  1. In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a box. Two boxes intersect if they have a common point in their interior or on their boundary. Find the largest $ n$ for which there exist $ n$ boxes $ B_1$, $ \ldots$, $ B_n$ such that $ B_i$ and $ B_j$ intersect if and only if $ i\not\equiv j\pm 1\pmod n$.
  2. Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which \[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\] Find the number of elements of the set $A_n$.
  3. In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $ a$, $ B\in S$ will be called $ k$-friends if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-clique if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements.
  4. Let $ n$ and $ k$ be positive integers with $ k \geq n$ and $ k - n$ an even number. Let $ 2n$ lamps labelled $ 1$, $ 2$, ..., $ 2n$ be given, each of which can be either on or off. Initially all the lamps are off. We consider sequences of steps: at each step one of the lamps is switched (from on to off or from off to on). Let $ N$ be the number of such sequences consisting of $ k$ steps and resulting in the state where lamps $ 1$ through $ n$ are all on, and lamps $ n + 1$ through $ 2n$ are all off. Let $ M$ be number of such sequences consisting of $ k$ steps, resulting in the state where lamps $ 1$ through $ n$ are all on, and lamps $ n + 1$ through $ 2n$ are all off, but where none of the lamps $ n + 1$ through $ 2n$ is ever switched on. Determine $ \frac {N}{M}$.
  5. Let $ S = \{x_1, x_2, \ldots, x_{k + l}\}$ be a $ (k + l)$-element set of real numbers contained in the interval $ [0, 1]$; $ k$ and $ l$ are positive integers. A $ k$-element subset $ A\subset S$ is called nice if \[ \left |\frac {1}{k}\sum_{x_i\in A} x_i - \frac {1}{l}\sum_{x_j\in S\setminus A} x_j\right |\le \frac {k + l}{2kl}.\] Prove that the number of nice subsets is at least $ \dfrac{2}{k + l}\dbinom{k + l}{k}$.
  6. For $ n\ge 2$, let $ S_1$, $ S_2$, $ \ldots$, $ S_{2^n}$ be $ 2^n$ subsets of $ A = \{1, 2, 3, \ldots, 2^{n + 1}\}$ that satisfy the following property: There do not exist indices $ a$ and $ b$ with $ a < b$ and elements $ x$, $ y$, $ z\in A$ with $ x < y < z$ and $ y$, $ z\in S_a$, and $ x$, $ z\in S_b$. Prove that at least one of the sets $ S_1$, $ S_2$, $ \ldots$, $ S_{2^n}$ contains no more than $ 4n$ elements.


  1. Let $ H$ be the orthocenter of an acute-angled triangle $ ABC$. The circle $ \Gamma_{A}$ centered at the midpoint of $ BC$ and passing through $ H$ intersects the sideline $ BC$ at points $ A_{1}$ and $ A_{2}$. Similarly, define the points $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$. Prove that the six points $ A_{1}$, $ A_{2}$, $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$ are concyclic.
  2. Given trapezoid $ ABCD$ with parallel sides $ AB$ and $ CD$, assume that there exist points $ E$ on line $ BC$ outside segment $ BC$, and $ F$ inside segment $ AD$ such that $ \angle DAE = \angle CBF$. Denote by $ I$ the point of intersection of $ CD$ and $ EF$, and by $ J$ the point of intersection of $ AB$ and $ EF$. Let $ K$ be the midpoint of segment $ EF$, assume it does not lie on line $ AB$. Prove that $ I$ belongs to the circumcircle of $ ABK$ if and only if $ K$ belongs to the circumcircle of $ CDJ$.
  3. Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE = \angle QDE$ and $ \angle PBE = \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic.
  4. In an acute triangle $ ABC$ segments $ BE$ and $ CF$ are altitudes. Two circles passing through the point $ A$ anf $ F$ and tangent to the line $ BC$ at the points $ P$ and $ Q$ so that $ B$ lies between $ C$ and $ Q$. Prove that lines $ PE$ and $ QF$ intersect on the circumcircle of triangle $ AEF$.
  5. Let $ k$ and $ n$ be integers with $ 0\le k\le n - 2$. Consider a set $ L$ of $ n$ lines in the plane such that no two of them are parallel and no three have a common point. Denote by $ I$ the set of intersections of lines in $ L$. Let $ O$ be a point in the plane not lying on any line of $ L$. A point $ X\in I$ is colored red if the open line segment $ OX$ intersects at most $ k$ lines in $ L$. Prove that $ I$ contains at least $ \dfrac{1}{2}(k + 1)(k + 2)$ red points.
  6. There is given a convex quadrilateral $ ABCD$. Prove that there exists a point $ P$ inside the quadrilateral such that $\angle PAB + \angle PDC$ $= \angle PBC + \angle PAD$ $= \angle PCD + \angle PBA$ $= \angle PDA + \angle PCB$ $= 90^{\circ}$ if and only if the diagonals $ AC$ and $ BD$ are perpendicular.
  7. Let $ ABCD$ be a convex quadrilateral with $ BA\neq BC$. Denote the incircles of triangles $ ABC$ and $ ADC$ by $ \omega_{1}$ and $ \omega_{2}$ respectively. Suppose that there exists a circle $ \omega$ tangent to ray $ BA$ beyond $ A$ and to the ray $ BC$ beyond $ C$, which is also tangent to the lines $ AD$ and $ CD$. Prove that the common external tangents to $ \omega_{1}$ and $\omega_{2}$ intersect on $ \omega$.

Number Theory

  1. Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$.
  2. Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i + a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$.
  3. Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i + 1}) > a_{i - 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$.
  4. Let $ n$ be a positive integer. Show that the numbers \[ \binom{2^n - 1}{0},\; \binom{2^n - 1}{1},\; \binom{2^n - 1}{2},\; \ldots,\; \binom{2^n - 1}{2^{n - 1} - 1}\] are congruent modulo $ 2^n$ to $ 1$, $ 3$, $ 5$, $ \ldots$, $ 2^n - 1$ in some order.
  5. For every $ n\in\mathbb{N}$ let $ d(n)$ denote the number of (positive) divisors of $ n$. Find all functions $ f: \mathbb{N}\to\mathbb{N}$ with the following properties
    • $ d\left(f(x)\right) = x$ for all $ x\in\mathbb{N}$.
    • $ f(xy)$ divides $ (x - 1)y^{xy - 1}f(x)$ for all $ x$, $ y\in\mathbb{N}$.
  6. Prove that there are infinitely many positive integers $ n$ such that $ n^{2} + 1$ has a prime divisor greater than $ 2n + \sqrt {2n}$.

Post a Comment





Kỷ Yếu$cl=violet$type=three$count=6$sr=random$t=oot$h=1$l=0$meta=hide$rm=hide$sn=0$hide=mobile



Ả-rập Xê-út,1,Abel,5,Albania,2,AMM,3,Amsterdam,5,Ấn Độ,2,An Giang,23,Andrew Wiles,1,Anh,2,Áo,1,APMO,19,Ba Đình,2,Ba Lan,1,Bà Rịa Vũng Tàu,53,Bắc Giang,50,Bắc Kạn,1,Bạc Liêu,9,Bắc Ninh,48,Bắc Trung Bộ,7,Bài Toán Hay,5,Balkan,38,Baltic Way,30,BAMO,1,Bất Đẳng Thức,66,Bến Tre,46,Benelux,14,Bình Định,46,Bình Dương,23,Bình Phước,38,Bình Thuận,34,Birch,1,Booklet,11,Bosnia Herzegovina,3,BoxMath,3,Brazil,2,Bùi Đắc Hiên,1,Bùi Thị Thiện Mỹ,1,Bùi Văn Tuyên,1,Bùi Xuân Diệu,1,Bulgaria,6,Buôn Ma Thuột,1,BxMO,13,Cà Mau,14,Cần Thơ,14,Canada,40,Cao Bằng,7,Cao Quang Minh,1,Câu Chuyện Toán Học,36,Caucasus,2,CGMO,10,China,10,Chọn Đội Tuyển,355,Chu Tuấn Anh,1,Chuyên Đề,124,Chuyên Sư Phạm,31,Chuyên Trần Hưng Đạo,3,Collection,8,College Mathematic,1,Concours,1,Cono Sur,1,Contest,618,Correspondence,1,Cosmin Poahata,1,Crux,2,Czech-Polish-Slovak,26,Đà Nẵng,39,Đa Thức,2,Đại Số,20,Đắk Lắk,56,Đắk Nông,7,Đan Phượng,1,Danube,7,Đào Thái Hiệp,1,ĐBSCL,2,Đề Thi,1,Đề Thi HSG,1773,Đề Thi JMO,1,Điện Biên,8,Định Lý,1,Định Lý Beaty,1,Đỗ Hữu Đức Thịnh,1,Do Thái,3,Doãn Quang Tiến,4,Đoàn Quỳnh,1,Đoàn Văn Trung,1,Đống Đa,4,Đồng Nai,50,Đồng Tháp,52,Du Hiền Vinh,1,Đức,1,Duyên Hải Bắc Bộ,25,E-Book,33,EGMO,17,ELMO,19,EMC,9,Epsilon,1,Estonian,5,Euler,1,Evan Chen,1,Fermat,3,Finland,4,Forum Of Geometry,2,Furstenberg,1,G. Polya,3,Gặp Gỡ Toán Học,26,Gauss,1,GDTX,3,Geometry,12,Gia Lai,26,Gia Viễn,2,Giải Tích Hàm,1,Giảng Võ,1,Giới hạn,2,Goldbach,1,Hà Giang,2,Hà Lan,1,Hà Nam,29,Hà Nội,232,Hà Tĩnh,73,Hà Trung Kiên,1,Hải Dương,50,Hải Phòng,42,Hàn Quốc,5,Hậu Giang,4,Hậu Lộc,1,Hilbert,1,Hình Học,33,HKUST,7,Hòa Bình,13,Hoài Nhơn,1,Hoàng Bá Minh,1,Hoàng Minh Quân,1,Hodge,1,Hojoo Lee,2,HOMC,5,HongKong,8,HSG 10,101,HSG 11,91,HSG 12,588,HSG 9,425,HSG Cấp Trường,78,HSG Quốc Gia,106,HSG Quốc Tế,16,Hứa Lâm Phong,1,Hứa Thuần Phỏng,1,Hùng Vương,2,Hưng Yên,33,Hương Sơn,2,Huỳnh Kim Linh,1,Hy Lạp,1,IMC,26,IMO,56,IMT,1,India,45,Inequality,13,InMC,1,International,315,Iran,11,Jakob,1,JBMO,41,Jewish,1,Journal,20,Junior,38,K2pi,1,Kazakhstan,1,Khánh Hòa,17,KHTN,54,Kiên Giang,64,Kim Liên,1,Kon Tum,18,Korea,5,Kvant,2,Kỷ Yếu,42,Lai Châu,4,Lâm Đồng,33,Lạng Sơn,21,Langlands,1,Lào Cai,17,Lê Hải Châu,1,Lê Hải Khôi,1,Lê Hoành Phò,4,Lê Khánh Sỹ,3,Lê Minh Cường,1,Lê Phúc Lữ,1,Lê Phương,1,Lê Quý Đôn,1,Lê Viết Hải,1,Lê Việt Hưng,1,Leibniz,1,Long An,42,Lớp 10,10,Lớp 10 Chuyên,455,Lớp 10 Không Chuyên,229,Lớp 11,1,Lục Ngạn,1,Lượng giác,1,Lương Tài,1,Lưu Giang Nam,2,Lý Thánh Tông,1,Macedonian,1,Malaysia,1,Margulis,2,Mark Levi,1,Mathematical Excalibur,1,Mathematical Reflections,1,Mathematics Magazine,1,Mathematics Today,1,Mathley,1,MathLinks,1,MathProblems Journal,1,Mathscope,8,MathsVN,5,MathVN,1,MEMO,11,Metropolises,4,Mexico,1,MIC,1,Michael Guillen,1,Mochizuki,1,Moldova,1,Moscow,1,Mỹ,10,MYM,227,MYTS,4,Nam Định,33,Nam Phi,1,Nam Trung Bộ,1,National,249,Nesbitt,1,Newton,4,Nghệ An,52,Ngô Bảo Châu,2,Ngô Việt Hải,1,Ngọc Huyền,2,Nguyễn Anh Tuyến,1,Nguyễn Bá Đang,1,Nguyễn Đình Thi,1,Nguyễn Đức Tấn,1,Nguyễn Đức Thắng,1,Nguyễn Duy Khương,1,Nguyễn Duy Tùng,1,Nguyễn Hữu Điển,3,Nguyễn Mình Hà,1,Nguyễn Minh Tuấn,8,Nguyễn Phan Tài Vương,1,Nguyễn Phú Khánh,1,Nguyễn Phúc Tăng,1,Nguyễn Quản Bá Hồng,1,Nguyễn Quang Sơn,1,Nguyễn Tài Chung,5,Nguyễn Tăng Vũ,1,Nguyễn Tất Thu,1,Nguyễn Thúc Vũ Hoàng,1,Nguyễn Trung Tuấn,8,Nguyễn Tuấn Anh,2,Nguyễn Văn Huyện,3,Nguyễn Văn Mậu,25,Nguyễn Văn Nho,1,Nguyễn Văn Quý,2,Nguyễn Văn Thông,1,Nguyễn Việt Anh,1,Nguyễn Vũ Lương,2,Nhật Bản,4,Nhóm $\LaTeX$,4,Nhóm Toán,1,Ninh Bình,43,Ninh Thuận,15,Nội Suy Lagrange,2,Nội Suy Newton,1,Nordic,19,Olympiad Corner,1,Olympiad Preliminary,2,Olympic 10,99,Olympic 10/3,5,Olympic 11,92,Olympic 12,30,Olympic 24/3,7,Olympic 27/4,20,Olympic 30/4,69,Olympic KHTN,6,Olympic Sinh Viên,73,Olympic Tháng 4,12,Olympic Toán,304,Olympic Toán Sơ Cấp,3,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,Philippines,8,Phú Thọ,30,Phú Yên,29,Phùng Hồ Hải,1,Phương Trình Hàm,11,Phương Trình Pythagoras,1,Pi,1,Polish,32,Problems,1,PT-HPT,14,PTNK,45,Putnam,25,Quảng Bình,44,Quảng Nam,32,Quảng Ngãi,34,Quảng Ninh,43,Quảng Trị,27,Quỹ Tích,1,Riemann,1,RMM,12,RMO,24,Romania,36,Romanian Mathematical,1,Russia,1,Sách Thường Thức Toán,7,Sách Toán,69,Sách Toán Cao Học,1,Sách Toán THCS,7,Saudi Arabia,7,Scholze,1,Serbia,17,Sharygin,24,Shortlists,56,Simon Singh,1,Singapore,1,Số Học - Tổ Hợp,27,Sóc Trăng,28,Sơn La,12,Spain,8,Star Education,5,Stars of Mathematics,11,Swinnerton-Dyer,1,Talent Search,1,Tăng Hải Tuân,2,Tạp Chí,14,Tập San,6,Tây Ban Nha,1,Tây Ninh,29,Thạch Hà,1,Thái Bình,39,Thái Nguyên,49,Thái Vân,2,Thanh Hóa,62,THCS,2,Thổ Nhĩ Kỳ,5,Thomas J. Mildorf,1,THPT Chuyên Lê Quý Đôn,1,THPTQG,15,THTT,7,Thừa Thiên Huế,36,Tiền Giang,19,Tin Tức Toán Học,1,Titu Andreescu,2,Toán 12,7,Toán Cao Cấp,3,Toán Chuyên,2,Toán Rời Rạc,5,Toán Tuổi Thơ,3,Tôn Ngọc Minh Quân,2,TOT,1,TPHCM,126,Trà Vinh,6,Trắc Nghiệm,1,Trắc Nghiệm Toán,2,Trại Hè,34,Trại Hè Hùng Vương,25,Trại Hè Phương Nam,5,Trần Đăng Phúc,1,Trần Minh Hiền,2,Trần Nam Dũng,9,Trần Phương,1,Trần Quang Hùng,1,Trần Quốc Anh,2,Trần Quốc Luật,1,Trần Quốc Nghĩa,1,Trần Tiến Tự,1,Trịnh Đào Chiến,2,Trung Quốc,14,Trường Đông,19,Trường Hè,7,Trường Thu,1,Trường Xuân,2,TST,56,Tuyên Quang,6,Tuyển Sinh,3,Tuyển Sinh 10,680,Tuyển Tập,44,Tuymaada,4,Undergraduate,67,USA,44,USAJMO,10,USATST,7,Uzbekistan,1,Vasile Cîrtoaje,4,Vật Lý,1,Viện Toán Học,2,Vietnam,4,Viktor Prasolov,1,VIMF,1,Vinh,27,Vĩnh Long,21,Vĩnh Phúc,64,Virginia Tech,1,VLTT,1,VMEO,4,VMF,12,VMO,47,VNTST,22,Võ Anh Khoa,1,Võ Quốc Bá Cẩn,26,Võ Thành Văn,1,Vojtěch Jarník,6,Vũ Hữu Bình,7,Vương Trung Dũng,1,WFNMC Journal,1,Wiles,1,Yên Bái,20,Yên Định,1,Yên Thành,1,Zhautykov,11,Zhou Yuan Zhe,1,
MOlympiad: [Shortlists] International Mathematical Olympiad 2008
[Shortlists] International Mathematical Olympiad 2008
Loaded All Posts Not found any posts VIEW ALL Readmore Reply Cancel reply Delete By Home PAGES POSTS View All RECOMMENDED FOR YOU LABEL ARCHIVE SEARCH ALL POSTS Not found any post match with your request Back Home Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sun Mon Tue Wed Thu Fri Sat January February March April May June July August September October November December Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec just now 1 minute ago $$1$$ minutes ago 1 hour ago $$1$$ hours ago Yesterday $$1$$ days ago $$1$$ weeks ago more than 5 weeks ago Followers Follow THIS PREMIUM CONTENT IS LOCKED Please share to unlock Copy All Code Select All Code All codes were copied to your clipboard Can not copy the codes / texts, please press [CTRL]+[C] (or CMD+C with Mac) to copy