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[Shortlists] International Mathematical Olympiad 2003

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Geometry

  1. Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.
  2. Given three fixed pairwisely distinct points $A$, $B$, $C$ lying on one straight line in this order. Let $G$ be a circle passing through $A$ and $C$ whose center does not lie on the line $AC$. The tangents to $G$ at $A$ and $C$ intersect each other at a point $P$. The segment $PB$ meets the circle $G$ at $Q$. Show that the point of intersection of the angle bisector of the angle $AQC$ with the line $AC$ does not depend on the choice of the circle $G$.
  3. Let $ABC$ be a triangle, and $P$ a point in the interior of this triangle. Let $D$, $E$, $F$ be the feet of the perpendiculars from the point $P$ to the lines $BC$, $CA$, $AB$, respectively. Assume that $AP^{2}+PD^{2}$ $=BP^{2}+PE^{2}$ $=CP^{2}+PF^{2}$. Furthermore, let $I_{a}$, $I_{b}$, $I_{c}$ be the excenters of triangle $ABC$. Show that the point $P$ is the circumcenter of triangle $I_{a}I_{b}I_{c}$.
  4. Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, $\Gamma_4$ be distinct circles such that $\Gamma_1$, $\Gamma_3$ are externally tangent at $P$, and $\Gamma_2$, $\Gamma_4$ are externally tangent at the same point $P$. Suppose that $\Gamma_1$ and $\Gamma_2$; $\Gamma_2$ and $\Gamma_3$; $\Gamma_3$ and $\Gamma_4$; $\Gamma_4$ and $\Gamma_1$ meet at $A$, $B$, $C$, $D$, respectively, and that all these points are different from $P$. Prove that \[ \frac{AB\cdot BC}{AD\cdot DC}=\frac{PB^2}{PD^2}. \]
  5. Let $ABC$ be an isosceles triangle with $AC=BC$, whose incentre is $I$. Let $P$ be a point on the circumcircle of the triangle $AIB$ lying inside the triangle $ABC$. The lines through $P$ parallel to $CA$ and $CB$ meet $AB$ at $D$ and $E$, respectively. The line through $P$ parallel to $AB$ meets $CA$ and $CB$ at $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ intersect on the circumcircle of the triangle $ABC$.
  6. Each pair of opposite sides of a convex hexagon has the following property: the distance between their midpoints is equal to $\dfrac{\sqrt{3}}{2}$ times the sum of their lengths. Prove that all the angles of the hexagon are equal.
  7. Let $ABC$ be a triangle with semiperimeter $s$ and inradius $r$. The semicircles with diameters $BC$, $CA$, $AB$ are drawn on the outside of the triangle $ABC$. The circle tangent to all of these three semicircles has radius $t$. Prove that \[\frac{s}{2}<t\le\frac{s}{2}+\left(1-\frac{\sqrt{3}}{2}\right)r. \]

Number Theory

  1. Let $m$ be a fixed integer greater than $1$. The sequence $x_0$, $x_1$, $x_2$, $\ldots$ is defined as follows: $x_i= 2^i$ if $0 \leq i\leq m-1$, $x_i = \sum_{j=1}^{m}x_{i-j},$ if $i\geq m$. Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by $m$ .
  2. Each positive integer $a$ is subjected to the following procedure, yielding the number $d = d\left(a\right)$:
    a) The last digit of $a$ is moved to the first position. The resulting number is called $b$.
    b) The number $b$ is squared. The resulting number is called $c$.
    c) The first digit of $c$ is moved to the last position. The resulting number is called $d$. (All numbers are considered in the decimal system.) For instance, $a = 2003$ gives $b = 3200$, $c = 10240000$ and $d = 02400001$ $= 2400001$ $= d\left(2003\right)$.  Find all integers a such that $d\left( a\right) =a^2$.
  3. Determine all pairs of positive integers $(a,b)$ such that \[ \dfrac{a^2}{2ab^2-b^3+1} \] is a positive integer.
  4. Let $ b$ be an integer greater than $ 5$. For each positive integer $ n$, consider the number \[ x_n = \underbrace{11\cdots1}_{n - 1}\underbrace{22\cdots2}_{n}5, \] written in base $ b$. Prove that the following condition holds if and only if $ b = 10$: there exists a positive integer $ M$ such that for any integer $ n$ greater than $ M$, the number $ x_n$ is a perfect square.
  5. An integer $n$ is said to be good if $|n|$ is not the square of an integer. Determine all integers $m$ with the following property: $m$ can be represented, in infinitely many ways, as a sum of three distinct good integers whose product is the square of an odd integer.
  6. Let $p$ be a prime number. Prove that there exists a prime number $q$ such that for every integer $n$, the number $n^p-p$ is not divisible by $q$.
  7. The sequence $a_0$, $a_1$, $a_2,$ $\ldots$ is defined as follows: $$a_0=2, \quad a_{k+1}=2a_k^2-1 \quad$ for $k \geq 0.$$ Prove that if an odd prime $p$ divides $a_n$, then $2^{n+3}$ divides $p^2-1$.
  8. Let $p$ be a prime number and let $A$ be a set of positive integers that satisfies the following conditions: (1) the set of prime divisors of the elements in $A$ consists of $p-1$ elements; (2) for any nonempty subset of $A$, the product of its elements is not a perfect $p$-th power. What is the largest possible number of elements in $A$ ?

Algebra

  1. Let $a_{ij}$ (with the indices $i$ and $j$ from the set $\left\{1,\ 2,\ 3\right\}$) be real numbers such that $a_{ij}>0$ for $i = j$; $a_{ij}<0$ for $i\neq j$. Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers $$a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},$$ $$a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},$$ $$a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}$$ are either all negative, or all zero, or all positive.
  2. Find all nondecreasing functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that
    • $f(0) = 0, f(1) = 1;$
    • $f(a) + f(b) = f(a)f(b) + f(a + b - ab)$ for all real numbers $a, b$ such that $a < 1 < b$.
  3. Consider two monotonically decreasing sequences $ \left( a_k\right)$ and $ \left( b_k\right)$, where $ k \geq 1$, and $ a_k$ and $ b_k$ are positive real numbers for every k. Now, define the sequences $$ c_k = \min \left( a_k, b_k \right)$$ $$ A_k = a_1 + a_2 + ... + a_k$$ $$ B_k = b_1 + b_2 + ... + b_k$$ $$ C_k = c_1 + c_2 + ... + c_k$$ for all natural numbers k.
    a) Do there exist two monotonically decreasing sequences $ \left( a_k\right)$ and $ \left( b_k\right)$ of positive real numbers such that the sequences $ \left( A_k\right)$ and $ \left( B_k\right)$ are not bounded, while the sequence $ \left( C_k\right)$ is bounded?
    b) Does the answer to problem (a) change if we stipulate that the sequence $ \left( b_k\right)$ must be $ \displaystyle b_k = \frac {1}{k}$ for all k ?
  4. Let $n$ be a positive integer and let $x_1\le x_2\le\cdots\le x_n$ be real numbers. Prove that \[ \left(\sum_{i,j=1}^{n}|x_i-x_j|\right)^2\le\frac{2(n^2-1)}{3}\sum_{i,j=1}^{n}(x_i-x_j)^2. \] Show that the equality holds if and only if $x_1, \ldots, x_n$ is an arithmetic sequence.
  5. Let $\mathbb{R}^+$ be the set of all positive real numbers. Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ that satisfy the following conditions: - $f(xyz)+f(x)+f(y)+f(z)=f(\sqrt{xy})f(\sqrt{yz})f(\sqrt{zx})$ for all $x,y,z\in\mathbb{R}^+$; - $f(x)<f(y)$ for all $1\le x<y$.
  6. Let $n$ be a positive integer and let $(x_1,\ldots,x_n)$, $(y_1,\ldots,y_n)$ be two sequences of positive real numbers. Suppose $(z_2,\ldots,z_{2n})$ is a sequence of positive real numbers such that $z_{i+j}^2 \geq x_iy_j$ for all $1\le i,j \leq n$. Let $M=\max\{z_2,\ldots,z_{2n}\}$. Prove that \[ \left( \frac{M+z_2+\dots+z_{2n}}{2n} \right)^2 \ge \left( \frac{x_1+\dots+x_n}{n} \right) \left( \frac{y_1+\dots+y_n}{n} \right). \]

Combinatorics

  1. Let $A$ be a $101$-element subset of the set $S=\{1,2,\ldots,1000000\}$. Prove that there exist numbers $t_1$, $t_2, \ldots, t_{100}$ in $S$ such that the sets \[ A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100 \] are pairwise disjoint.
  2. Let $D_1$, $D_2$, ..., $D_n$ be closed discs in the plane. (A closed disc is the region limited by a circle, taken jointly with this circle.) Suppose that every point in the plane is contained in at most $2003$ discs $D_i$. Prove that there exists a disc $D_k$ which intersects at most $7\cdot 2003 - 1 = 14020$ other discs $D_i$.
  3. Let $n \geq 5$ be an integer. Find the maximal integer $k$ such that there exists a polygon with $n$ vertices (convex or not, but not self-intersecting!) having $k$ internal $90^{\circ}$ angles.
  4. Given $n$ real numbers $x_1$, $x_2$, ..., $x_n$, and $n$ further real numbers $y_1$, $y_2$, ..., $y_n$. The entries $a_{ij}$ (with $1\leq i,\;j\leq n$) of an $n\times n$ matrix $A$ are defined as follows: $$a_{ij}=\left\{ \begin{array}{c} 1\text{ if }x_{i}+y_{j}\geq 0; \\ 0\text{ if }x_{i}+y_{j}<0. \end{array} \right.$$ Further, let $B$ be an $n\times n$ matrix whose elements are numbers from the set $\left\{0;\ 1\right\}$ satisfying the following condition: The sum of all elements of each row of $B$ equals the sum of all elements of the corresponding row of $A$; the sum of all elements of each column of $B$ equals the sum of all elements of the corresponding column of $A$. Show that in this case, $A = B$.
  5. Regard a plane with a Cartesian coordinate system; for each point with integer coordinates, draw a circular disk centered at this point and having the radius $\frac{1}{1000}$.
    a) Prove the existence of an equilateral triangle whose vertices lie in the interior of different disks;
    b) Show that every equilateral triangle whose vertices lie in the interior of different disks has a sidelength > 96.
  6. Let $f(k)$ be the number of all non-negative integers $n$ satisfying the following conditions
    • The integer $n$ has exactly $k$ digits in the decimal representation (where the first digit is not necessarily non-zero!), i. e. we have $0 \leq n <10^k$.
    • These $k$ digits of n can be permuted in such a way that the resulting number is divisible by $11$.
    Show that for any positive integer number $m,$ we have $$f\left(2m\right) = 10 f\left(2m - 1\right).$$

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Name

Abel,5,Albania,2,AMM,2,Amsterdam,4,An Giang,46,Andrew Wiles,1,Anh,2,APMO,21,Austria (Áo),1,Ba Lan,1,Bà Rịa Vũng Tàu,77,Bắc Bộ,2,Bắc Giang,62,Bắc Kạn,4,Bạc Liêu,18,Bắc Ninh,54,Bắc Trung Bộ,3,Bài Toán Hay,5,Balkan,41,Baltic Way,32,BAMO,1,Bất Đẳng Thức,69,Bến Tre,72,Benelux,16,Bình Định,65,Bình Dương,38,Bình Phước,52,Bình Thuận,42,Birch,1,BMO,41,Booklet,12,Bosnia Herzegovina,3,BoxMath,3,Brazil,2,British,16,Bùi Đắc Hiên,1,Bùi Thị Thiện Mỹ,1,Bùi Văn Tuyên,1,Bùi Xuân Diệu,1,Bulgaria,6,Buôn Ma Thuột,2,BxMO,15,Cà Mau,22,Cần Thơ,28,Canada,40,Cao Bằng,12,Cao Quang Minh,1,Câu Chuyện Toán Học,43,Caucasus,3,CGMO,11,China - Trung Quốc,25,Chọn Đội Tuyển,528,Chu Tuấn Anh,1,Chuyên Đề,125,Chuyên SPHCM,7,Chuyên SPHN,30,Chuyên Trần Hưng Đạo,3,Collection,8,College Mathematic,1,Concours,1,Cono Sur,1,Contest,675,Correspondence,1,Cosmin Poahata,1,Crux,2,Czech-Polish-Slovak,28,Đà Nẵng,50,Đa Thức,2,Đại Số,20,Đắk Lắk,76,Đắk Nông,15,Danube,7,Đào Thái Hiệp,1,ĐBSCL,2,Đề Thi,1,Đề Thi HSG,2262,Đề Thi JMO,1,DHBB,30,Điện Biên,15,Định Lý,1,Định Lý Beaty,1,Đỗ Hữu Đức Thịnh,1,Do Thái,3,Doãn Quang Tiến,5,Đoàn Quỳnh,1,Đoàn Văn Trung,1,Đồng Nai,64,Đồng Tháp,64,Du Hiền Vinh,1,Đức,1,Dương Quỳnh Châu,1,Dương Tú,1,Duyên Hải Bắc Bộ,30,E-Book,31,EGMO,30,ELMO,19,EMC,11,Epsilon,1,Estonian,5,Euler,1,Evan Chen,1,Fermat,3,Finland,4,Forum Of Geometry,2,Furstenberg,1,G. 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2011-2012,44,HSG 12 2012-2013,58,HSG 12 2013-2014,53,HSG 12 2014-2015,44,HSG 12 2015-2016,37,HSG 12 2016-2017,46,HSG 12 2017-2018,55,HSG 12 2018-2019,43,HSG 12 2019-2020,43,HSG 12 2020-2021,52,HSG 12 2021-2022,35,HSG 12 2022-2023,42,HSG 12 2023-2024,23,HSG 12 2023-2041,1,HSG 12 An Giang,8,HSG 12 Bà Rịa Vũng Tàu,13,HSG 12 Bắc Giang,18,HSG 12 Bạc Liêu,3,HSG 12 Bắc Ninh,13,HSG 12 Bến Tre,19,HSG 12 Bình Định,17,HSG 12 Bình Dương,8,HSG 12 Bình Phước,9,HSG 12 Bình Thuận,8,HSG 12 Cà Mau,7,HSG 12 Cần Thơ,7,HSG 12 Cao Bằng,5,HSG 12 Chuyên SPHN,11,HSG 12 Đà Nẵng,3,HSG 12 Đắk Lắk,21,HSG 12 Đắk Nông,1,HSG 12 Điện Biên,3,HSG 12 Đồng Nai,20,HSG 12 Đồng Tháp,18,HSG 12 Gia Lai,14,HSG 12 Hà Nam,5,HSG 12 Hà Nội,17,HSG 12 Hà Tĩnh,16,HSG 12 Hải Dương,16,HSG 12 Hải Phòng,20,HSG 12 Hậu Giang,4,HSG 12 Hòa Bình,10,HSG 12 Hưng Yên,10,HSG 12 Khánh Hòa,4,HSG 12 KHTN,26,HSG 12 Kiên Giang,12,HSG 12 Kon Tum,3,HSG 12 Lai Châu,4,HSG 12 Lâm Đồng,11,HSG 12 Lạng Sơn,8,HSG 12 Lào Cai,17,HSG 12 Long An,18,HSG 12 Nam 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MOlympiad.NET: [Shortlists] International Mathematical Olympiad 2003
[Shortlists] International Mathematical Olympiad 2003
MOlympiad.NET
https://www.molympiad.net/2017/08/imo-2002-shortlists_6.html
https://www.molympiad.net/
https://www.molympiad.net/
https://www.molympiad.net/2017/08/imo-2002-shortlists_6.html
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