Navvye/Olympiad-Problems-CSV huggingface dataset:
2009 IMO Shortlist
Consider $2009$ cards, each having one gold side and one black side, lying on parallel on a long table. Initially all cards show their gold sides. Two player, standing by the same long side of the table, play a game with alternating moves. Each move consists of choosing a block of $50$ consecutive cards, the leftmost of which is showing gold, and turning them all over, so those which showed gold now show black and vice versa. The last player who can make a legal move wins.

(a) Does the game necessarily end?

(b) Does there exist a winning strategy for the starting player?



Proposed by Michael Albert, Richard Guy, New Zealand
2009 IMO Shortlist
Let $n$ be a positive integer.  Given a sequence $\varepsilon_1$, $\dots$, $\varepsilon_{n - 1}$ with $\varepsilon_i = 0$ or $\varepsilon_i = 1$ for each $i = 1$, $\dots$, $n - 1$, the sequences $a_0$, $\dots$, $a_n$ and $b_0$, $\dots$, $b_n$ are constructed by the following rules: \[a_0 = b_0 = 1, \quad a_1 = b_1 = 7,\] \[\begin{array}{lll}

	a_{i+1} = 

	\begin{cases}

		2a_{i-1} + 3a_i, \\
		3a_{i-1} + a_i, 

	\end{cases} & 

        \begin{array}{l} 

                \text{if } \varepsilon_i = 0, \\  

                \text{if } \varepsilon_i = 1, \end{array} 

         & \text{for each } i = 1, \dots, n - 1, \\[15pt]

        b_{i+1}= 

        \begin{cases}

		2b_{i-1} + 3b_i, \\
		3b_{i-1} + b_i, 

	\end{cases} & 

        \begin{array}{l} 

                \text{if } \varepsilon_{n-i} = 0, \\  

                \text{if } \varepsilon_{n-i} = 1, \end{array} 

         & \text{for each } i = 1, \dots, n - 1.

	\end{array}\]  Prove that $a_n = b_n$.



Proposed by Ilya Bogdanov, Russia
2009 IMO Shortlist
For any integer $n\geq 2$, we compute the integer $h(n)$ by applying the following procedure to its decimal representation. Let $r$ be the rightmost digit of $n$.



If $r=0$, then the decimal representation of $h(n)$ results from the decimal representation of $n$ by removing this rightmost digit $0$.

If $1\leq r \leq 9$ we split the decimal representation of $n$ into a maximal right part $R$ that solely consists of digits not less than $r$ and into a left part $L$ that either is empty or ends with a digit strictly smaller than $r$. Then the decimal representation of $h(n)$ consists of the decimal representation of $L$, followed by two copies of the decimal representation of $R-1$. For instance, for the number $17,151,345,543$, we will have $L=17,151$, $R=345,543$ and $h(n)=17,151,345,542,345,542$.



Prove that, starting with an arbitrary integer $n\geq 2$, iterated application of $h$ produces the integer $1$ after finitely many steps.



Proposed by Gerhard Woeginger, Austria
2009 IMO Shortlist
Let $k$ be a positive integer. Show that if there exists a sequence $a_0,a_1,\ldots$ of integers satisfying the condition $$a_n=\frac{a_{n-1}+n^k}{n}\text{ for all } n\geq 1,$$then $k-2$ is divisible by $3$.



Proposed by Okan Tekman, Turkey
2009 IMO Shortlist
Let $a$ and $b$ be distinct integers greater than $1$. Prove that there exists a positive integer $n$ such that $(a^n-1)(b^n-1)$ is not a perfect square.



Proposed by Mongolia
2008 IMO Shortlist
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$.



Proposed by Gerhard Woeginger, Austria
2008 IMO Shortlist
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$.



Proposed by Žymantas Darbėnas (Zymantas Darbenas), Lithuania
2008 IMO Shortlist
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.



Proposed by Gerhard Woeginger, Netherlands
2008 IMO Shortlist
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.



Proposed by Duskan Dukic, Serbia
2007 IMO Shortlist
Let $ c > 2,$ and let $ a(1), a(2), \ldots$ be a sequence of nonnegative real numbers such that

$$ a(m + n) \leq 2 \cdot a(m) + 2 \cdot a(n) \text{ for all } m,n \geq 1,

$$

and $ a\left(2^k \right) \leq \frac {1}{(k + 1)^c} \text{ for all } k \geq 0.$ Prove that the sequence $ a(n)$ is bounded.



Author: Vjekoslav Kovač, Croatia
2007 IMO Shortlist
In the Cartesian coordinate plane define the strips $ S_n = \{(x,y)|n\le x < n + 1\}$, $ n\in\mathbb{Z}$ and color each strip black or white. Prove that any rectangle which is not a square can be placed in the plane so that its vertices have the same color.



IMO Shortlist 2007 Problem C5 as it appears in the official booklet:

In the Cartesian coordinate plane define the strips $ S_n = \{(x,y)|n\le x < n + 1\}$ for every integer $ n.$ Assume each strip $ S_n$ is colored either red or blue, and let $ a$ and $ b$ be two distinct positive integers. Prove that there exists a rectangle with side length $ a$ and $ b$ such that its vertices have the same color.

(Edited by Orlando Döhring)



Author: Radu Gologan and Dan Schwarz, Romania
2007 IMO Shortlist
Let $ \alpha < \frac {3 - \sqrt {5}}{2}$ be a positive real number. Prove that there exist positive integers $ n$ and $ p > \alpha \cdot 2^n$ for which one can select $ 2 \cdot p$ pairwise distinct subsets $ S_1, \ldots, S_p, T_1, \ldots, T_p$ of the set $ \{1,2, \ldots, n\}$ such that $ S_i \cap T_j \neq \emptyset$ for all $ 1 \leq i,j \leq p$



Author: Gerhard Wöginger, Austria
2007 IMO Shortlist
Given an acute triangle $ ABC$ with $ \angle B > \angle C$. Point $ I$ is the incenter, and $ R$ the circumradius. Point $ D$ is the foot of the altitude from vertex $ A$. Point $ K$ lies on line $ AD$ such that $ AK = 2R$, and $ D$ separates $ A$ and $ K$. Lines $ DI$ and $ KI$ meet sides $ AC$ and $ BC$ at $ E,F$ respectively. Let $ IE = IF$.



Prove that $ \angle B\leq 3\angle C$.



Author: Davoud Vakili, Iran
2007 IMO Shortlist
Let $ k$ be a positive integer. Prove that the number $ (4 \cdot k^2 - 1)^2$ has a positive divisor of the form $ 8kn - 1$ if and only if $ k$ is even.



Actual IMO 2007 Problem, posed as question 5 in the contest, which was used as a lemma in the official solutions for problem N6 as shown above.



Author: Kevin Buzzard and Edward Crane, United Kingdom
2007 IMO Shortlist
For a prime $ p$ and a given integer $ n$ let $ \nu_p(n)$ denote the exponent of $ p$ in the prime factorisation of $ n!$. Given $ d \in \mathbb{N}$ and $ \{p_1,p_2,\ldots,p_k\}$ a set of $ k$ primes, show that there are infinitely many positive integers $ n$ such that $ d\mid \nu_{p_i}(n)$ for all $ 1 \leq i \leq k$.



Author: Tejaswi Navilarekkallu, India
2006 IMO Shortlist
The sequence $c_{0}, c_{1}, . . . , c_{n}, . . .$ is defined by $c_{0}= 1, c_{1}= 0$, and $c_{n+2}= c_{n+1}+c_{n}$ for $n \geq 0$. Consider the set $S$ of ordered pairs $(x, y)$ for which there is a finite set $J$ of positive integers such that $x=\textstyle\sum_{j \in J}{c_{j}}$, $y=\textstyle\sum_{j \in J}{c_{j-1}}$. Prove that there exist real numbers $\alpha$, $\beta$, and $M$ with the following property: An ordered pair of nonnegative integers $(x, y)$ satisfies the inequality $$m < \alpha x+\beta y < M$$ if and only if $(x, y) \in S$.



Remark: A sum over the elements of the empty set is assumed to be $0$.
2006 IMO Shortlist
If $a,b,c$ are the sides of a triangle, prove that

$$\frac{\sqrt{b+c-a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}+\frac{\sqrt{c+a-b}}{\sqrt{c}+\sqrt{a}-\sqrt{b}}+\frac{\sqrt{a+b-c}}{\sqrt{a}+\sqrt{b}-\sqrt{c}}\leq 3 $$



Proposed by Hojoo Lee, Korea
2006 IMO Shortlist
A holey triangle is an upward equilateral triangle of side length $n$ with $n$ upward unit triangular holes cut out. A diamond is a $60^\circ-120^\circ$ unit rhombus.

Prove that a holey triangle $T$ can be tiled with diamonds if and only if the following condition holds: Every upward equilateral triangle of side length $k$ in $T$ contains at most $k$ holes, for $1\leq k\leq n$.



Proposed by Federico Ardila, Colombia
2006 IMO Shortlist
Consider a convex polyhedron without parallel edges and without an edge parallel to any face other than the two faces adjacent to it. Call a pair of points of the polyhedron antipodal if there exist two parallel planes passing through these points and such that the polyhedron is contained between these planes. Let $A$ be the number of antipodal pairs of vertices, and let $B$ be the number of antipodal pairs of midpoint edges. Determine the difference $A-B$ in terms of the numbers of vertices, edges, and faces.



Proposed by Kei Irei, Japan
2006 IMO Shortlist
In triangle $ABC$, let $J$ be the center of the excircle tangent to side $BC$ at $A_{1}$ and to the extensions of the sides $AC$ and $AB$ at $B_{1}$ and $C_{1}$ respectively. Suppose that the lines $A_{1}B_{1}$ and $AB$ are perpendicular and intersect at $D$. Let $E$ be the foot of the perpendicular from $C_{1}$ to line $DJ$. Determine the angles $\angle{BEA_{1}}$ and $\angle{AEB_{1}}$.



Proposed by Dimitris Kontogiannis, Greece
2006 IMO Shortlist
In a triangle $ ABC$, let $ M_{a}$, $ M_{b}$, $ M_{c}$ be the midpoints of the sides $ BC$, $ CA$, $ AB$, respectively, and $ T_{a}$, $ T_{b}$, $ T_{c}$ be the midpoints of the arcs $ BC$, $ CA$, $ AB$ of the circumcircle of $ ABC$, not containing the vertices $ A$, $ B$, $ C$, respectively. For $ i \in \left\{a, b, c\right\}$, let $ w_{i}$ be the circle with $ M_{i}T_{i}$ as diameter. Let $ p_{i}$ be the common external common tangent to the circles $ w_{j}$ and $ w_{k}$ (for all $ \left\{i, j, k\right\}= \left\{a, b, c\right\}$) such that $ w_{i}$ lies on the opposite side of $ p_{i}$ than $ w_{j}$ and $ w_{k}$ do.

Prove that the lines $ p_{a}$, $ p_{b}$, $ p_{c}$ form a triangle similar to $ ABC$ and find the ratio of similitude.



Proposed by Tomas Jurik, Slovakia
2006 IMO Shortlist
Let $ABCD$ be a convex quadrilateral. A circle passing through the points $A$ and $D$ and a circle passing through the points $B$ and $C$ are externally tangent at a point $P$ inside the quadrilateral. Suppose that $$\angle{PAB}+\angle{PDC}\leq  90^\circ\qquad\text{and}\qquad\angle{PBA}+\angle{PCD}\leq  90^\circ.$$ Prove that $AB+CD \geq  BC+AD$.



Proposed by Waldemar Pompe, Poland
2005 IMO Shortlist
This ISL 2005 problem has not been used in any TST I know. A pity, since it is a nice problem, but in its shortlist formulation, it is absolutely incomprehensible. Here is a mathematical restatement of the problem:



Let $k$ be a nonnegative integer.



A forest consists of rooted (i. e. oriented) trees. Each vertex of the forest is either a leaf or has two successors. A vertex $v$ is called an extended successor of a vertex $u$ if there is a chain of vertices $u_{0}=u$, $u_{1}$, $u_{2}$, ..., $u_{t-1}$, $u_{t}=v$ with $t>0$ such that the vertex $u_{i+1}$ is a successor of the vertex $u_{i}$ for every integer $i$ with $0\leq i\leq t-1$. A vertex is called dynastic if it has two successors and each of these successors has at least $k$ extended successors.



Prove that if the forest has $n$ vertices, then there are at most $\frac{n}{k+2}$ dynastic vertices.
2005 IMO Shortlist
Let $ABC$ be a triangle, and $M$ the midpoint of its side $BC$. Let $\gamma$ be the incircle of triangle $ABC$. The median $AM$ of triangle $ABC$ intersects the incircle $\gamma$ at two points $K$ and $L$. Let the lines passing through $K$ and $L$, parallel to $BC$, intersect the incircle $\gamma$ again in two points $X$ and $Y$. Let the lines $AX$ and $AY$ intersect $BC$ again at the points $P$ and $Q$. Prove that $BP = CQ$.
2005 IMO Shortlist
Let $P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots+a_{0}$, where $a_{0},\ldots,a_{n}$ are integers, $a_{n}>0$, $n\geq 2$. Prove that there exists a positive integer $m$ such that $P(m!)$ is a composite number.
2004 IMO Shortlist
Let $A_1A_2A_3\ldots A_n$ be a regular $n$-gon. Let $B_1$ and $B_{n-1}$ be the midpoints of its sides $A_1A_2$ and $A_{n-1}A_n$. Also, for every $i\in\left\{2,3,4,\ldots ,n-2\right\}$. Let $S$ be the point of intersection of the lines $A_1A_{i+1}$ and $A_nA_i$, and let $B_i$ be the point of intersection of the angle bisector bisector of the angle $\measuredangle A_iSA_{i+1}$ with the segment $A_iA_{i+1}$.



Prove that $\sum_{i=1}^{n-1} \measuredangle A_1B_iA_n=180^{\circ}$.



Proposed by Dusan Dukic, Serbia and Montenegro
2004 IMO Shortlist
Let $P$ be a convex polygon. Prove that there exists a convex hexagon that is contained in $P$ and whose area is at least $\frac34$ of the area of the polygon $P$.



Alternative version. Let $P$ be a convex polygon with $n\geq 6$ vertices. Prove that there exists a convex hexagon with



a) vertices on the sides of the polygon (or)

b) vertices among the vertices of the polygon



such that the area of the hexagon is at least $\frac{3}{4}$ of the area of the polygon.



Proposed by Ben Green and Edward Crane, United Kingdom
2004 IMO Shortlist
Given a cyclic quadrilateral $ABCD$, let $M$ be the midpoint of the side $CD$, and let $N$ be a point on the circumcircle of triangle $ABM$. Assume that the point $N$ is different from the point $M$ and satisfies $\frac{AN}{BN}=\frac{AM}{BM}$. Prove that the points $E$, $F$, $N$ are collinear, where $E=AC\cap BD$ and $F=BC\cap DA$.



Proposed by Dusan Dukic, Serbia and Montenegro
2004 IMO Shortlist
The function $f$ from the set $\mathbb{N}$ of positive integers into itself is defined by the equality $$f(n)=\sum_{k=1}^{n} \gcd(k,n),\qquad n\in \mathbb{N}.$$

a) Prove that $f(mn)=f(m)f(n)$ for every two relatively prime ${m,n\in\mathbb{N}}$.



b) Prove that for each $a\in\mathbb{N}$ the equation $f(x)=ax$ has a solution.



c) Find all ${a\in\mathbb{N}}$ such that the equation $f(x)=ax$ has a unique solution.
2004 IMO Shortlist
Given an integer ${n>1}$, denote by $P_{n}$ the product of all positive integers $x$ less than $n$ and such that $n$ divides ${x^2-1}$. For each ${n>1}$, find the remainder of $P_{n}$ on division by $n$.



Proposed by John Murray, Ireland
2004 IMO Shortlist
Let $p$ be an odd prime and $n$ a positive integer. In the coordinate plane, eight distinct points with integer coordinates lie on a circle with diameter of length $p^{n}$. Prove that there exists a triangle with vertices at three of the given points such that the squares of its side lengths are integers divisible by $p^{n+1}$.



Proposed by Alexander Ivanov, Bulgaria
2004 IMO Shortlist
Does there exist a function $s\colon \mathbb{Q} \rightarrow \{-1,1\}$ such that if $x$ and $y$ are distinct rational numbers satisfying ${xy=1}$ or ${x+y\in \{0,1\}}$, then ${s(x)s(y)=-1}$? Justify your answer.



Proposed by Dan Brown, Canada
2004 IMO Shortlist
Find all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying the equation $$

	f(x^2+y^2+2f(xy)) = (f(x+y))^2.

$$ for all $x,y \in \mathbb{R}$.
2004 IMO Shortlist
Let ${a_1,a_2,\dots,a_n}$ be positive real numbers, ${n>1}$. Denote by $g_n$ their geometric mean, and by $A_1,A_2,\dots,A_n$ the sequence of arithmetic means defined by $$ A_k=\frac{a_1+a_2+\cdots+a_k}{k},\qquad k=1,2,\dots,n. $$  Let $G_n$ be the geometric mean of $A_1,A_2,\dots,A_n$. Prove the inequality $$

n \root n\of{\frac{G_n}{A_n}}+ \frac{g_n}{G_n}\le n+1 $$ and establish the cases of equality.



Proposed by Finbarr Holland, Ireland
2004 IMO Shortlist
The following operation is allowed on a finite graph: Choose an arbitrary cycle of length 4 (if there is any), choose an arbitrary edge in that cycle, and delete it from the graph. For a fixed integer ${n\ge 4}$, find the least number of edges of a graph that can be obtained by repeated applications of this operation from the complete graph on $n$ vertices (where each pair of vertices are joined by an edge).



Proposed by Norman Do, Australia
2004 IMO Shortlist
$A$ and $B$ play a game, given an integer $N$, $A$ writes down $1$ first, then every player sees the last number written and if it is $n$ then in his turn he writes $n+1$ or $2n$, but his number cannot be bigger than $N$. The player who writes $N$ wins. For which values of $N$ does $B$ win?



Proposed by A. Slinko & S. Marshall, New Zealand
2004 IMO Shortlist
For a finite graph $G$, let $f(G)$ be the number of triangles and $g(G)$ the number of tetrahedra formed by edges of $G$. Find the least constant $c$ such that $$g(G)^3\le c\cdot f(G)^4$$ for every graph $G$.



Proposed by Marcin Kuczma, Poland
2003 IMO Shortlist
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}.

 $$
2003 IMO Shortlist
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 $ABC$.



Proposed by Hojoo Lee
2003 IMO Shortlist
The sequence $a_0$, $a_1$, $a_2,$ $\ldots$ is defined as follows: $$a_0=2, \qquad a_{k+1}=2a_k^2-1 \quad\text{for }k \geq 0.$$Prove that if an odd prime $p$ divides $a_n$, then $2^{n+3}$ divides $p^2-1$.



commentHi guys ,



Here is a nice problem:



Let be given a sequence $a_n$ such that $a_0=2$ and $a_{n+1}=2a_n^2-1$ . Show that if $p$ is an odd prime such that $p|a_n$ then we have $p^2\equiv 1\pmod{2^{n+3}}$



Here are some futher question proposed by me :Prove or disprove that :

1) $gcd(n,a_n)=1$

2) for every odd prime number $p$ we have $a_m\equiv \pm 1\pmod{p}$ where $m=\frac{p^2-1}{2^k}$ where $k=1$ or $2$



Thanks kiu si u





Edited by Orl.
2003 IMO Shortlist
Let $p$ be a prime number and let $A$ be a set of positive integers that satisfies the following conditions:



(i) the set of prime divisors of the elements in $A$ consists of $p-1$ elements;



(ii) 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 $A$ ?
2003 IMO Shortlist
Find all nondecreasing functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that

(i) $f(0) = 0, f(1) = 1;$

(ii) $f(a) + f(b) = f(a)f(b) + f(a + b - ab)$ for all real numbers $a, b$ such that $a < 1 < b$.



Proposed by A. Di Pisquale & D. Matthews, Australia
2003 IMO Shortlist
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 $1\le x<y$.



Proposed by Hojoo Lee, Korea
2003 IMO Shortlist
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). $$



commentEdited by Orl.



Proposed by Reid Barton, USA
2003 IMO Shortlist
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 $D_i$.
2002 IMO Shortlist
Let $ABC$ be a triangle for which there exists an interior point $F$ such that $\angle AFB=\angle BFC=\angle CFA$. Let the lines $BF$ and $CF$ meet the sides $AC$ and $AB$ at $D$ and $E$ respectively. Prove that $$ AB+AC\geq4DE. $$
2002 IMO Shortlist
Circles $S_1$ and $S_2$ intersect at points $P$ and $Q$. Distinct points $A_1$ and $B_1$ (not at $P$ or $Q$) are selected on $S_1$. The lines $A_1P$ and $B_1P$ meet $S_2$ again at $A_2$ and $B_2$ respectively, and the lines $A_1B_1$ and $A_2B_2$ meet at $C$.  Prove that, as $A_1$ and $B_1$ vary, the circumcentres of triangles $A_1A_2C$ all lie on one fixed circle.
2002 IMO Shortlist
For any set $S$ of five points in the plane, no three of which are collinear, let $M(S)$ and $m(S)$ denote the greatest and smallest areas, respectively, of triangles determined by three points from $S$.  What is the minimum possible value of $M(S)/m(S)$ ?
2002 IMO Shortlist
The incircle $ \Omega$ of the acute-angled triangle $ ABC$ is tangent to its side $ BC$ at a point $ K$. Let $ AD$ be an altitude of triangle $ ABC$, and let $ M$ be the midpoint of the segment $ AD$. If $ N$ is the common point of the circle $ \Omega$ and the line $ KM$ (distinct from $ K$), then prove that the incircle $ \Omega$ and the circumcircle of triangle $ BCN$ are tangent to each other at the point $ N$.
2002 IMO Shortlist
Let two circles $S_{1}$ and $S_{2}$ meet at the points $A$ and $B$. A line through $A$ meets $S_{1}$ again at $C$ and $S_{2}$ again at $D$. Let $M$, $N$, $K$ be three points on the line segments $CD$, $BC$, $BD$ respectively, with $MN$ parallel to $BD$ and $MK$ parallel to $BC$. Let $E$ and $F$ be points on those arcs $BC$ of $S_{1}$ and $BD$ of $S_{2}$ respectively that do not contain $A$. Given that $EN$ is perpendicular to $BC$ and $FK$ is perpendicular to $BD$ prove that $\angle EMF=90^{\circ}$.
2002 IMO Shortlist
What is the smallest positive integer $t$ such that there exist integers $x_1,x_2,\ldots,x_t$ with  $$x^3_1+x^3_2+\,\ldots\,+x^3_t=2002^{2002}\,?$$
2002 IMO Shortlist
Is there a positive integer $m$ such that the equation $$ {1\over a}+{1\over b}+{1\over c}+{1\over abc}={m\over a+b+c}  $$ has infinitely many solutions in positive integers $a,b,c$?
2002 IMO Shortlist
Let $m,n\geq2$ be positive integers, and let $a_1,a_2,\ldots ,a_n$ be integers, none of which is a multiple of $m^{n-1}$.  Show that there exist integers $e_1,e_2,\ldots,e_n$, not all zero, with $\left|{\,e}_i\,\right|<m$ for all $i$, such that $e_1a_1+e_2a_2+\,\ldots\,+e_na_n$ is a multiple of $m^n$.
2002 IMO Shortlist
Let $P$ be a cubic polynomial given by $P(x)=ax^3+bx^2+cx+d$, where $a,b,c,d$ are integers and $a\ne0$. Suppose that $xP(x)=yP(y)$ for infinitely many pairs $x,y$ of integers with $x\ne y$. Prove that the equation $P(x)=0$ has an integer root.
2002 IMO Shortlist
Let $n$ be a positive integer that is not a perfect cube. Define real numbers $a,b,c$ by



$$a=\root3\of n\kern1.5pt,\qquad b={1\over a-[a]}\kern1pt,\qquad c={1\over b-[b]}\kern1.5pt,$$



where $[x]$ denotes the integer part of $x$. Prove that there are infinitely many such integers $n$ with the property that there exist integers $r,s,t$, not all zero, such that $ra+sb+tc=0$.
2002 IMO Shortlist
Let $A$ be a non-empty set of positive integers. Suppose that there are positive integers $b_1,\ldots b_n$ and $c_1,\ldots,c_n$ such that



- for each $i$ the set $b_iA+c_i=\left\{b_ia+c_i\colon a\in A\right\}$ is a subset of $A$, and



- the sets $b_iA+c_i$ and $b_jA+c_j$ are disjoint whenever $i\ne j$



Prove that $${1\over b_1}+\,\ldots\,+{1\over b_n}\leq1.$$