Navvye/Olympiad-Problems-CSV huggingface dataset:
2006 IMO
Assign to each side $b$ of a convex polygon $P$ the maximum area of a triangle that has $b$ as a side and is contained in $P$. Show that the sum of the areas assigned to the sides of $P$ is at least twice the area of $P$.
2005 IMO
Six points are  chosen on the sides of an equilateral triangle $ABC$: $A_1$, $A_2$ on $BC$, $B_1$, $B_2$ on $CA$ and $C_1$, $C_2$ on $AB$, such that they are the vertices of a convex hexagon $A_1A_2B_1B_2C_1C_2$ with equal side lengths.



Prove that the lines $A_1B_2$, $B_1C_2$ and $C_1A_2$ are concurrent.



Bogdan Enescu, Romania
2005 IMO
Let $a_1,a_2,\ldots$ be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer $n$ the numbers $a_1,a_2,\ldots,a_n$ leave $n$ different remainders upon division by $n$.



Prove that every integer occurs exactly once in the sequence $a_1,a_2,\ldots$.
2005 IMO
Let $x,y,z$ be three positive reals such that $xyz\geq 1$. Prove that

$$ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . $$

Hojoo Lee, Korea
2005 IMO
Determine all positive integers relatively prime to all the terms of the infinite sequence $$ a_n=2^n+3^n+6^n -1,\ n\geq 1. $$
2005 IMO
Let $ABCD$ be a fixed convex quadrilateral with $BC=DA$ and $BC$ not parallel with $DA$. Let two variable points $E$ and $F$ lie of the sides $BC$ and $DA$, respectively and satisfy $BE=DF$. The lines $AC$ and $BD$ meet at $P$, the lines $BD$ and $EF$ meet at $Q$, the lines $EF$ and $AC$ meet at $R$.



Prove that the circumcircles of the triangles $PQR$, as $E$ and $F$ vary, have a common point other than $P$.
2005 IMO
In a mathematical competition, in which $6$ problems were posed to the participants, every two of these problems were solved by more than $\frac 25$ of the contestants. Moreover, no contestant solved all the $6$ problems. Show that there are at least $2$ contestants who solved exactly $5$ problems each.



Radu Gologan and Dan Schwartz
2004 IMO
1. Let $ABC$ be an acute-angled triangle with $AB\neq AC$. The circle with diameter $BC$ intersects the sides $AB$ and $AC$ at $M$ and $N$ respectively. Denote by $O$ the midpoint of the side $BC$. The bisectors of the angles $\angle BAC$ and $\angle MON$ intersect at $R$. Prove that the circumcircles of the triangles $BMR$ and $CNR$ have a common point lying on the side $BC$.
2004 IMO
Find all polynomials $f$ with real coefficients such that for all reals $a,b,c$ such that $ab+bc+ca = 0$ we have the following relations



$$ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). $$
2004 IMO
Define a "hook" to be a figure made up of six unit squares as shown below in the picture, or any of the figures obtained by applying rotations and reflections to this figure.



[asy]

unitsize(0.5 cm);



draw((0,0)--(1,0));

draw((0,1)--(1,1));

draw((2,1)--(3,1));

draw((0,2)--(3,2));

draw((0,3)--(3,3));

draw((0,0)--(0,3));

draw((1,0)--(1,3));

draw((2,1)--(2,3));

draw((3,1)--(3,3));

[/asy]

Determine all $ m\times n$ rectangles that can  be covered without gaps and without overlaps with hooks such that



- the rectangle is covered without gaps and without overlaps

- no part of a hook covers area outside the rectangle.
2004 IMO
Let $n \geq 3$ be an integer. Let $t_1$, $t_2$, ..., $t_n$ be positive real numbers such that $$n^2 + 1 > \left( t_1 + t_2 + \cdots + t_n \right) \left( \frac{1}{t_1} + \frac{1}{t_2} + \cdots + \frac{1}{t_n} \right).$$ Show that $t_i$, $t_j$, $t_k$ are side lengths of a triangle for all $i$, $j$, $k$ with $1 \leq i < j < k \leq n$.
2004 IMO
In a convex quadrilateral $ABCD$, the diagonal $BD$ bisects neither the angle $ABC$ nor the angle $CDA$. The point $P$ lies inside $ABCD$ and satisfies $$\angle PBC=\angle DBA\quad\text{and}\quad \angle PDC=\angle BDA.$$ Prove that $ABCD$ is a cyclic quadrilateral if and only if $AP=CP$.
2004 IMO
We call a positive integer alternating if every two consecutive digits in its decimal representation are of different parity.



Find all positive integers $n$ such that $n$ has a multiple which is alternating.
2003 IMO
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.
2003 IMO
Determine all pairs of positive integers $(a,b)$ such that  $$ \dfrac{a^2}{2ab^2-b^3+1}  $$  is a positive integer.
2003 IMO
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.
2003 IMO
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.
2003 IMO
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 $q$.
2002 IMO
Let $n$ be a positive integer. Each point $(x,y)$ in the plane, where $x$ and $y$ are non-negative integers with $x+y<n$, is coloured red or blue, subject to the following condition: if a point $(x,y)$ is red, then so are all points $(x',y')$ with $x'\leq x$ and $y'\leq y$. Let $A$ be the number of ways to choose $n$ blue points with distinct $x$-coordinates, and let $B$ be the number of ways to choose $n$ blue points with distinct $y$-coordinates. Prove that $A=B$.
2002 IMO
The circle $S$ has centre $O$, and $BC$ is a diameter of $S$. Let $A$ be a point of $S$ such that $\angle AOB<120{{}^\circ}$.  Let $D$ be the midpoint of the arc $AB$ which does not contain $C$. The line through $O$ parallel to $DA$ meets the line $AC$ at $I$. The perpendicular bisector of $OA$ meets $S$ at $E$ and at $F$. Prove that $I$ is the incentre of the triangle $CEF.$
2002 IMO
Find all pairs of positive integers $m,n\geq3$ for which there exist infinitely many positive integers $a$ such that $$ \frac{a^m+a-1}{a^n+a^2-1}  $$ is itself an integer.



Laurentiu Panaitopol, Romania
2002 IMO
Let $n\geq2$ be a positive integer, with divisors $1=d_1<d_2<\,\ldots<d_k=n$.  Prove that $d_1d_2+d_2d_3+\,\ldots\,+d_{k-1}d_k$ is always less than $n^2$, and determine when it is a divisor of $n^2$.
2002 IMO
Find all functions $f$ from the reals to the reals such that $$ \left(f(x)+f(z)\right)\left(f(y)+f(t)\right)=f(xy-zt)+f(xt+yz)  $$ for all real $x,y,z,t$.
2002 IMO
Let $n\geq3$ be a positive integer. Let $C_1,C_2,C_3,\ldots,C_n$ be unit circles in the plane, with centres $O_1,O_2,O_3,\ldots,O_n$ respectively. If no line meets more than two of the circles, prove that $$ \sum\limits^{}_{1\leq i<j\leq n}{1\over O_iO_j}\leq{(n-1)\pi\over 4}.  $$
2001 IMO
Consider an acute-angled triangle $ABC$. Let $P$ be the foot of the altitude of triangle $ABC$ issuing from the vertex $A$, and let $O$ be the circumcenter of triangle $ABC$. Assume that $\angle C \geq \angle B+30^{\circ}$. Prove that $\angle A+\angle COP < 90^{\circ}$.
2001 IMO
Prove that for all positive real numbers $a,b,c$, $$ \frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1.  $$
2001 IMO
Let $n$ be an odd integer greater than 1 and let $c_1, c_2, \ldots, c_n$ be integers. For each permutation $a = (a_1, a_2, \ldots, a_n)$ of $\{1,2,\ldots,n\}$, define $S(a) = \sum_{i=1}^n c_i a_i$. Prove that there exist permutations $a \neq b$ of $\{1,2,\ldots,n\}$ such that $n!$ is a divisor of $S(a)-S(b)$.
2001 IMO
Let $ABC$ be a triangle with $\angle BAC = 60^{\circ}$. Let $AP$ bisect $\angle BAC$ and let $BQ$ bisect  $\angle ABC$, with $P$ on $BC$ and $Q$ on $AC$. If $AB + BP = AQ + QB$, what are the angles of the triangle?
2001 IMO
Let $a > b > c > d$ be positive integers and suppose that $$ ac + bd = (b+d+a-c)(b+d-a+c).  $$ Prove that $ab + cd$ is not prime.
2000 IMO
Let $ a, b, c$ be positive real numbers so that $ abc = 1$. Prove that

$$ \left( a - 1 + \frac 1b \right) \left( b - 1 + \frac 1c \right) \left( c - 1 + \frac 1a \right) \leq 1.

$$
2000 IMO
Let $ n \geq 2$ be a positive integer and $ \lambda$ a positive real number. Initially there are $ n$ fleas on a horizontal line, not all at the same point. We define a move as choosing two fleas at some points $ A$ and $ B$, with $ A$ to the left of $ B$, and letting the flea from $ A$ jump over the flea from $ B$ to the point $ C$ so that $ \frac {BC}{AB} = \lambda$.



Determine all values of $ \lambda$ such that, for any point $ M$ on the line and for any initial position of the $ n$ fleas, there exists a sequence of moves that will take them all to the position right of $ M$.
2000 IMO
A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two different boxes and announces the sum of numbers on those cards. Given this information, the magician locates the box from which no card has been drawn.



How many ways are there to put the cards in the three boxes so that the trick works?
2000 IMO
Does there exist a positive integer $ n$ such that $ n$ has exactly 2000 prime divisors and $ n$ divides $ 2^n + 1$?
2000 IMO
Let $ AH_1, BH_2, CH_3$ be the altitudes of an acute angled triangle $ ABC$. Its incircle touches the sides $ BC, AC$ and $ AB$ at $ T_1, T_2$ and $ T_3$ respectively. Consider the symmetric images  of  the  lines $ H_1H_2, H_2H_3$ and $ H_3H_1$ with  respect  to  the  lines $ T_1T_2, T_2T_3$ and $ T_3T_1$.  Prove  that  these  images  form a  triangle  whose  vertices  lie  on  the incircle of $ ABC$.
1999 IMO
A set $ S$ of points from the space will be called completely symmetric if it has at least three elements and fulfills the condition that for every two distinct points $ A$ and $ B$ from $ S$, the perpendicular bisector plane of the segment $ AB$ is a plane of symmetry for $ S$. Prove that if a completely symmetric set is finite, then it consists of the vertices of either a regular polygon, or a regular tetrahedron or a regular octahedron.
1999 IMO
Let $n \geq 2$ be a fixed integer. Find the least constant $C$ such the inequality



$$\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C

\left(\sum_{i}x_{i} \right)^4$$



holds for any $x_{1}, \ldots ,x_{n} \geq 0$ (the sum on the left consists of $\binom{n}{2}$ summands). For this constant $C$, characterize the instances of equality.
1999 IMO
Let $n$ be an even positive integer. We say that two different cells of a $n \times n$ board are neighboring if they have a common side. Find the minimal number of cells on the $n \times n$ board that must be marked so that any cell (marked or not marked) has a marked neighboring cell.
1999 IMO
Find all the pairs of positive integers $(x,p)$ such that p is a prime, $x \leq 2p$ and $x^{p-1}$ is a divisor of $ (p-1)^{x}+1$.
1999 IMO
Two circles $\Omega_{1}$ and $\Omega_{2}$ touch internally the circle $\Omega$ in M and N and the center of $\Omega_{2}$ is on $\Omega_{1}$. The common chord of the circles $\Omega_{1}$ and $\Omega_{2}$ intersects $\Omega$ in $A$ and $B$. $MA$ and $MB$ intersects $\Omega_{1}$ in $C$ and $D$. Prove that $\Omega_{2}$ is tangent to $CD$.
1999 IMO
Find all the functions $f: \mathbb{R} \to\mathbb{R}$ such that

$$f(x-f(y))=f(f(y))+xf(y)+f(x)-1$$for all $x,y \in \mathbb{R} $.
1998 IMO
A convex quadrilateral $ABCD$ has perpendicular diagonals. The perpendicular bisectors of the sides $AB$ and $CD$ meet at a unique point $P$ inside $ABCD$. Prove that the quadrilateral $ABCD$ is cyclic if and only if triangles $ABP$ and $CDP$ have equal areas.
1998 IMO
In a contest, there are $m$ candidates and $n$ judges, where $n\geq 3$ is an odd integer. Each candidate is evaluated by each judge as either pass or fail. Suppose that each pair of judges agrees on at most $k$ candidates. Prove that $${\frac{k}{m}} \geq {\frac{n-1}{2n}}. $$
1998 IMO
For any positive integer $n$, let $\tau (n)$ denote the number of its positive divisors (including 1 and itself). Determine all positive integers $m$ for which there exists a positive integer $n$ such that $\frac{\tau (n^{2})}{\tau (n)}=m$.
1998 IMO
Determine all pairs $(x,y)$ of positive integers such that $x^{2}y+x+y$ is divisible by $xy^{2}+y+7$.
1998 IMO
Let $I$ be the incenter of triangle $ABC$. Let $K,L$ and $M$ be the points of tangency of the incircle of $ABC$ with $AB,BC$ and $CA$, respectively. The line $t$ passes through $B$ and is parallel to $KL$. The lines $MK$ and $ML$ intersect $t$ at the points $R$ and $S$. Prove that $\angle RIS$ is acute.
1998 IMO
Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$,



$$f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. $$
1997 IMO
In the plane the points with integer coordinates are the vertices of unit squares. The squares are coloured alternately black and white (as on a chessboard). For any pair of positive integers $ m$ and $ n$, consider a right-angled triangle whose vertices have integer coordinates and whose legs, of lengths $ m$ and $ n$, lie along edges of the squares. Let $ S_1$ be the total area of the black part of the triangle and $ S_2$ be the total area of the white part. Let $ f(m,n) = | S_1 - S_2 |$.



a) Calculate $ f(m,n)$ for all positive integers $ m$ and $ n$ which are either both even or both odd.



b) Prove that $ f(m,n) \leq \frac 12 \max \{m,n \}$ for all $ m$ and $ n$.



c) Show that there is no constant $ C\in\mathbb{R}$ such that $ f(m,n) < C$ for all $ m$ and $ n$.
1997 IMO
It is known that $ \angle BAC$ is the smallest angle in the triangle $ ABC$. The points $ B$ and $ C$ divide the circumcircle of the triangle into two arcs. Let $ U$ be an interior point of the arc between $ B$ and $ C$ which does not contain $ A$. The perpendicular bisectors of $ AB$ and $ AC$ meet the line $ AU$ at $ V$ and $ W$, respectively. The lines $ BV$ and $ CW$ meet at $ T$.



Show that $ AU = TB + TC$.





Alternative formulation:



Four different points $ A,B,C,D$ are chosen on a circle $ \Gamma$ such that the triangle $ BCD$ is not right-angled. Prove that:



(a) The perpendicular bisectors of $ AB$ and $ AC$ meet the line $ AD$ at certain points $ W$ and $ V,$ respectively, and that the lines $ CV$ and $ BW$ meet at a certain point $ T.$



(b) The length of one of the line segments $ AD, BT,$ and $ CT$ is the sum of the lengths of the other two.
1997 IMO
Let $ x_1$, $ x_2$, $ \ldots$, $ x_n$  be real numbers satisfying the conditions:

$$ \left\{\begin{array}{cccc} |x_1 + x_2 + \cdots + x_n | & = & 1 & \ \\
|x_i| & \leq & \displaystyle \frac {n + 1}{2} & \ \textrm{ for }i = 1, 2, \ldots , n. \end{array} \right.

$$

Show that there exists a permutation   $ y_1$, $ y_2$, $ \ldots$, $ y_n$  of $ x_1$, $ x_2$, $ \ldots$, $ x_n$  such that

$$ | y_1 + 2 y_2 + \cdots + n y_n | \leq \frac {n + 1}{2}.

$$
1997 IMO
An $ n \times n$ matrix whose entries come from the set $ S = \{1, 2, \ldots , 2n - 1\}$ is called a silver matrix if, for each $ i = 1, 2, \ldots , n$, the $ i$-th row and the $ i$-th column together contain all elements of $ S$. Show that:



(a)  there is no silver matrix for $ n = 1997$;



(b)  silver matrices exist for infinitely many values of $ n$.
1997 IMO
Find all pairs $ (a,b)$ of positive integers that satisfy the equation: $ a^{b^2} = b^a$.
1997 IMO
For each positive integer $ n$, let $ f(n)$ denote the number of ways of representing $ n$ as a sum of powers of 2 with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance, $ f(4) = 4$, because the number 4 can be represented in the following four ways: 4; 2+2; 2+1+1; 1+1+1+1.



Prove that, for any integer $ n \geq 3$ we have $ 2^{\frac {n^2}{4}} < f(2^n) < 2^{\frac {n^2}2}$.
1996 IMO
We are given a positive integer $ r$ and a rectangular board $ ABCD$ with dimensions $ AB = 20, BC = 12$. The rectangle is divided into a grid of $ 20 \times 12$ unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of the two squares is $ \sqrt {r}$. The task is to find a sequence of moves leading from the square with $ A$ as a vertex to the square with $ B$ as a vertex.



(a) Show that the task cannot be done if $ r$ is divisible by 2 or 3.



(b) Prove that the task is possible when $ r = 73$.



(c) Can the task be done when $ r = 97$?
1996 IMO
Let $ P$ be a point inside a triangle $ ABC$ such that

$$ \angle APB - \angle ACB = \angle APC - \angle ABC.

$$

Let $ D$, $ E$ be the incenters of triangles $ APB$, $ APC$, respectively. Show that the lines $ AP$, $ BD$, $ CE$ meet at a point.
1996 IMO
Let $ \mathbb{N}_0$ denote the set of nonnegative integers. Find all  functions $ f$ from $ \mathbb{N}_0$ to itself such that

$$ f(m + f(n)) = f(f(m)) + f(n)\qquad \text{for all} \; m, n \in \mathbb{N}_0.

$$
1996 IMO
The positive integers $ a$ and $ b$ are such that the numbers  $ 15a + 16b$ and $ 16a - 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?
1996 IMO
Let $ ABCDEF$ be a convex hexagon such that $ AB$ is parallel to  $ DE$, $ BC$ is parallel to $ EF$, and $ CD$ is parallel to $ FA$. Let  $ R_{A},R_{C},R_{E}$ denote the circumradii of triangles $ FAB,BCD,DEF$, respectively, and let $ P$ denote the perimeter of the hexagon. Prove that

$$ R_{A} + R_{C} + R_{E}\geq \frac {P}{2}.

$$
1996 IMO
Let $ p,q,n$ be three positive integers with $ p + q < n$. Let $ (x_{0},x_{1},\cdots ,x_{n})$ be an $ (n + 1)$-tuple of integers satisfying the following conditions :



(a) $ x_{0} = x_{n} = 0$, and



(b) For each $ i$ with $ 1\leq i\leq n$, either $ x_{i} - x_{i - 1} = p$ or $ x_{i} - x_{i - 1} = - q$.



Show that there exist indices $ i < j$ with $ (i,j)\neq (0,n)$, such that $ x_{i} = x_{j}$.
1995 IMO
Let $ A,B,C,D$ be four distinct points on a line, in that order.  The circles with diameters $ AC$ and $ BD$ intersect at $ X$ and $ Y$. The line $ XY$ meets $ BC$ at $ Z$. Let $ P$ be a point on the line $ XY$ other than $ Z$. The line $ CP$ intersects the circle with diameter $ AC$ at $ C$ and $ M$, and the line $ BP$ intersects the circle with diameter $ BD$ at $ B$ and $ N$. Prove that the lines $ AM,DN,XY$ are concurrent.
1995 IMO
Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc = 1$. Prove that

$$ \frac {1}{a^{3}\left(b + c\right)} + \frac {1}{b^{3}\left(c + a\right)} + \frac {1}{c^{3}\left(a + b\right)}\geq \frac {3}{2}.

$$
1995 IMO
Determine all integers $ n > 3$ for which there exist $ n$ points $ A_{1},\cdots ,A_{n}$ in the plane, no three collinear, and real numbers $ r_{1},\cdots ,r_{n}$ such that for $ 1\leq i < j < k\leq n$, the area of $ \triangle A_{i}A_{j}A_{k}$ is $ r_{i} + r_{j} + r_{k}$.
1995 IMO
Find the maximum value of $ x_{0}$ for which there exists a sequence $ x_{0},x_{1}\cdots ,x_{1995}$ of positive reals with $ x_{0} = x_{1995}$, such that

$$ x_{i - 1} + \frac {2}{x_{i - 1}} = 2x_{i} + \frac {1}{x_{i}},

$$

for all $ i = 1,\cdots ,1995$.
1995 IMO
Let $ ABCDEF$ be a convex hexagon with $ AB = BC = CD$ and $ DE = EF = FA$,  such that $ \angle BCD = \angle EFA = \frac {\pi}{3}$. Suppose $ G$ and $ H$ are points in the interior of the hexagon such that $ \angle AGB = \angle DHE = \frac {2\pi}{3}$. Prove that $ AG + GB + GH + DH + HE \geq CF$.
1995 IMO
Let $ p$ be an odd prime number. How many $ p$-element subsets $ A$ of $ \{1,2,\dots,2p\}$ are there, the sum of whose elements is divisible by $ p$?
1994 IMO
Let $ m$ and $ n$ be two positive integers. Let $ a_1$, $ a_2$, $ \ldots$, $ a_m$ be $ m$ different numbers from the set $ \{1, 2,\ldots, n\}$ such that for any two indices $ i$ and $ j$ with $ 1\leq i \leq j \leq m$ and $ a_i + a_j \leq n$, there exists an index $ k$ such that $ a_i + a_j = a_k$. Show that

$$ \frac {a_1 + a_2 + ... + a_m}{m} \geq \frac {n + 1}{2}.

$$
1994 IMO
Let $ ABC$ be an isosceles triangle with $ AB = AC$. $ M$ is the midpoint of $ BC$ and $ O$ is the point on the line $ AM$ such that $ OB$ is perpendicular to $ AB$. $ Q$ is an arbitrary point on $ BC$ different from $ B$ and $ C$. $ E$ lies on the line $ AB$ and $ F$ lies on the line $ AC$ such that $ E, Q, F$ are distinct and collinear. Prove that $ OQ$ is perpendicular to $ EF$ if and only if $ QE = QF$.
1994 IMO
For any positive integer $ k$, let $ f_k$ be the number of elements in the set $ \{ k + 1, k + 2, \ldots, 2k\}$ whose base 2 representation contains exactly three 1s.



(a) Prove that for any positive integer $ m$, there exists at least one positive integer $ k$ such that $ f(k) = m$.



(b) Determine all positive integers $ m$ for which there exists exactly one $ k$ with $ f(k) = m$.
1994 IMO
Find all ordered pairs $ (m,n)$ where $ m$ and $ n$ are positive integers such that $ \frac {n^3 + 1}{mn - 1}$ is an integer.
1994 IMO
Let $ S$ be the set of all real numbers strictly greater than −1. Find all functions $ f: S \to S$ satisfying the two conditions:



(a) $ f(x + f(y) + xf(y)) = y + f(x) + yf(x)$ for all $ x, y$ in $ S$;



(b) $ \frac {f(x)}{x}$ is strictly increasing on each of the two intervals $ - 1 < x < 0$ and $ 0 < x$.
1994 IMO
Show that there exists a set $ A$ of positive integers with the following property: for any infinite set $ S$ of primes, there exist two positive integers $ m$ in $ A$ and $ n$ not in $ A$, each of which is a product of $ k$ distinct elements of $ S$ for some $ k \geq 2$.
1993 IMO
Let $n > 1$ be an integer and let $f(x) = x^n + 5 \cdot x^{n-1} + 3.$ Prove that there do not exist polynomials $g(x),h(x),$ each having integer coefficients and degree at least one, such that $f(x) = g(x) \cdot h(x).$
1993 IMO
Let $A$, $B$, $C$, $D$ be four points in the plane, with $C$ and $D$ on the same side of the line $AB$, such that $AC \cdot BD = AD \cdot BC$ and $\angle ADB = 90^{\circ}+\angle ACB$. Find the ratio

$$\frac{AB \cdot CD}{AC \cdot BD}, $$

and prove that the circumcircles of the triangles $ACD$ and $BCD$ are orthogonal. (Intersecting circles are said to be orthogonal if at either common point their tangents are perpendicuar. Thus, proving that the circumcircles of the triangles $ACD$ and $BCD$ are orthogonal is equivalent to proving that the tangents to the circumcircles of the triangles $ACD$ and $BCD$ at the point $C$ are perpendicular.)
1993 IMO
On an infinite chessboard, a solitaire game is played as follows: at the start, we have $n^2$ pieces occupying a square of side $n.$ The only allowed move is to jump over an occupied square to an unoccupied one, and the piece which has been jumped over is removed. For which $n$ can the game end with only one piece remaining on the board?
1993 IMO
For three points $A,B,C$ in the plane, we define $m(ABC)$ to be the smallest length of the three heights of the triangle $ABC$, where in the case $A$, $B$, $C$ are collinear, we set $m(ABC) = 0$. Let $A$, $B$, $C$ be given points in the plane. Prove that for any point $X$ in the plane,



$$ m(ABC) \leq m(ABX) + m(AXC) + m(XBC).  $$
1993 IMO
Let $\mathbb{N} = \{1,2,3, \ldots\}$. Determine if there exists a strictly increasing function $f: \mathbb{N} \mapsto \mathbb{N}$ with the following properties:



(i) $f(1) = 2$;



(ii) $f(f(n)) = f(n) + n, (n \in \mathbb{N})$.
1993 IMO
Let $n > 1$ be an integer. In a circular arrangement of $n$ lamps $L_0, \ldots, L_{n-1},$ each of of which can either ON or OFF, we start with the situation where all lamps are ON, and then carry out a sequence of steps, $Step_0, Step_1, \ldots .$  If $L_{j-1}$ ($j$ is taken mod $n$) is ON then $Step_j$ changes the state of $L_j$ (it goes from ON to OFF or from OFF to ON) but does not change the state of any of the other lamps. If $L_{j-1}$ is OFF then $Step_j$ does not change anything at all. Show that:



(i) There is a positive integer $M(n)$ such that after $M(n)$ steps all lamps are ON again,



(ii) If $n$ has the form $2^k$ then all the lamps are ON after $n^2-1$ steps,



(iii) If $n$ has the form $2^k + 1$ then all lamps are ON after $n^2 - n + 1$ steps.