Navvye/Olympiad-Problems-CSV huggingface dataset:
2002 IMO Shortlist
Let $n$ be a positive integer. A sequence of $n$ positive integers (not necessarily distinct) is called full if it satisfies the following condition: for each positive integer $k\geq2$, if the number $k$ appears in the sequence then so does the number $k-1$, and moreover the first occurrence of $k-1$ comes before the last occurrence of $k$. For each $n$, how many full sequences are there ?
2002 IMO Shortlist
Let $T$ be the set of ordered triples $(x,y,z)$, where $x,y,z$ are integers with $0\leq x,y,z\leq9$. Players $A$ and $B$ play the following guessing game. Player $A$ chooses a triple $(x,y,z)$ in $T$, and Player $B$ has to discover $A$'s triple in as few moves as possible.  A move consists of the following: $B$ gives $A$ a triple $(a,b,c)$ in $T$, and $A$ replies by giving $B$ the number $\left|x+y-a-b\right |+\left|y+z-b-c\right|+\left|z+x-c-a\right|$. Find the minimum number of moves that $B$ needs to be sure of determining $A$'s  triple.
2002 IMO Shortlist
Let $r\geq2$ be a fixed positive integer, and let $F$ be an infinite family of sets, each of size $r$, no two of which are disjoint. Prove that there exists a set of size $r-1$ that meets each set in $F$.
2002 IMO Shortlist
Let $n$ be an even positive integer. Show that there is a permutation $\left(x_{1},x_{2},\ldots,x_{n}\right)$ of $\left(1,\,2,\,\ldots,n\right)$ such that for every $i\in\left\{1,\ 2,\ ...,\ n\right\}$, the number $x_{i+1}$ is one of the numbers $2x_{i}$, $2x_{i}-1$, $2x_{i}-n$, $2x_{i}-n-1$. Hereby, we use the cyclic subscript convention, so that $x_{n+1}$ means $x_{1}$.
2002 IMO Shortlist
Among a group of 120 people, some pairs are friends. A weak quartet is a set of four people containing exactly one pair of friends. What is the maximum possible number of weak quartets ?
2001 IMO Shortlist
Let $ABC$ be a triangle with centroid $G$. Determine, with proof, the position of the point $P$ in the plane of $ABC$ such that $AP{\cdot}AG + BP{\cdot}BG + CP{\cdot}CG$ is a minimum, and express this minimum value in terms of the side lengths of $ABC$.
2001 IMO Shortlist
Let $M$ be a point in the interior of triangle $ABC$. Let $A'$ lie on $BC$ with $MA'$ perpendicular to $BC$. Define $B'$ on $CA$ and $C'$ on $AB$ similarly.  Define



$$

p(M) = \frac{MA' \cdot MB' \cdot MC'}{MA \cdot MB \cdot MC}.

$$



Determine, with proof, the location of $M$ such that $p(M)$ is maximal.  Let $\mu(ABC)$ denote this maximum value.  For which triangles $ABC$ is the value of $\mu(ABC)$ maximal?
2001 IMO Shortlist
Let $O$ be an interior point of acute triangle $ABC$. Let $A_1$ lie on $BC$ with $OA_1$ perpendicular to $BC$. Define $B_1$ on $CA$ and $C_1$ on $AB$ similarly. Prove that $O$ is the circumcenter of $ABC$ if and only if the perimeter of $A_1B_1C_1$ is not less than any one of the perimeters of $AB_1C_1, BC_1A_1$, and $CA_1B_1$.
2001 IMO Shortlist
Consider the system \begin{align*}x + y  &= z + u,\\2xy & = zu.\end{align*}Find the greatest value of the real constant $m$ such that $m \leq x/y$ for any positive integer solution $(x,y,z,u)$ of the system, with $x \geq y$.
2001 IMO Shortlist
Let $ a_1 = 11^{11}, \, a_2 = 12^{12}, \, a_3 = 13^{13}$, and $ a_n = |a_{n - 1} - a_{n - 2}| + |a_{n - 2} - a_{n - 3}|, n \geq 4.$ Determine $ a_{14^{14}}$.
2001 IMO Shortlist
Let $p \geq 5$ be a prime number. Prove that there exists an integer $a$ with $1 \leq a \leq p-2$ such that neither $a^{p-1}-1$ nor $(a+1)^{p-1}-1$ is divisible by $p^2$.
2001 IMO Shortlist
Let $a_0, a_1, a_2, \ldots$ be an arbitrary infinite sequence of positive numbers.  Show that the inequality $1 + a_n > a_{n-1} \sqrt[n]{2}$ holds for infinitely many positive integers $n$.
2001 IMO Shortlist
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$, satisfying $$

f(xy)(f(x) - f(y)) = (x-y)f(x)f(y)

$$ for all $x,y$.
2001 IMO Shortlist
Find all positive integers $a_1, a_2, \ldots, a_n$ such that



$$

\frac{99}{100} = \frac{a_0}{a_1} + \frac{a_1}{a_2} + \cdots +

\frac{a_{n-1}}{a_n},

$$

where $a_0 = 1$ and $(a_{k+1}-1)a_{k-1} \geq a_k^2(a_k - 1)$ for $k = 1,2,\ldots,n-1$.
2001 IMO Shortlist
Let $A = (a_1, a_2, \ldots, a_{2001})$ be a sequence of positive integers. Let $m$ be the number of 3-element subsequences $(a_i,a_j,a_k)$ with $1 \leq i < j < k \leq 2001$, such that $a_j = a_i + 1$ and $a_k = a_j + 1$.  Considering all such sequences $A$, find the greatest value of $m$.
2001 IMO Shortlist
A set of three nonnegative integers $\{x,y,z\}$ with $x < y < z$ is called historic if $\{z-y,y-x\} = \{1776,2001\}$.  Show that the set of all nonnegative integers can be written as the union of pairwise disjoint historic sets.
2001 IMO Shortlist
For a positive integer $n$ define a sequence of zeros and ones to be balanced if it contains $n$ zeros and $n$ ones. Two balanced sequences $a$ and $b$ are neighbors if you can move one of the $2n$ symbols of $a$ to another position to form $b$. For instance, when $n = 4$, the balanced sequences $01101001$ and $00110101$ are neighbors because the third (or fourth) zero in the first sequence can be moved to the first or second position to form the second sequence. Prove that there is a set $S$ of at most $\frac{1}{n+1} \binom{2n}{n}$ balanced sequences such that every balanced sequence is equal to or is a neighbor of at least one sequence in $S$.
2001 IMO Shortlist
A pile of $n$ pebbles is placed in a vertical column. This configuration is modified according to the following rules. A pebble can be moved if it is at the top of a column which contains at least two more pebbles than the column immediately to its right. (If there are no pebbles to the right, think of this as a column with 0 pebbles.) At each stage, choose a pebble from among those that can be moved (if there are any) and place it at the top of the column to its right. If no pebbles can be moved, the configuration is called a final configuration. For each $n$, show that, no matter what choices are made at each stage, the final configuration obtained is unique. Describe that configuration in terms of $n$.



IMO ShortList 2001, combinatorics problem 7, alternative
2000 IMO Shortlist
Let $ a, b, c$ be positive integers satisfying the conditions $ b > 2a$ and $ c > 2b.$ Show that there exists a real number $ \lambda$ with the property that all the three numbers $ \lambda a, \lambda b, \lambda c$ have their fractional parts lying in the interval $ \left(\frac {1}{3}, \frac {2}{3} \right].$
2000 IMO Shortlist
Find all pairs of functions $ f : \mathbb R \to \mathbb R$, $g : \mathbb R \to \mathbb R$ such that $$f \left( x + g(y) \right) = xf(y) - y  f(x) + g(x) \quad\text{for all } x, y\in\mathbb{R}.$$
2000 IMO Shortlist
The function $ F$ is defined on the set of nonnegative integers and takes nonnegative integer values satisfying the following conditions: for every $ n \geq 0,$



(i) $ F(4n) = F(2n) + F(n),$

(ii) $ F(4n + 2) = F(4n) + 1,$

(iii) $ F(2n + 1) = F(2n) + 1.$



Prove that for each positive integer $ m,$ the number of integers $ n$ with $ 0 \leq n < 2^m$ and $ F(4n) = F(3n)$ is $ F(2^{m + 1}).$
2000 IMO Shortlist
A nonempty set $ A$ of real numbers is called a $ B_3$-set if the conditions $ a_1, a_2, a_3, a_4, a_5, a_6 \in A$ and $ a_1 + a_2 + a_3 = a_4 + a_5 + a_6$ imply that the sequences $ (a_1, a_2, a_3)$ and $ (a_4, a_5, a_6)$ are identical up to a permutation. Let $A = \{a_0 = 0 < a_1 < a_2 < \cdots \}$, $B = \{b_0 = 0 < b_1 < b_2 < \cdots \}$ be infinite sequences of real numbers with $ D(A) = D(B),$ where, for a set $ X$ of real numbers, $ D(X)$ denotes the difference set $ \{|x-y|\mid x, y \in X \}.$ Prove that if $ A$ is a $ B_3$-set, then $ A = B.$
2000 IMO Shortlist
For a polynomial $ P$ of degree 2000 with distinct real coefficients let $ M(P)$ be the set of all polynomials that can be produced from $ P$ by permutation of its coefficients. A polynomial $ P$ will be called $ n$-independent if $ P(n) = 0$ and we can get from any $ Q \in M(P)$ a polynomial $ Q_1$ such that $ Q_1(n) = 0$ by interchanging at most one pair of coefficients of $ Q.$ Find all integers $ n$ for which $ n$-independent polynomials exist.
2000 IMO Shortlist
In the plane we are given two circles intersecting at $ X$ and $ Y$. Prove that there exist four points with the following property:



(P) For every circle touching the two given circles at $ A$ and $ B$, and meeting the line $ XY$ at $ C$ and $ D$, each of the lines $ AC$, $ AD$, $ BC$, $ BD$ passes through one of these points.
2000 IMO Shortlist
Let $ A_1A_2 \ldots A_n$ be a convex polygon, $ n \geq 4.$ Prove that $ A_1A_2 \ldots A_n$ is cyclic if and only if to each vertex $ A_j$ one can assign a pair $ (b_j, c_j)$ of real numbers, $ j = 1, 2, \ldots, n,$ so that $ A_iA_j = b_jc_i - b_ic_j$ for all $ i, j$ with $ 1 \leq i < j \leq n.$
2000 IMO Shortlist
The tangents at $B$ and $A$ to the circumcircle of an acute angled triangle $ABC$ meet the tangent at $C$ at $T$ and $U$ respectively.  $AT$ meets $BC$ at $P$, and $Q$ is the midpoint of $AP$; $BU$ meets $CA$ at $R$, and $S$ is the midpoint of $BR$.  Prove that $\angle ABQ=\angle BAS$.  Determine, in terms of ratios of side lengths, the triangles for which this angle is a maximum.
2000 IMO Shortlist
Let $ ABCD$ be a convex quadrilateral. The perpendicular bisectors of its sides $ AB$ and $ CD$ meet at $ Y$. Denote by $ X$ a point inside the quadrilateral $ ABCD$ such that $ \measuredangle ADX = \measuredangle BCX < 90^{\circ}$ and $ \measuredangle DAX = \measuredangle CBX < 90^{\circ}$. Show that $ \measuredangle AYB = 2\cdot\measuredangle ADX$.
2000 IMO Shortlist
Ten gangsters are standing on a flat surface, and the distances between them are all distinct. At twelve o’clock, when the church bells start chiming, each of them fatally shoots the one among the other nine gangsters who is the nearest. At least how many gangsters will be killed?
2000 IMO Shortlist
Determine all positive integers $ n\geq 2$ that satisfy the following condition: for all $ a$ and $ b$ relatively prime to $ n$ we have $$a \equiv b \pmod n\qquad\text{if and only if}\qquad ab\equiv 1 \pmod n.$$
2000 IMO Shortlist
For a positive integer $n$, let $d(n)$ be the number of all positive divisors of $n$. Find all positive integers $n$ such that $d(n)^3=4n$.
2000 IMO Shortlist
Find all triplets of positive integers $ (a,m,n)$ such that  $ a^m + 1 \mid (a + 1)^n$.
2000 IMO Shortlist
Prove that there exist infinitely many positive integers $ n$ such that $ p = nr,$ where $ p$ and $ r$ are respectively the semiperimeter and the inradius of a triangle with integer side lengths.
2000 IMO Shortlist
A staircase-brick with 3 steps of width 2 is made of 12 unit cubes. Determine all integers $ n$ for which it is possible to build a cube of side $ n$ using such bricks.
2000 IMO Shortlist
Let $ n \geq 4$ be a fixed positive integer. Given a set $ S = \{P_1, P_2, \ldots, P_n\}$ of $ n$ points in the plane such that no three are collinear and no four concyclic, let $ a_t,$ $ 1 \leq t \leq n,$ be the number of circles $ P_iP_jP_k$ that contain $ P_t$ in their interior, and let $$m(S)=a_1+a_2+\cdots + a_n.$$Prove that there exists a positive integer $ f(n),$ depending only on $ n,$ such that the points of $ S$ are the vertices of a convex polygon if and only if $ m(S) = f(n).$
2000 IMO Shortlist
Let $ n$ and $ k$ be positive integers such that $ \frac{1}{2} n < k \leq \frac{2}{3} n.$ Find the least number $ m$ for which it is possible to place $ m$ pawns on $ m$ squares of an $ n \times n$ chessboard so that no column or row contains a block of $ k$ adjacent unoccupied squares.
2000 IMO Shortlist
In the plane we have $n$ rectangles with parallel sides.  The sides of distinct rectangles lie on distinct lines.  The boundaries of the rectangles cut the plane into connected regions.  A region is nice if it has at least one of the vertices of the $n$ rectangles on the boundary.  Prove that the sum of the numbers of the vertices of all nice regions is less than $40n$.  (There can be nonconvex regions as well as regions with more than one boundary curve.)
1999 IMO Shortlist
A circle is called a separator for a set of five points in a plane if it passes through three of these points, it contains a fourth point inside and the fifth point is outside the circle. Prove that every set of five points such that no three are collinear and no four are concyclic has exactly four separators.
1999 IMO Shortlist
For a triangle $T = ABC$ we take the point $X$ on the side $(AB)$ such that $AX/AB=4/5$, the point $Y$ on the segment $(CX)$ such that $CY = 2YX$ and, if possible, the point $Z$ on the ray ($CA$ such that $\widehat{CXZ} = 180 - \widehat{ABC}$. We denote by $\Sigma$ the set of all triangles $T$ for which

$\widehat{XYZ} = 45$. Prove that all triangles from $\Sigma$ are similar and find the measure of their smallest angle.
1999 IMO Shortlist
Let $ABC$ be a triangle, $\Omega$ its incircle and $\Omega_{a}, \Omega_{b}, \Omega_{c}$ three circles orthogonal to $\Omega$ passing through $(B,C),(A,C)$ and $(A,B)$ respectively. The circles $\Omega_{a}$ and $\Omega_{b}$ meet again in $C'$; in the same way we obtain the points $B'$ and $A'$. Prove that the radius of the circumcircle  of $A'B'C'$ is half the radius of $\Omega$.
1999 IMO Shortlist
The point $M$ is inside the convex quadrilateral $ABCD$, such that



$$ MA = MC, \hspace{0,2cm} \widehat{AMB} = \widehat{MAD} + \widehat{MCD} \quad \textnormal{and} \quad \widehat{CMD} = \widehat{MCB} + \widehat{MAB}. $$



Prove that $AB \cdot CM = BC \cdot MD$ and $BM \cdot AD = MA \cdot CD.$
1999 IMO Shortlist
Denote by S the set of all primes such the decimal representation of $\frac{1}{p}$ has the fundamental period divisible by 3. For every $p \in S$ such that $\frac{1}{p}$ has the fundamental period $3r$ one may write



$$\frac{1}{p}=0,a_{1}a_{2}\ldots a_{3r}a_{1}a_{2} \ldots a_{3r} \ldots , $$



where $r=r(p)$; for every $p \in S$ and every integer $k \geq 1$ define $f(k,p)$ by $$ f(k,p)= a_{k}+a_{k+r(p)}+a_{k+2.r(p)}$$



a) Prove that $S$ is infinite.

b) Find the highest value of $f(k,p)$ for $k \geq 1$ and $p \in S$
1999 IMO Shortlist
The numbers from 1 to $n^2$  are randomly arranged in the cells of a $n \times n$ square ($n \geq 2$). For any pair of numbers situated on the same row or on the same column the ratio of the greater number to the smaller number is calculated.  Let us call the characteristic of the arrangement the smallest of these $n^2\left(n-1\right)$ fractions. What is the highest possible value of the characteristic ?
1999 IMO Shortlist
For $n \geq 3$ and $a_{1} \leq a_{2} \leq \ldots \leq a_{n}$ given real numbers we have the following instructions:



- place out the numbers in some order in a ring;

- delete one of the numbers from the ring;

- if just two numbers are remaining in the ring: let $S$ be the sum of these two numbers. Otherwise, if there are more the two numbers in the ring, replace



Afterwards start again with the step (2). Show that the largest sum $S$ which can result in this way is given by the formula



$$S_{max}= \sum^n_{k=2} \begin{pmatrix} n -2 \\
[\frac{k}{2}] - 1\end{pmatrix}a_{k}.$$
1999 IMO Shortlist
Let $n \geq 1$ be an integer. A path from $(0,0)$ to $(n,n)$ in the $xy$ plane is a chain of consecutive unit moves either to the right (move denoted by $E$) or upwards (move denoted by $N$), all the moves being made inside the half-plane $x \geq y$. A step in a path is the occurence of two consecutive moves of the form $EN$. Show that the number of paths from $(0,0)$ to $(n,n)$ that contain exactly $s$ steps $(n \geq s \geq 1)$ is



$$\frac{1}{s} \binom{n-1}{s-1} \binom{n}{s-1}.$$
1999 IMO Shortlist
If a $5 \times n$ rectangle can be tiled using $n$ pieces like those shown in the diagram, prove that $n$ is even. Show that there are more than $2 \cdot 3^{k-1}$ ways to file a fixed $5 \times 2k$ rectangle $(k \geq 3)$ with $2k$ pieces. (symmetric constructions are supposed to be different.)
1999 IMO Shortlist
A biologist watches a chameleon. The chameleon catches flies and rests after each catch. The biologist notices that:



the first fly is caught after a resting period of one minute;

the resting period before catching the $2m^\text{th}$ fly is the same as the resting period before catching the $m^\text{th}$ fly and one minute shorter than the resting period before catching the $(2m+1)^\text{th}$ fly;

when the chameleon stops resting, he catches a fly instantly.







How many flies were caught by the chameleon before his first resting period of $9$ minutes in a row?

After how many minutes will the chameleon catch his $98^\text{th}$ fly?

How many flies were caught by the chameleon after 1999 minutes have passed?
1999 IMO Shortlist
Let $A$ be a set of $N$ residues $\pmod{N^{2}}$. Prove that there exists a set $B$ of of $N$ residues $\pmod{N^{2}}$ such that $A + B = \{a+b|a \in A, b \in B\}$ contains at least half of all the residues $\pmod{N^{2}}$.
1999 IMO Shortlist
Let $p >3$ be a prime number. For each nonempty subset $T$ of $\{0,1,2,3, \ldots , p-1\}$, let $E(T)$ be the set of all $(p-1)$-tuples $(x_1, \ldots ,x_{p-1} )$, where each $x_i \in T$ and $x_1+2x_2+ \ldots + (p-1)x_{p-1}$ is divisible by $p$ and let $|E(T)|$ denote the number of elements in $E(T)$. Prove that



$$|E(\{0,1,3\})| \geq |E(\{0,1,2\})|$$



with equality if and only if $p = 5$.
1998 IMO Shortlist
Let $ABC$ be a triangle such that $\angle ACB=2\angle ABC$. Let $D$ be the point on the side $BC$ such that $CD=2BD$. The segment $AD$ is extended to $E$ so that $AD=DE$. Prove that $$ \angle ECB+180^{\circ }=2\angle EBC. $$
1998 IMO Shortlist
Determine all pairs $(a,b)$ of real numbers such that $a \lfloor bn \rfloor =b \lfloor an \rfloor $ for all positive integers $n$. (Note that $\lfloor x\rfloor $ denotes the greatest integer less than or equal to $x$.)
1998 IMO Shortlist
Determine the smallest integer $n\geq 4$ for which one can choose four different numbers $a,b,c$ and $d$ from any $n$ distinct integers such that $a+b-c-d$ is divisible by $20$.
1998 IMO Shortlist
A sequence of integers $ a_{1},a_{2},a_{3},\ldots$ is defined as follows: $ a_{1} = 1$ and for $ n\geq 1$, $ a_{n + 1}$ is the smallest integer greater than $ a_{n}$ such that $ a_{i} + a_{j}\neq 3a_{k}$ for any $ i,j$ and $ k$ in $ \{1,2,3,\ldots ,n + 1\}$, not necessarily distinct. Determine $ a_{1998}$.
1998 IMO Shortlist
Prove that for each positive integer $n$, there exists a positive integer with the following properties: It has exactly $n$ digits. None of the digits is 0. It is divisible by the sum of its digits.
1998 IMO Shortlist
Let $a_{0},a_{1},a_{2},\ldots $ be an increasing sequence of nonnegative integers such that every nonnegative integer can be expressed uniquely in the form $a_{i}+2a_{j}+4a_{k}$, where $i,j$ and $k$ are not necessarily distinct. Determine $a_{1998}$.
1998 IMO Shortlist
Let $a_{1},a_{2},\ldots ,a_{n}$ be positive real numbers such that $a_{1}+a_{2}+\cdots +a_{n}<1$. Prove that



$$ \frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_{1})(1 - a_{2}) \cdots (1 - a_{n})} \leq \frac{1}{ n^{n+1}}. $$
1998 IMO Shortlist
Let $r_{1},r_{2},\ldots ,r_{n}$ be real numbers greater than or equal to 1. Prove that



$$ \frac{1}{r_{1} + 1} + \frac{1}{r_{2} + 1} + \cdots +\frac{1}{r_{n}+1} \geq \frac{n}{ \sqrt[n]{r_{1}r_{2} \cdots r_{n}}+1}. $$
1998 IMO Shortlist
Let $x,y$ and $z$ be positive real numbers such that $xyz=1$. Prove that





$$

 \frac{x^{3}}{(1 + y)(1 + z)}+\frac{y^{3}}{(1 + z)(1 + x)}+\frac{z^{3}}{(1 + x)(1 + y)}

 \geq \frac{3}{4}. 

$$
1998 IMO Shortlist
A rectangular array of numbers is given. In each row and each column, the sum of all numbers is an integer. Prove that each nonintegral number $x$ in the array can be changed into either $\lceil x\rceil $ or $\lfloor x\rfloor $ so that the row-sums and column-sums remain unchanged. (Note that $\lceil x\rceil $ is the least integer greater than or equal to $x$, while $\lfloor x\rfloor $ is the greatest integer less than or equal to $x$.)
1998 IMO Shortlist
Cards numbered 1 to 9 are arranged at random in a row. In a move, one may choose any block of consecutive cards whose numbers are in ascending or descending order, and switch the block around. For example, 9 1 $\underline{6\ 5\ 3}$ $2\ 7\ 4\ 8$ may be changed to $9 1$ $\underline{3\ 5\ 6}$ $2\ 7\ 4\ 8$. Prove that in at most 12 moves, one can arrange the 9 cards so that their numbers are in ascending or descending order.
1998 IMO Shortlist
Let $U=\{1,2,\ldots ,n\}$, where $n\geq 3$. A subset $S$ of $U$ is said to be split by an arrangement of the elements of $U$ if an element not in $S$ occurs in the arrangement somewhere between two elements of $S$. For example, 13542 splits $\{1,2,3\}$ but not $\{3,4,5\}$. Prove that for any $n-2$ subsets of $U$, each containing at least 2 and at most $n-1$ elements, there is an arrangement of the elements of $U$ which splits all of them.
1998 IMO Shortlist
Ten points are marked in the plane so that no three of them lie on a line. Each pair of points is connected with a segment. Each of these segments is painted with one of $k$ colors, in such a way that for any $k$ of the ten points, there are $k$ segments each joining two of them and no two being painted with the same color. Determine all integers $k$, $1\leq k\leq 10$, for which this is possible.
1998 IMO Shortlist
A solitaire game is played on an $m\times n$ rectangular board, using $mn$ markers which are white on one side and black on the other. Initially, each  square of the board contains a marker with its white side up, except for one corner square, which contains a marker with its black side up. In each move, one may take away one marker with its black side up, but must then turn over  all markers which are in squares having an edge in common with the square of the removed marker. Determine all pairs $(m,n)$ of positive integers such that all markers can be removed from the board.
1997 IMO Shortlist
Let $ ABCD$ be a regular tetrahedron and $ M,N$ distinct points in the planes $ ABC$ and $ ADC$ respectively. Show that the segments $ MN,BN,MD$ are the sides of a triangle.
1997 IMO Shortlist
(a) Let $ n$ be a positive integer. Prove that there exist distinct positive integers $ x, y, z$ such that



$$ x^{n-1} + y^n = z^{n+1}.$$



(b) Let $ a, b, c$ be positive integers such that $ a$ and $ b$ are relatively prime and $ c$ is relatively prime either to $ a$ or to $ b.$ Prove that there exist infinitely many triples $ (x, y, z)$ of distinct positive integers $ x, y, z$ such that



$$ x^a + y^b = z^c.$$
1997 IMO Shortlist
Let $ P(x)$ be a polynomial with real coefficients such that $ P(x) > 0$ for all $ x \geq 0.$ Prove that there exists a positive integer n such that $ (1 + x)^n \cdot P(x)$ is a polynomial with nonnegative coefficients.
1997 IMO Shortlist
In an acute-angled triangle $ ABC,$ let $ AD,BE$ be altitudes and $ AP,BQ$ internal bisectors. Denote by $ I$ and $ O$ the incenter and the circumcentre of the triangle, respectively. Prove that the points $ D, E,$ and $ I$ are collinear if and only if the points $ P, Q,$ and $ O$ are collinear.
1997 IMO Shortlist
Let $ a_1\geq \cdots \geq a_n \geq a_{n + 1} = 0$ be real numbers. Show that

$$ \sqrt {\sum_{k = 1}^n a_k} \leq \sum_{k = 1}^n \sqrt k (\sqrt {a_k} - \sqrt {a_{k + 1}}).

$$

Proposed by Romania
1997 IMO Shortlist
Let $ ABC$ be a triangle. $ D$ is a point on the side $ (BC)$. The line $ AD$ meets the circumcircle again at $ X$. $ P$ is the foot of the perpendicular from $ X$ to $ AB$, and $ Q$ is the foot of the perpendicular from $ X$ to $ AC$. Show that the line $ PQ$ is a tangent to the circle on diameter $ XD$ if and only if $ AB = AC$.
1997 IMO Shortlist
Does there exist functions $ f,g: \mathbb{R}\to\mathbb{R}$ such that  $ f(g(x)) = x^2$ and $ g(f(x)) = x^k$ for all real numbers $ x$



a) if $ k = 3$?



b) if $ k = 4$?
1997 IMO Shortlist
Let $ ABCD$ be a convex quadrilateral. The diagonals $ AC$ and $ BD$ intersect at $ K$. Show that $ ABCD$ is cyclic if and only if $ AK \sin A + CK \sin C = BK \sin B + DK \sin D$.