Navvye/Olympiad-Problems-CSV huggingface dataset:
2020 Benelux
Let $ABC$ be a triangle.  The circle $\omega_A$ through $A$ is tangent to line $BC$ at $B$. The circle $\omega_C$ through $C$ is tangent to line $AB$ at $B$.  Let $\omega_A$ and $\omega_C$ meet again at $D$. Let $M$ be the midpoint of line segment $[BC]$, and let $E$ be the intersection of lines $MD$ and $AC$. Show that $E$ lies on $\omega_A$.
2020 Benelux
A divisor $d$ of a positive integer $n$ is said to be a close divisor of $n$ if $\sqrt{n}<d<2\sqrt{n}$. Does there exist a positive integer with exactly $2020$ close divisors?
2019 Benelux




Let $a,b,c,d$ be real numbers with $0\leqslant a,b,c,d\leqslant 1$. Prove that

$$ab(a-b)+bc(b-c)+cd(c-d)+da(d-a)\leqslant \frac{8}{27}.$$

Find all quadruples $(a,b,c,d)$ of real numbers with $0\leqslant a,b,c,d\leqslant 1$ for which equality holds in the above inequality.
2019 Benelux
Pawns and rooks are placed on a $2019\times 2019$ chessboard, with at most one piece on each of the $2019^2$ squares. A rook can see another rook if they are in the same row or column and all squares between them are empty. What is the maximal number $p$ for which $p$ pawns and $p+2019$ rooks can be placed on the chessboard in such a way that no two rooks can see each other?
2019 Benelux
Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $A$ and $Z$ (with $A\neq Z$). Let $B$ be the centre of $\Gamma_1$ and let $C$ be the centre of $\Gamma_2$. The exterior angle bisector of $\angle{BAC}$ intersects $\Gamma_1$ again at $X$ and $\Gamma_2$ again at $Y$. Prove that the interior angle bisector of $\angle{BZC}$ passes through the circumcenter of $\triangle{XYZ}$.



For  points $P,Q,R$ that lie on a line $\ell$ in that order, and a point $S$ not on $\ell$, the interior angle bisector of $\angle{PQS}$ is the line that divides $\angle{PQS}$ into two equal angles, while the exterior angle bisector of $\angle{PQS}$ is the line that divides $\angle{RQS}$ into two equal angles.
2019 Benelux
An integer $m>1$ is rich if for any positive integer $n$, there exist positive integers $x,y,z$ such that $n=mx^2-y^2-z^2$. An integer $m>1$ is poor if it is not rich.



Find a poor integer.

Find a rich integer.
2018 Benelux
(a) Determine the minimal value of

$\displaystyle\left(x+\dfrac{1}{y}\right)\left(x+\dfrac{1}{y}-2018\right)+\left(y+\dfrac{1}{x}\right)\left(y+\dfrac{1}{x}-2018\right), $

where $x$ and $y$ vary over the positive reals.



(b) Determine the minimal value of

$\displaystyle\left(x+\dfrac{1}{y}\right)\left(x+\dfrac{1}{y}+2018\right)+\left(y+\dfrac{1}{x}\right)\left(y+\dfrac{1}{x}+2018\right), $

where $x$ and $y$ vary over the positive reals.
2018 Benelux
In the land of Heptanomisma, four different coins and three different banknotes are used, and their denominations are seven different natural numbers. The denomination of the smallest banknote is greater than the sum of the denominations of the four different coins. A tourist has exactly one coin of each denomination and exactly one banknote of each denomination, but he cannot afford the book on numismatics he wishes to buy. However, the mathematically inclined shopkeeper offers to sell the book to the tourist at a price of his choosing, provided that he can pay this price in more than one way.

(The tourist can pay a price in more than one way if there are two different subsets of his coins and notes, the denominations of which both add up to this price.)

(a) Prove that the tourist can purchase the book if the denomination of each banknote is smaller than $49$.

(b) Show that the tourist may have to leave the shop empty-handed if the denomination of the largest banknote is $49$.
2018 Benelux
Let $ABC$ be a triangle with orthocentre $H$, and let $D$, $E$, and $F$ denote the respective midpoints of line segments $AB$, $AC$, and $AH$. The reflections of $B$ and $C$ in $F$ are $P$ and $Q$, respectively.

(a) Show that lines $PE$ and $QD$ intersect on the circumcircle of triangle $ABC$.

(b) Prove that lines $PD$ and $QE$ intersect on line segment $AH$.
2018 Benelux
An integer $n\geq 2$ having exactly $s$ positive divisors $1=d_1<d_2<\cdots<d_s=n$ is said to be good if there exists an integer $k$, with $2\leq k\leq s$, such that $d_k>1+d_1+\cdots+d_{k-1}$. An integer $n\geq 2$ is said to be bad if it is not good.

(a) Show that there are infinitely many bad integers.

(b) Prove that, among any seven consecutive integers all greater than $2$, there are always at least four good integers.

(c) Show that there are infinitely many sequences of seven consecutive good integers.
2017 Benelux
Find all functions $f : \Bbb{Q}_{>0}\to \Bbb{Z}_{>0}$ such that $$f(xy)\cdot \gcd\left( f(x)f(y), f(\frac{1}{x})f(\frac{1}{y})\right)

= xyf(\frac{1}{x})f(\frac{1}{y}),$$for all $x, y \in  \Bbb{Q}_{>0,}$ where $\gcd(a, b)$ denotes the greatest common divisor of $a$ and $b.$
2017 Benelux
Let $n\geq 2$ be an integer. Alice and Bob play a game concerning a country made of $n$ islands. Exactly two of those $n$ islands have a factory. Initially there is no bridge in the country. Alice and Bob take turns in the following way. In each turn, the player must build a bridge between two different islands $I_1$ and $I_2$ such that:

$\bullet$ $I_1$ and $I_2$ are not already connected by a bridge.

$\bullet$ at least one of the two islands $I_1$ and $I_2$ is connected by a series of bridges to an island with a factory (or has a factory itself). (Indeed, access to a factory is needed for the construction.)

As soon as a player builds a bridge that makes it possible to go from one factory to the other, this player loses the game. (Indeed, it triggers an industrial battle between both factories.) If Alice starts, then determine (for each $n\geq  2$) who has a winning strategy.

(Note: It is allowed to construct a bridge passing above another bridge.)
2017 Benelux
In the convex quadrilateral $ABCD$ we have $\angle B = \angle C$ and $\angle D = 90^{\circ}.$ Suppose that $|AB| = 2|CD|.$ Prove that the angle bisector of $\angle ACB$ is perpendicular to $CD.$
2017 Benelux
A Benelux n-square (with $n\geq  2$) is an $n\times  n$ grid consisting of $n^2$ cells, each of them containing a positive integer, satisfying the following conditions:

$\bullet$ the $n^2$ positive integers are pairwise distinct.

$\bullet$ if for each row and each column we compute the greatest common divisor of the $n$ numbers in that row/column, then we obtain $2n$ different outcomes.



(a) Prove that, in each Benelux n-square (with $n \geq 2$), there exists a cell containing a number which is at least $2n^2.$

(b) Call a Benelux n-square minimal if all $n^2$ numbers in the cells are at most $2n^2.$ Determine all $n\geq 2$ for which there exists a minimal Benelux n-square.
2016 Benelux
Find the greatest positive integer $N$ with the following property: there exist integers $x_1, . . . , x_N$ such that $x^2_i - x_ix_j$ is not divisible by $1111$ for any $i\ne j.$
2016 Benelux
Let $n$ be a positive integer. Suppose that its positive divisors can be partitioned into pairs (i.e. can be split in groups of two) in such a way that the sum of each pair is a prime number. Prove that these prime numbers are distinct and that none of these are a divisor of $n.$
2016 Benelux
Find all functions $f :\Bbb{ R}\to \Bbb{Z}$ such that $$\left( f(f(y) - x) \right)^2+ f(x)^2 + f(y)^2 = f(y) \cdot \left( 1 + 2f(f(y)) \right),$$for all $x, y \in \Bbb{R}.$
2016 Benelux
A circle $\omega$ passes through the two vertices $B$ and $C$ of a triangle $ABC.$ Furthermore, $\omega$  intersects segment $AC$ in $D\ne C$ and segment $AB$ in $E\ne B.$ On the ray from $B$ through $D$ lies a point $K$ such that $|BK| = |AC|,$ and on the ray from $C$  through $E$ lies a point $L$ such that $|CL| = |AB|.$ Show that the circumcentre $O$ of triangle $AKL$ lies on $\omega$.
2015 Benelux
Determine the smallest positive integer $q$ with the following property: for  every  integer $m$ with  $1\leqslant m\leqslant 1006$, there  exists  an  integer $n$ such that

$$\dfrac{m}{1007}q<n<\dfrac{m+1}{1008}q$$.
2015 Benelux
Let $ABC$ be  an  acute  triangle  with  circumcentre $O$. Let $\mathit{\Gamma}_B$ be  the  circle through $A$ and $B$ that is tangent to $AC$, and let $\mathit{\Gamma}_C$ be the circle through $A$ and $C$ that is tangent to $AB$.  An arbitrary line through $A$ intersects $\mathit{\Gamma}_B$ again in $X$ and $\mathit{\Gamma}_C$ again in $Y$. Prove that $|OX|=|OY|$.
2015 Benelux
Does there exist a prime number whose decimal representation is of the form $3811\cdots11$ (that is, consisting of the digits $3$ and $8$ in that order, followed by one or more digits $1$)?
2015 Benelux
Let $n$ be a positive integer.  For each partition of the set $\{1,2,\dots,3n\}$ into arithmetic  progressions,  we  consider  the  sum $S$ of  the  respective  common differences  of  these  arithmetic  progressions.   What  is  the  maximal  value that $S$ can attain?



(An arithmetic progression is a set of the form $\{a,a+d,\dots,a+kd\}$, where $a,d,k$ are positive integers, and $k\geqslant 2$; thus an arithmetic progression has at least three elements, and successive elements have difference $d$, called the common difference of the arithmetic progression.)
2014 Benelux
Find the smallest possible value of the expression $$\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{b+c+d}{a}\right\rfloor+\left\lfloor\frac{c+d+a}{b}\right\rfloor+\left\lfloor\frac{d+a+b}{c}\right\rfloor$$

in which $a,~ b,~ c$, and $d$ vary over the set of positive integers.



(Here $\lfloor x\rfloor$ denotes the biggest integer which is smaller than or equal to $x$.)
2014 Benelux
Let $k\ge 1$ be a positive integer.



We consider $4k$ chips, $2k$ of which are red and $2k$ of which are blue. A sequence of those $4k$ chips can be transformed into another sequence by a so-called move, consisting of interchanging a number (possibly one) of consecutive red chips with an

equal number of consecutive blue chips.  For example, we can move from $r\underline{bb}br\underline{rr}b$ to $r\underline{rr}br\underline{bb}b$ where $r$ denotes a red chip and $b$ denotes a blue chip.



Determine the smallest number $n$ (as a function of $k$) such that starting from any initial sequence of the $4k$ chips, we need at most $n$ moves to reach the state in which the first $2k$ chips are red.
2014 Benelux
For all integers $n\ge 2$ with the following property:



for each pair of positive divisors $k,~\ell <n$, at least one of the numbers $2k-\ell$ and $2\ell-k$ is a (not necessarily positive) divisor of $n$ as well.
2014 Benelux
Let $ABCD$ be a square. Consider a variable point $P$ inside the square for which $\angle BAP \ge 60^\circ.$ Let $Q$ be the intersection of the line $AD$ and the perpendicular to $BP$ in $P$. Let $R$ be the intersection of the line $BQ$ and the perpendicular to $BP$ from $C$.



(a) Prove that $|BP|\ge |BR|$



(b) For which point(s) $P$ does the inequality in (a) become an equality?
2013 Benelux
Let $n \ge 3$ be an integer. A frog is to jump along the real axis, starting at the point $0$ and making $n$ jumps: one of length $1$, one of length $2$, $\dots$ , one of length $n$. It may perform these $n$ jumps in any order. If at some point the frog is sitting on a number $a \le 0$, its next jump must be to the right (towards the positive numbers). If at some point the frog is sitting on a number $a > 0$, its next jump must be to the left (towards the negative numbers). Find the largest positive integer $k$ for which the frog can perform its jumps in such an order that it never lands on any of the numbers $1, 2, \dots , k$.
2013 Benelux
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that

$$f(x + y) + y \le f(f(f(x)))$$

holds for all $x, y \in \mathbb{R}$.
2013 Benelux
Let $\triangle ABC$ be a triangle with circumcircle $\Gamma$, and let $I$ be the center of the incircle of $\triangle ABC$. The lines $AI$, $BI$ and $CI$ intersect $\Gamma$ in $D \ne A$, $E \ne B$ and $F \ne C$. The tangent lines to $\Gamma$ in $F$, $D$ and $E$ intersect the lines $AI$, $BI$ and $CI$ in $R$, $S$ and $T$, respectively. Prove that

$$\vert AR\vert \cdot \vert BS\vert \cdot \vert CT\vert = \vert ID\vert \cdot \vert IE\vert \cdot \vert IF\vert.$$
2013 Benelux
a) Find all positive integers $g$ with the following property: for each odd prime number $p$ there exists a positive integer $n$ such that $p$ divides the two integers

$$g^n - n\quad\text{ and }\quad g^{n+1} - (n + 1).$$

b) Find all positive integers $g$ with the following property: for each odd prime number $p$ there exists a positive integer $n$ such that $p$ divides the two integers

$$g^n - n^2\quad\text{ and }g^{n+1} - (n + 1)^2.$$
2012 Benelux
A sequence $a_1,a_2,\ldots ,a_n,\ldots$ of natural numbers is defined by the rule

$$a_{n+1}=a_n+b_n\ (n=1,2,\ldots)$$

where $b_n$ is the last digit of $a_n$. Prove that such a sequence contains infinitely many powers of $2$ if and only if $a_1$ is not divisible by $5$.
2012 Benelux
Find all quadruples $(a,b,c,d)$ of positive real numbers such that $abcd=1,a^{2012}+2012b=2012c+d^{2012}$ and $2012a+b^{2012}=c^{2012}+2012d$.
2012 Benelux
In triangle $ABC$ the midpoint of $BC$ is called $M$. Let $P$ be a variable interior point of the triangle such that $\angle CPM=\angle PAB$. Let $\Gamma$ be the circumcircle of triangle $ABP$. The line $MP$ intersects $\Gamma$ a second time at $Q$. Define $R$ as the reflection of $P$ in the tangent to $\Gamma$ at $B$. Prove that the length $|QR|$ is independent of the position of $P$ inside the triangle.
2012 Benelux
Yesterday, $n\ge 4$ people sat around a round table. Each participant remembers only who his two neighbours were, but not necessarily which one sat on his left and which one sat on his right. Today, you would like the same people to sit around the same round table so that each participant has the same two neighbours as yesterday (it is possible that yesterday’s left-hand side neighbour is today’s right-hand side neighbour). You are allowed to query some of the participants: if anyone is asked, he will answer by pointing at his two neighbours from yesterday.



a) Determine the minimal number $f(n)$ of participants you have to query in order to be certain to succeed, if later questions must not depend on the outcome of the previous questions. That is, you have to choose in advance the list of people you are going to query, before effectively asking any question.



b) Determine the minimal number $g(n)$ of participants you have to query in order to be certain to succeed, if later questions may depend on the outcome of previous questions. That is, you can wait until you get the first answer to choose whom to ask the second question, and so on.
2011 Benelux
An ordered pair of integers $(m,n)$ with $1<m<n$ is said to be a Benelux couple if the following two conditions hold: $m$ has the same prime divisors as $n$, and $m+1$ has the same prime divisors as $n+1$.

(a) Find three Benelux couples $(m,n)$ with $m\leqslant 14$.

(b) Prove that there are infinitely many Benelux couples
2011 Benelux
Let $ABC$ be a triangle with incentre $I$. The angle bisectors $AI$, $BI$ and $CI$ meet $[BC]$, $[CA]$ and $[AB]$ at $D$, $E$ and $F$, respectively. The perpendicular bisector of $[AD]$ intersects the lines $BI$ and $CI$ at $M$ and $N$, respectively. Show that $A$, $I$, $M$ and $N$ lie on a circle.
2011 Benelux
If $k$ is an integer, let $\mathrm{c}(k)$ denote the largest cube that is less than or equal to $k$. Find all positive integers $p$ for which the following sequence is bounded:

$a_0 = p$ and $a_{n+1} = 3a_n-2\mathrm{c}(a_n)$ for $n \geqslant 0$.
2011 Benelux
Abby and Brian play the following game: They first choose a positive integer $N$. Then they write numbers on a blackboard in turn. Abby starts by writing a $1$. Thereafter, when one of them has written the number $n$, the other writes down either $n + 1$ or $2n$, provided that the number is not greater than $N$. The player who writes $N$ on the blackboard wins.

(a) Determine which player has a winning strategy if $N = 2011$.

(b) Find the number of positive integers $N\leqslant2011$ for which Brian has a winning strategy.



(This is based on ISL 2004, Problem C5.)
2010 Benelux
A finite set of integers is called bad if its elements add up to $2010$. A finite set of integers is a Benelux-set if none of its subsets is bad. Determine the smallest positive integer $n$ such that the set $\{502, 503, 504, . . . , 2009\}$ can be partitioned into $n$ Benelux-sets.

(A partition of a set $S$ into $n$ subsets is a collection of $n$ pairwise disjoint subsets of $S$, the union of which equals $S$.)



(2nd Benelux Mathematical Olympiad 2010, Problem 1)
2010 Benelux
Find all polynomials $p(x)$ with real coeffcients such that

$$p(a + b - 2c) + p(b + c - 2a) + p(c + a - 2b) = 3p(a - b) + 3p(b - c) + 3p(c - a)$$

for all $a, b, c\in\mathbb{R}$.



(2nd Benelux Mathematical Olympiad 2010, Problem 2)
2010 Benelux
On a line $l$ there are three different points $A$, $B$ and $P$ in that order. Let $a$ be the line through $A$ perpendicular to $l$, and let $b$ be the line through $B$ perpendicular to $l$. A line through $P$, not coinciding with $l$, intersects $a$ in $Q$ and $b$ in $R$. The line through $A$ perpendicular to $BQ$ intersects $BQ$ in $L$ and $BR$ in $T$. The line through $B$ perpendicular to $AR$ intersects $AR$ in $K$ and $AQ$ in $S$.

(a) Prove that $P$, $T$, $S$ are collinear.

(b) Prove that $P$, $K$, $L$ are collinear.



(2nd Benelux Mathematical Olympiad 2010, Problem 3)
2010 Benelux
Find all quadruples $(a, b, p, n)$ of positive integers, such that $p$ is a prime and

$$a^3 + b^3 = p^n\mbox{.}$$



(2nd Benelux Mathematical Olympiad 2010, Problem 4)
2009 Benelux
Find all functions $f:\mathbb{Z}_{>0}\rightarrow\mathbb{Z}_{>0}$ that satisfy the following two conditions:





$\bullet\ f(n)$ is a perfect square for all $n\in\mathbb{Z}_{>0}$

$\bullet\ f(m+n)=f(m)+f(n)+2mn$ for all $m,n\in\mathbb{Z}_{>0}$.
2009 Benelux
Let $n$ be a positive integer and let $k$ be an odd positive integer. Moreover, let $a,b$ and $c$ be integers (not necessarily positive) satisfying the equations

$$a^n+kb=b^n+kc=c^n+ka $$

Prove that $a=b=c$.
2009 Benelux
Let $n\ge 1$ be an integer. In town $X$ there are $n$ girls and $n$ boys, and each girl knows each boy. In town $Y$ there are $n$ girls, $g_1,g_2,\ldots ,g_n$, and $2n-1$ boys, $b_1,b_2,\ldots  ,b_{2n-1}$. For $i=1,2,\ldots ,n$, girl $g_i$ knows boys $b_1,b_2,\ldots ,b_{2i-1}$ and no other boys. Let $r$ be an integer with $1\le r\le n$. In each of the towns a party will be held where $r$ girls from that town and $r$ boys from the same town are supposed to dance with each other in $r$ dancing pairs. However, every girl only wants to dance with a boy she knows. Denote by $X(r)$ the number of ways in which we can choose $r$ dancing pairs from town $X$, and by $Y(r)$ the number of ways in which we can choose $r$ dancing pairs from town $Y$. Prove that $X(r)=Y(r)$ for $r=1,2,\ldots ,n$.
2009 Benelux
Given trapezoid $ABCD$ with parallel sides $AB$ and $CD$, let $E$ be a point on line $BC$ outside segment $BC$, such that segment $AE$ intersects segment $CD$. Assume that there exists a point $F$ inside segment $AD$ such that $\angle EAD=\angle CBF$. 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 $EF$, and assume that $K$ is different from $I$ and $J$.



Prove that $K$ belongs to the circumcircle of $\triangle ABI$ if and only if $K$ belongs to the circumcircle of $\triangle CDJ$.
2021 Caucasus Mathematical Olympiad
Let  $a$, $b$, $c$ be real numbers such that $a^2+b=c^2$, $b^2+c=a^2$, $c^2+a=b^2$. Find all possible values of $abc$.
2021 Caucasus Mathematical Olympiad
In a triangle $ABC$ let $K$ be a point on the median $BM$  such that $CK=CM$. It appears that $\angle CBM = 2 \angle ABM$. Prove that $BC=MK$.
2021 Caucasus Mathematical Olympiad
We have $n>2$ non-zero integers such that each one of them is divisible by the sum of the other $n-1$ numbers. Prove that the sum of all the given numbers is zero.
2021 Caucasus Mathematical Olympiad
A square grid $2n \times 2n$ is constructed of matches (each match is a segment of length 1). By one move Peter can choose a vertex which (at this moment) is the endpoint of  3 or 4 matches and delete two matches whose union is a segment of length 2. Find the least possible number of matches that could remain after a number of Peter's moves.
2021 Caucasus Mathematical Olympiad
Let $a, b, c$ be positive integers such that the product $$\gcd(a,b) \cdot \gcd(b,c) \cdot \gcd(c,a) $$is a perfect square. Prove that the product $$\operatorname{lcm}(a,b) \cdot \operatorname{lcm}(b,c) \cdot \operatorname{lcm}(c,a) $$is also a perfect square.
2021 Caucasus Mathematical Olympiad
A row of 2021 balls is given. Pasha and Vova play a game, taking turns to perform moves; Pasha begins. On each turn a boy should paint a non-painted ball in one of the three available colors: red, yellow, or green (initially all balls are non-painted). When all the balls are colored, Pasha wins,  if there are three consecutive balls of different colors; otherwise Vova wins. Who has a winning strategy?
2021 Caucasus Mathematical Olympiad
An acute triangle $ABC$ is given. Let $AD$ be its altitude, let $H$ and $O$ be its orthocenter and its circumcenter, respectively. Let $K$ be the point on the segment $AH$ with $AK=HD$; let $L$ be the point on the segment $CD$ with $CL=DB$. Prove that line $KL$ passes through $O$.
2021 Caucasus Mathematical Olympiad
Let us call a set of positive integers nice, if its number of elements is equal to the average of all its elements. Call a number $n$ amazing, if one can partition the set $\{1,2,\ldots,n\}$ into nice subsets.



a) Prove that any perfect square is amazing.



b) Prove that there exist infinitely many positive integers which are not amazing.
2021 Caucasus Mathematical Olympiad
Integers from 1 to 100 are placed in a row in some order. Let us call a number large-right, if it is greater than each number to the right of it; let us call a number large-left, is it is greater than each number to the left of it. It appears that in the row there are exactly $k$ large-right numbers and exactly $k$ large-left numbers. Find the maximal possible value of $k$.
2021 Caucasus Mathematical Olympiad
Let $n\ge 3$ be a positive integer. In the plane $n$ points which are not all collinear are marked. Find the least possible number of triangles whose vertices are all marked.



(Recall that the vertices of a triangle are not collinear.)
2021 Caucasus Mathematical Olympiad
In an acute triangle $ABC$ let $AH_a$ and $BH_b$ be altitudes. Let $H_aH_b$ intersect the circumcircle of  $ABC$ at $P$ and $Q$. Let $A'$ be the reflection of $A$ in $BC$, and let $B'$ be the reflection of $B$ in $CA$. Prove that $A', B'$, $P$, $Q$ are concyclic.
2021 Caucasus Mathematical Olympiad
A triangle $\Delta$ with sidelengths $a\leq b\leq c$ is given. It appears that it is impossible to construct a triangle from three segments whose lengths are equal to the altitudes of $\Delta$. Prove that $b^2>ac$.
2021 Caucasus Mathematical Olympiad
4 tokens are placed in the plane. If the tokens are now at the vertices of a convex quadrilateral $P$, then the following move could be performed: choose one of the tokens and shift it in the direction perpendicular to the diagonal of $P$ not containing this token; while shifting tokens it is prohibited to get three collinear tokens.



Suppose that initially tokens were at the vertices of a rectangle $\Pi$, and after a number of moves tokens were at the vertices of one another rectangle  $\Pi'$ such that $\Pi'$ is similar to $\Pi$ but not equal to $\Pi $. Prove that  $\Pi$ is a square.
2021 Caucasus Mathematical Olympiad
An infinite table whose rows and columns are numbered with positive integers, is given. For a sequence of functions

$f_1(x), f_2(x), \ldots $ let us place the number $f_i(j)$ into the cell  $(i,j)$ of the table (for all $i, j\in \mathbb{N}$).

A sequence $f_1(x), f_2(x), \ldots $ is said to be {\it nice}, if all the numbers in the table are positive integers, and each positive integer appears exactly once. Determine if there exists a nice sequence of functions  $f_1(x), f_2(x), \ldots $, such that each $f_i(x)$ is a polynomial of degree 101 with integer coefficients and its leading coefficient equals to 1.