KöMaL 2025

Marine biologists are studying a new species of shellfish whose first generation consists of 100 shellfish, and their colony reproduces as follows: if a given generation consists of $N$ shellfish (where $5 \mid N$ always holds), they divide themselves into $N/5$ groups of 5 shellfish each. Each group collectively produces 15 offspring, who form the next generation. Some of the shellfish contain a pearl, but a shellfish can only contain a pearl if none of its direct ancestors contained a pearl. The value of a pearl is determined by the generation of the shellfish containing it: in the $n$-th generation, its value is $1 / 3^n$. Find the maximum possible total value of the pearls in the colony.

Let $O$ denote the origin and let $\gamma$ be the circle with center $(1,0)$ and radius 1 in the Cartesian system of coordinates. Let $\lambda$ be a real number from the interval $(0,2)$, and let the line $x=\lambda$ intersect the circle $\gamma$ at points $P$ and $Q$. The lines $OP$ and $OQ$ intersect the line $x=2-\lambda$ at the points $P'$ and $Q'$, respectively. Let $\mathcal{G}$ denote the locus of such points $P'$ and $Q'$ as $\lambda$ varies over the interval $(0,2)$. Prove that there exist points $R$ and $S$ different from the origin in the plane such that for every $A\in \mathcal{G}$ there exists a point $A'$ on line $OA$ satisfying $A'R^2=(A'S-OS)^2=A'A\cdot A'O.$

Let $n$ be a given positive integer. Ana and Bob play the following game: Ana chooses a polynomial $p$ of degree $n$ with integer coefficients. In each round, Bob can choose a finite set $S$ of positive integers, and Ana responds with a list containing the values of the polynomial $p$ evaluated at the elements of $S$ with multiplicity (sorted in increasing order). Determine, in terms of $n$, the smallest positive integer $k$ such that Bob can always determine the polynomial $p$ in at most $k$ rounds.

The world famous infinite hotel with infinitely many floors (where the floors and the rooms on each floor are numbered with the positive integers) is full of guests: each room is occupied by exactly one guest. The manager of the hotel wants to carpet the corridor on each floor, and an infinite set of carpets of finite length (numbered with the positive integers) was obtained. Every guest marked an infinite number of carpets that they liked. Luckily, any two guests living on a different floor share only a finite number of carpets that they both like. Prove that the carpets can be distributed among the floors in a way that for every guest there are only finitely many carpets they like that are placed on floors different from the one where the guest is.

In a room, there are $n$ lights numbered with positive integers $1$, $2$, $\ldots$, $n$. At the beginning of the game subsets $S_1$, $S_2$, $\ldots$, $S_k$ of $\{1,2,\ldots,n\}$ can be chosen. For every integer $1\leq i\leq k$, there is a button that turns on the lights corresponding to the elements of $S_i$ and also a button that turns off all the lights corresponding to the elements of $S_i$. For any positive integer $n$, determine the smallest $k$ for which it is possible to choose the sets $S_1$, $S_2$, $\ldots$, $S_n$ in such a way that allows any combination of the $n$ lights to be turned on, starting from the state where all the lights are off.

Let $A'B'C'$ denote the reflection of scalene and acute triangle $ABC$ across its Euler-line. Let $P$ be an arbitrary point of the nine-point circle of $ABC$. For every point $X$, let $p(X)$ denote the reflection of $X$ across $P$. a) Let $e_{AB}$ denote the line connecting the orthogonal projection of $A$ to line $BB'$ and the orthogonal projection of $B$ to line $AA'$. Lines $e_{BC}$ and $e_{CA}$ are defined analogously. Prove that these three lines are concurrent (and denote their intersection by $K$). b) Prove that there are two choices of $P$ such that lines $Ap(A')$, $Bp(B')$ and $Cp(C')$ are concurrent, and the four points $p(A)p(A')\cap BC$, $p(B)p(B')\cap CA$, $p(C)p(C')\cap AB$, and $K$ are collinear.

In triangle $ABC$, interior point $D$ is chosen such that triangle $BCD$ is equilateral. Let $E$ be the isogonal conjugate of point $D$ with respect to triangle $ABC$. Define point $P$ on the ray $AB$ such that $AP = BE$. Similarly, define point $Q$ on the ray $AC$ such that $AQ = CE$. Prove that line $AD$ bisects segment $PQ$.

Let the irrational number $\alpha={1-\frac1{2a_1-\frac1{2a_2-\frac1{2a_3-\frac1{\dots}}}}}$, where coefficients $a_1$, $a_2$, $\ldots$ are positive integers, infinitely many of which are greater than $1$. Prove that for every positive integer $N$ at least half of the numbers $\lfloor \alpha \rfloor, \lfloor 2\alpha \rfloor,\ldots , \lfloor N\alpha \rfloor$ are even. $\lfloor x \rfloor$ denotes the floor function of $x$, which is the greatest integer less than or equal to $x$.

Let $n$ be a given positive integer. Luca, the lazy flea sits on one of the vertices of a regular $2n$-gon. For each jump, Luca picks an axis of symmetry of the polygon, and reflects herself on the chosen axis of symmetry. Let $P(n)$ denote the number of different ways Luca can make $2n$ jumps such that she returns to her original position in the end, and does not pick the same axis twice. (It is possible that Luca's jump does not change her position, however, it still counts as a jump.) a) Find the value of $P(n)$ if $n$ is odd. b) Prove that if $n$ is even, then $P(n)=(n-1)!\cdot n!\cdot \sum_{d\mid n,d\in \mathbb{N}}\left(\varphi\left (\dfrac{n}{d}\right) \cdot \binom{2d}{d} \right)$, where $\varphi(k)$ denotes the number of positive integers not greater than $k$ that are coprime to $k$.

We say that a strictly increasing sequence of positive integers $n_1$, $n_2$, $\ldots$ is non-decelerating if $n_{k+1}-n_k\le n_{k+2}-n_{k+1}$ holds for all positive integers $k$. We say that a strictly increasing sequence $n_1$, $n_2$, $\ldots$ is convergence-inducing , if the following statement is true for all real sequences $a_1$, $a_2$, $\ldots$: if subsequence $a_{m+n_1}$, $a_{m+n_2}$, $\ldots$ is convergent and tends to 0 for all positive integers $m$, then sequence $a_1$, $a_2$, $\ldots$ is also convergent and tends to 0. Prove that a non-decelerating sequence $n_1$, $n_2$, $\ldots$ is convergence-inducing if and only if sequence $n_2-n_1$, $n_3-n_2$, $\ldots$ is bounded from above.

Let $\mathcal{V}_c$ denote the infinite parallel ruler with the parallel edges being at distance $c$ from each other. The following construction steps are allowed using ruler $\mathcal{V}_c$: $\bullet$ the line through two given points; $\bullet$ line $l'$ parallel to a given line $l$ at distance $c$ (there are two such lines, both of which can be constructed using this step); $\bullet$ for given points $A$ and $B$ with $|AB|\ge c$ two parallel lines at distance $c$ such that one of them passes through $A$, and the other one passes through $B$ (if $|AB|>c$, there exists two such pairs of parallel lines, and both can be constructed using this step). On the perimeter of a circular piece of paper three points are given that form a scalene triangle. Let $n$ be a given positive integer. Prove that based on the three points and $n$ there exists $C>0$ such that for any $0<c\le C$ it is possible to construct $n$ points using only $\mathcal{V}_c$ on one of the excircles of the triangle. We are not allowed to draw anything outside our circular paper. We can construct on the boundary of the paper; it is allowed to take the intersection point of a line with the boundary of the paper.

2025 light bulbs are operated by some switches. Each switch works on a subset of the light bulbs. When we use a switch, all the light bulbs in the subset change their state: bulbs that were on turn off, and bulbs that were off turn on. We know that every light bulb is operated by at least one of the switches. Initially, all lamps were off. Find the biggest number $k$ for which we can surely turn on at least $n$ light bulbs.

Let $n \geq 2$ be an integer. Little Red Riding Hood and the wolf are playing a game in which the numbers $\{1, 2, \ldots, n\}$ are being colored red and blue. In each round, Little Red Riding Hood selects a pair $(i, j)$ such that $1 \leq i < j \leq n$, and both $i$ and $j$ are still uncolored. Then the wolf colors one of them blue and the other red. The game ends when all but at most one of the numbers from $\{1, 2, \ldots, n\}$ have been colored. Show that there exists a constant $c>0$ independent of $n$, and a strategy for Little Red Riding Hood such that she can ensure that at some point during the game, there exist numbers $1 \leq a \leq b \leq n$ such that the set $\{a, a+1, \ldots, b\}$ contains at least $c \cdot \sqrt[3]{n}$ more red numbers than blue numbers.

Prove that for any positive integer $N$, there exists a positive integer $n>2$ such that the difference between any two distinct prime factors of $n^3-1$ is at least $N$.

For which integers $n \ge 4$ is it true that the area of the regular $n$-gon is equal to the product of two of its diagonals?

Let $ABC$ be a triangle, and let the symmedian through $A$ intersect the circumcircle of the triangle at a second point $A'$. Prove that the following three points lie on a line:

• the reflection of point $A$ over the line $BC$;
• the inverse image of point $A$ under inversion with respect to the Feuerbach circle of triangle $ABC$;
• the midpoint of the chord $AA'$.

Let $n$ be a positive integer. Marci writes down the first $n$ positive integers in some order in his notebook, which we cannot see. Let this order be $m(1),m(2),\ldots,m(n)$; we would like to determine this order. In one step we may also list the first $n$ positive integers for Marci in some order, let this be $a(1),\ldots,a(n)$. Then Marci, in his notebook (which we cannot see), draws a directed graph on $n$ vertices, labelled with $1,2,\ldots,n$, respectively. After this, for every integer $1\leq i\leq n$, he draws a directed edge from the vertex labelled $i$ to the vertex labelled $m(a(i))$. Finally, the resulting graph with $n$ edges decomposes into a disjoint union of directed cycles (each of which may consist of a single loop), and Marci tells us the number of these cycles. a) Prove that the secret permutation can be determined in at most $n\log_2(n)$ steps. b) Does there exist a constant $c<1$ such that for every positive integer $n$ the secret permutation can be determined in at most $cn\log_2(n)$ steps?

We write the ordered pair $(1,0)$ on the board. In one step, if the ordered pair $(a,b)$ is on the board, we erase it and replace it with either $(a+b,b)$ or $(2a+b,a+b)$. (For example, after two steps, the pair on the board may be one of $(1,0)$, $(2,1)$, $(3,1)$, or $(5,3)$.) Prove that after $n$ steps, the ordered pair on the board can take exactly $2^n$ different values.

Given a circle with three points $A$, $B$, and $C$ on it, which do not form an isosceles triangle. For every point $P\notin\{A,B,C\}$ on the circle, let $A_P$, $B_P$ and $C_P$ denote the intersections of the tangent at $P$ with the tangents at $A$, $B$, and $C$, respectively. Prove that there exist exactly three points $P\notin\{A,B,C\}$ on the circle for which the points $A_P$, $B_P$ and $C_P$ are well-defined and the perpendiculars from $A_P$ to $BC$, from $B_P$ to $CA$, and from $C_P$ to $AB$ are concurrent. Furthermore, show that these three points $P$ form an equilateral triangle.

Let $a \geq 3$ be an integer, and define $f(n) = a^n - 1$ for every positive integer $n$. Denote by $f^{(k)}$ the $k$-iterate of $f$, that is, $f^{(1)}(n) = f(n)$ and $f^{(k+1)}(n)=f(f^{(k)}(n))$ for $k \geq 1$. a) Prove that for any positive integer $K$ there exists a positive integer $M$ such that for every integer $1 \leq k \leq K$, the number $f^{(k)}(M)$ is divisible by $M$ if and only if $k$ is divisible by $2025$. b) Does there exist a positive integer $N$ such that for every positive integer $k$, the number $f^{(k)}(N)$ is divisible by $N$ if and only if $k$ is divisible by $2025$?

Let a set $S$ of complex numbers be called symmetric if the complex conjugate of every element of $S$ also belongs to $S$. For each positive integer $n$, determine the largest positive integer $K_n$ for which there exists an $n$-element symmetric set $S = \{z_1, z_2, \ldots, z_n\}$ not containing $0$, such that $z_1^k + z_2^k + \ldots + z_n^k \leq 0$ holds true for every integer $1 \leq k \leq K_n$.

Given a tree with $n \ge 2$ vertices labelled $v_1$, $v_2$, $\dots$, $v_n$. Each vertex hosts a dwarf, and every dwarf has some number of coins, at least $n$. We then go through the vertices in the order of increasing indices: the dwarf on the current vertex takes one coin from its richest neighbour; if there are several richest neighbours, he takes one coin from each of them. Determine, as a function of $n$, the smallest integer $k$ such that for every tree with $n$ vertices there exists an initial coin distribution in which the numbers of coins of any two dwarfs differ by at most $k$, and after the process every dwarf ends up with exactly the same number of coins as at the beginning.

Let $\mathcal{P}$ be a convex cyclic polygon with at least four vertices such that no two diagonals of $\mathcal{P}$ are congruent. Prove that there is at most one triangulation of $\mathcal{P}$ where all triangles have the same perimeter.

Given a non-isosceles triangle $ABC$, let its circumcircle be denoted by $\Omega$. Let $\omega_A$ be the mixtilinear incircle corresponding to vertex $A$ (that is, the circle tangent to sides $AB$, $AC$, and internally tangent to $\Omega$). Define $\omega_B$ and $\omega_C$ analogously. Let the points where $\omega_A,\omega_B,\omega_C$ touch $\Omega$ be denoted by $T_A, T_B, T_C$, respectively. Show that the Monge line of the circles $\omega_A,\omega_B,\omega_C$ (that is, the line passing through the pairwise external homothety centers of the three circles) coincides with the Pascal line of the hexagon $AT_CBT_ACT_B$ (that is, the line passing through the points $AT_C \cap T_AC$, $T_CB \cap CT_B$, $BT_A \cap T_BA$).

Let $n \ge 2$ be an integer. Let $x_1,\ldots,x_n$ be positive real numbers whose product is $n-1$. Prove that $x_1+\ldots+x_n-\left(\frac{1}{x_1^{n-1}}+\ldots+\frac{1}{x_n^{n-1}}\right)<(n-1)^{1+\frac{1}{n}}.$

Fix two positive integers $a$ and $b$. Let $n$ be a positive integer, and let $T$ be a black-white coloring of the cells of an $an\times bn$ grid, containing at least two white cells. Csigusz the snail starts on a white cell, visits all white cells exactly once, and then returns to the starting cell, always moving between side-adjacent white cells. He then notices that he was able to do this in exactly one way (that is, once he made his first move, there was a unique way to complete the cycle). Let $\mathcal{T}_n$ denote the set of all colorings $T$ satisfying this property, and let $\phi(T)$ denote the number of white lattice points in $T$. Show that $L=\lim_{n\to \infty}\frac{\max_{T\in \mathcal{T}_n} \phi(T)}{abn^2}$ exists for every choice of positive integers $a$ and $b$, and determine the value of $L$.

At most how many positive integers can be written on the circumference of a circle such that the product of any two neighbouring numbers is less than $2025$?

Let $k$ denote the incircle of triangle $ABC$, and let $I$ denote its center. Line $AI$ intersects circle $k$ in points $D$ and $E$ such that $D$ is closer to $A$ than $E$. Let $k$ be tangent to $AC$ at $F$. Let the tangent of $K$ at point $E$ intersect $AC$ at $G$. Prove that $GI$ is parallel to $FD$.

Is it possible to find real numbers $a$, $b$ and $c$ such that none of the functions below has a root? $$\begin{gather*} p_1(x)=ax^2-(b^2+1)x+c,\\ p_2(x)=bx^2-(c^2+1)x+a,\\ p_3(x)=cx^2-(a^2+1)x+b. \end{gather*}$$

Prove that in any pyramid with a quadrilateral base the line segments connecting the centroids of the triangular faces to the midpoints of the opposite edges of the base are concurrent.

For which positive integers $m>1$ does there exist polynomial $f(x)$ with integer coefficients such that exactly one of $k$ and $f(k)$ is divisible by $m$? (5 points)

The incircle of triangle $ABC$ is tangent to the sides $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Let $P$ be the point on the segment $AD$ for which $2\cdot AP=PD$, and define points $Q$, $R$ on the segments $BE$ and $CF$, respectively in a similar manner. Show that at least one of the points $P$, $Q$, $R$ lies on or inside the incircle.

Let $p$ be an odd prime. Let $d_k(n)$ denote the number of positive divisors of $n$ with remainder $k$ modulo $p$ ($0\leq k<p$). Let $A\subseteq\{0,1,\dots,p-1\}=S$ satisfy $\sum_{k\in A}d_k(n)\geq\sum_{j\in S\setminus A}d_j(n)$ for every positive integer $n$. Find the minimum number of elements of $A$.

Let $ABC$ be a triangle satisfying $BC<AC$, and let $D$ be a point on side $AC$ satisfying $BC=DC$. Let circle $\Gamma$ be tangent to arc $AC$ not containing $B$ of the circumcircle of $ABC$ at point $X$, and to line segment $AC$ at $D$. Let the tangent line from $B$ to $\Gamma$ that is closer to $C$ intersect $AC$ at $Y$, and be tangent to $\Gamma$ at $Z$. Prove that if $A$, $Y$, $X$ and the circumcenter of triangle $ABC$ are concyclic, then $AZ=CZ$.

How many different results can we get by adding together two different $n$-digit numbers, if each digit in both numbers can only be $4$ or $7$?

Rectangle $ABCD$ satisfies $AD<AB<2AD$. Let point $O$ be chosen on side $AB$ satisfying $OB=AD$. The circle with center $O$ and radius $OB$ intersects side $AD$ at point $E$. Prove that the area of rectangle $ABCD$ equals $BE^2/2$.

We extended the sides of a triangle in both directions along the lines of the sides, and at each vertex the length of the two extensions equals the length of the side opposite the vertex. Prove the resulting six points are concyclic.

Let $\alpha$, $\beta$ and $\gamma$ denote the angles of a triangle. Prove that $\sin \frac{\alpha}{2}+\sin \frac{\beta}{2}+\sin \frac{\gamma}{2} \geq \cos \alpha+\cos \beta+\cos \gamma$.

Let $n$, $k$ and $\ell$ denote positive integers, and let $g(n,k,\ell)$ denote the number of ways that an $n\times k$ table can be filled with the elements of $\{1,2,\ldots, {\ell+1}\}$ such that in each row from left to right and in each column from top to bottom the numbers are monotonously increasing. Prove that $g(n,k,\ell)=g(k,\ell,n)$.

For an arbitrary positive integer $n$ let $a_n$ denote the number of positive perfect powers that are at most $n$ (for example, $a_9=4$). We call $n$ `interesting', if $a_n \mid n$. Prove that there exists infinitely many interesting positive integers.

In cyclic hexagon $ABCDEF$ let $P$ denote the intersection of diagonals $AD$ and $CF$, and let $Q$ denote the intersection of diagonals $AE$ and $BF$. Prove that if $BC=CP$ and $DP=DE$, then $PQ$ bisects angle $BQE$.

Decide whether the following statement is true: if an infinite arithmetic sequence of positive integers includes both a perfect square and a perfect cube, then it also includes a perfect $6^{\text{th}}$ power.

What can be the value of $\ell$, if up to congruence there are exactly three triangles with a side of unit length, another side of length $\ell$ and an angle of $60$ degrees?

The sum of positive real numbers $x$, $y$ and $z$ is $2025$. Find the smallest possible value of $x^2+y^2+z^2+20x+2y+5z$.

We have two congruent regular $n$-gon based pyramids. On the lateral faces of both pyramids, the numbers $1$, $2$, $\ldots$, $n$ are written in a random order. For which values of $n$ is it guaranteed that the bases of the pyramids can always be glued together in such a way that at least two edges of the resulting bipyramid satisfy the property that the two numbers on either side of the edges are equal to each other?

Find all pairs of positive integers $(a,b)$ satisfying $a^6=2b^2-1$.

Let positive integer $d$ be called a block-divisor of positive integer $n$ if $d\mid n$ and moreover, $d$ and $\frac{n}{d}$ are coprimes. Let $B(n)$ denote the sum of the block-divisors of $n$. Find all positive integers $n$ such that $n$ has at most three distinct prime factors and $B(n)=2n$.

Convex quadrilateral $ABCD$ satisfies properties $\angle BAC=2\angle CAD$, $\angle ADB=2\angle DBA$, and $\angle CBD=30^{\circ}$. What can be the measure of $\angle DCA$?

For positive integer $n\geq 3$ let points $P_1$, $P_2$, $\ldots$, $P_n$ be chosen on a parabola with focus $F$ such that $\angle P_1FP_2=\angle P_2FP_3=\ldots=\angle P_nFP_1=\frac{360^\circ}{n}$. Prove that the harmonic mean of the distances $FP_1$, $\ldots$, $FP_n$ eaquals the parameter of the parabola.

Positive integers $a_1$, $a_2$, $a_3$, $\ldots$, $a_{2024}$, $a_{2025}$ are chosen such that fractions $\frac{a_1}{a_2}$, $\frac{a_2}{a_3}$, $\frac{a_3}{a_4}$, $\ldots$, $\frac{a_{2024}}{a_{2025}}$ have pairwise different values. At least how many of the numbers must be different?

Let $P$ be the intersection of lines $AB$ and $CD$, where $ABCD$ is a convex quadrilateral. Let points $M$ and $N$ be the midpoints of the diagonals. Prove that the area of triangle $PMN$ equals one quarter of the area of quadrilateral $ABCD$.

Which can be the smallest and the largest of numbers $a^{\log_b c}$, $b^{\log_c a}$, $c^{\log_a b}$, provided that $1<a<b<c$?

The angles of a triangle are $\alpha \leq \beta \leq \gamma$, and its longest side is of unit length. Prove that the triangle is obtuse if and only if the reciprocal of its area is more than $2\left(\tan\alpha +\tan\beta\right)$.

Let $I$ be the incenter of triangle $ABC$, and let $AB<AC$. The perpendicular to $AI$ passing through $I$ intersects line $BC$ at $P$, and line segment $AP$ intersects the circumcircle of triangle $ABC$ at $Q$ for the second time. Prove that $IQ$ is perpendicular to $AP$.

Find all functions $f\colon\mathbb{Q}\rightarrow\mathbb{Q}$ satisfying $f(x)+f(y)=\frac{f(x+2y)+f(2x-y)}{5}$ for all rational numbers $x$, $y$.

In acute triangle $ABC$ let $D$, $E$ and $F$ denote the feet of the altitudes from $A$, $B$ and $C$, respectively. Let line $DE$ intersect circle $ABC$ at points $G$, $H$. Similarly, let line $DF$ intersect circle $ABC$ at points $I$, $J$. Prove that the radical axis of circles $EIJ$ and $FGH$ passes through the orthocenter of triangle $ABC$.

Prove that for any positive integers $a$, $b$, $c$ and $d$ there exists infinitely many positive integers $n$ for which $a^n+bc$ and $b^{n+d}-1$ are not relatively primes.

Abel has written down the positive integers from 1 to $n$ in a certain order. Any positive integer from 1 to $\frac{n(n+1)}{2}$ can be obtained as the sum of some consecutive terms from Ábel's sequence. For which values of $n$ is this possible?

The square $ABCD$ of positive orientation is given. Construct an equilateral triangle $BEF$ of positive orientation such that its vertex $F$ is on ray $BC$ and $\angle EAF=45^\circ$.

Let $k>1$ be a given integer. Prove that the sequence $a_n=n^2+n+1$ has infinitely many terms that equal the product of $k$ distinct terms of the same sequence.

Let $AA'$, $BB'$, and $CC'$ be diameters in the circumcircle of the non-equilateral acute triangle $ABC$. Let $A_1$ be the reflection of point $A'$ across line $BC$, $B_1$ be the reflection of $B'$ across line $AC$, and $C_1$ be the reflection of $C'$ across line $AB$. Prove that triangle $A_1B_1C_1$ is similar to triangle $ABC$.

The side lengths of a triangle are $a$, $b$ and $c$. Prove that $\frac{(b+c)^2}{a^2+bc}+\frac{(c+a)^2}{b^2+ca}+\frac{(a+b)^2}{c^2+ab}\ge 6$.

Prove that for any odd positive integers $a$ and $b$ the remainder of $(a+b+7)^7-a^7-b^7$ modulo 168 is 7.

Let $r$ be the inradius of the acute triangle $ABC$, let $R$ be the circumradius of the same triangle, and let the inradius of the orthic triangle of the triangle $ABC$ be $\varrho$. Prove that $r \ge \sqrt{R\varrho}$.

There are three cities in Bergengócia, $A$, $B$, and $C$, inhabited by a total of 2025 people. Starting from the current year, at the end of each year, half of the inhabitants of $A$ move to $B$, a third of the inhabitants of $B$ move to $C$, and a quarter of the inhabitants of $C$ move to $A$ (if any of the values of these fractions is not an integer, we round the value up). Prove that after a few years, the number of inhabitants of all three cities will remain unchanged.

1001, 2002 and 3003 are three consecutive elements of the $14^{\text{th}}$ row of Pascal's triangle. Does there exist another row where three consecutive elements are of the form $n$, $2n$ and $3n$ for some positive integer $n$?

At most how many positive integers can be chosen from the first 50 positive integers such that no two have a product which is a perfect $4^\text{th}$ power?

Convex quadrilateral $ABCD$ satisfies property $AB=BC=CD$. Prove that if $\angle BCD=2\angle DAB$, then $\angle ABC=2\angle CDA$.

The sum of some distinct positive integers is 1000. What is the largest possible value of their product?

Prove that squares with side lengths 3, 4, 5, 7, 8, 9, 11, 15, 16, 17, 19 and 23 cannot be arranged in a $45\times 45$ square without overlap.

Show an example of a regular polygon, the area of which equals the product of two of its diagonals.

Each of Ali and Baba has a random number generator (RNG). Ali's RNG chooses one of numbers 1, 2, $\ldots$, 100 with uniform distribution (each number is chosen with the same probability), and Baba's RNG chooses one of numbers 0, 1, 2, $\ldots$, 100 with uniform distribution. Ali calculated the expected number of times he has to use his RNG to get a sum which is at least 101, and Baba calculated the expected number of times he has to use his RNG to get a sum which is at least 100. Whose number is bigger?

Let $S$ denote the centroid of triangle $ABC$. Let $T$ denote the foot of the height from vertex $A$ in triangle $ABC$. Ray $TS$ (starting from point $T$) intersect the circumcircle of the triangle in $V$. Prove that $S$ trisects line segment $TV$.

Let $d$ be a given positive integer. Prove that there exist infinitely many pairs $(x,y)$ of positive integers such that the difference of the arithmetic and geometric mean of $x$ and $y$ is $d$.

In triangle $ABC$ angle $\angle ACB$ is $90^{\circ}$. Let $F$ denote the midpoint of leg $AC$, and let $G$ denote the midpoint of hypotenuse $BC$. Prove that the circle with diameter $FG$ is tangent to the incircle of triangle $ABC$ if and only if $AC=2BC$.

Prove the following divisibility for every positive integer $n$: $(n!)^n \mid n^2 \cdot (n^2-1) \cdot (n^2-2) \cdot \ldots \cdot (n+2) \cdot (n+1).$

We have selected an odd number of integers. We have changed one of them such that their average increased by 1, but their standard deviation did not change. Prove that the average of the originally selected numbers is an integer.

A deck of $2n$ cards consists of $n$ suits, each suit consisting of two values, 1 and 2. $n$ players play the following cooperative game: They start by shuffling the $2n$ cards, then each player is dealt two cards, which they reveal to the other players. Subsequently, they decide together the order they sit down around a round table, and they also choose the player starting the game. During the game, each consecutive player (in the clockwise direction) tries to put down a card, however, a card with value 2 can only be put down if the card of the same suit with value 1 is already on the table. They win the game if they manage to put all the cards on the table, however, they lose the game if somebody holding at least one card in their hand cannot put down a card on the table. Prove that the players can always win the game, regardless of the hand they were dealt.

Let $M$ denote the orthocenter of acute triangle $ABC$, and let $F$ denote the midpoint of side $BC$. Let $T$ denote the foot of the perpendicular from $M$ to the inner angle bisector at vertex $A$. Line $FT$ intersects side $AC$ at point $D$. Prove that the circumcircle of triangle $CDF$ contains the foot of the height of triangle $ABC$ from vertex $A$.

Prove that there exists a rational number arbitrarily close to $\frac1{2025}$ that cannot be obtained as the sum of the reciprocals of 2025 positive integers.

Let $E$ be the intersection of the diagonals of convex quadrilateral $ABCD$. Suppose that $AB=CD$, and the areas of triangles $ABE$ and $CDE$ are equal. Prove that the quadrilateral is a parallelogram or an isosceles trapezoid.

Positive numbers $a_1$, $a_2$, $\dots$, $a_{2025}$ satisfy $a_1=1$ and $\frac{1}{a_1+a_2}+\frac{1}{a_2+a_3}+\ldots+\frac{1}{a_{n-1}+a_n}=a_n-1$ for every $2\le n\le 2025$. Find the value of $a_{2025}$.

Is it possible to color the points of the plane with three given colors such that each color is used at least once, and if the vertices of a triangle are colored with a color chosen from only two of the three given colors, then every point inside and on the perimeter of this triangle is also colored with one of these two colors?

In right triangle $ABC$ the angles are the following: $\angle ABC=15^\circ$ and $\angle CAB=75^{\circ}$. Let $F$ be the midpoint of hypotenuse $AB$. Let point $D$ be chosen on leg $BC$ such that $BD=CA$, and let point $E$ be chosen on ray $CA$ beyond point $A$ such that $CE=BC$. Let $M$ be the point of intersection of lines $BE$ and $CF$. Prove that lines $DM$ and $CM$ are tangent to the circumcircle of triangle $AEF$.

Prove that there exist infinitely many positive integers $n$ such that numbers ${2^n-2025}$, ${2^n-2024}$, $\dots$, ${2^n+2025}$ are all composite.

Does there exist a set $H$ of polynomials with integer coefficients and degree at least two such that every integer value is taken by exactly one polynomial in $H$ at an integer place?

Kornél thinks about a closed subinterval of $I=[0, 2^k]$ (where $k$ is a positive integer) with integer endpoints and length at least $1$. Kristóf can ask the following question: he can choose an arbitrary closed subinterval with integer length, but not necessarily integer endpoints, and Kornél tells him the length of the intersection of the interval he picked and the interval chosen by Kristóf. (The answer is $0$ if the intersection of the two intervals is empty or consists of a single point.) Find the smallest number of questions with which Kristóf can guess the interval chosen by Kornél in all cases.

Show a function $f$ that assigns a non-negative value to every vector in the plane satisfying the following: for an arbitrary triangle $ABC$ with vertices on hyperbola $x^2-y^2=1$ the area of the triangle equals $f{(\overrightarrow{AB})\cdot f(\overrightarrow{BC})\cdot f(\overrightarrow{CA})}$.

Anna and Béla plays the following game on a $2\times 2026$ board. They take turns to place dominoes on the board until no new domino can be placed on the board. If the board is fully covered, the winner is Anna, otherwise Béla is the winner. Who has a winning strategy, if Anna is the first player? (A domino covers two adjacent squares on the board. The dominoes cannot extend beyond the edges of the board and cannot overlap each other.)

The legs of a right triangle are $a$ and $b$, its hypotenuse is $c$, and its inradius is $r$. Prove that $2r^2=(c-a)(c-b).$

Let $p(r)$ denote the product of the divisors of positive integer $r$. For a given positive integer $n$ determine those positive integers $k$ for which $p(k^n)$ is the $n^{\text{th}}$ power of an integer.

Circular arc $k$ with endpoints $AB$ is given. Let $F$ be the midpoint of the arc. A circle is tangent internally to $k$ at point $P$, and tangent internally to line segment $AB$ at point $Q$. Prove that the sum of the radii of circle $APQ$ and $BPQ$ equals the length of line segment $AF$.

Let $n\geq 2$ be a given positive integer. For which permutation $a_1,a_2,\dots,a_n$ of numbers $1,2,\dots,n$ will the value of $a_1a_2+a_2a_3+\dots+a_{n-1}a_n$ be maximal?

Each of the vertices of a regular 45-gon are colored red, yellow or green, and each color is used exactly 15 times. A triangle is red (yellow, green), if all three of its vertices is red (yellow, green). Prove that there exists a red, a yellow and a green triangle, all three of which are congruent to each other.

Find the smallest possible value of $|a|+|b|+|c|+|d|+|e|$, if equation $x^{6}+ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+1=0$ has a real root.

The incircle of triangle $ABC$ is tangent to sides $BC$, $CA$ and $AB$ at points $D$, $E$ and $F$, respectively. Let $G$ be the antipode of point $F$ on the incircle. Lines $GD$ and $GE$ intersect line $AB$ at points $D'$ and $E'$, respectively. Prove that circles $AED'$ and $BDE'$ intersect each other on the incircle.

We have written numbers 1, 2, 3, $\ldots$, 2024, 2025 on the blackboard. Farkas and Piroska take turns deleting the numbers from the blackboard until only two numbers remain on the blackboard. Piroska wins if the sum of the last two numbers is divisible by 11. How can Piroska win this game, if she starts the game? (Based on a German competition problem)

Two parallel lines and two points, $A$ and $B$ on one of them are given. Construct the midpoint of line segment $AB$ with only a straightedge.

Let $-20.25<K<0$. Solve the following system of equations for real numbers: $$\begin{gather*} x+y+z=K,\\ x^3+y^3+z^3=1,\\ xyz=-2024. \end{gather*}$$

In the convex quadrilateral $PQRS$, $QR=4$, $RS=6$, $SP=5$, and the internal angles at vertices $P$ and $Q$ are $60^{\circ}$. Knowing that $2PQ=a+\sqrt{b}$, where $a$ and $b$ are positive integers, find the value of sum $a+b$.

Solve equation $9^x+(6x-23)\cdot 3^x+5x^2-39x+76=0$ for real numbers.