KöMaL 2026

$2026$ people visited an exhibition where $1000$ paintings were displayed. Prove that it is possible to send some of the visitors to two rooms, with at least one visitor in each room, such that there is no painting that was liked by someone in one room but by nobody in the other, and there is no painting whose painter is personally known by someone in one room but by nobody in the other.

For which pairs $(k,l)$ of positive integers is the following statement true: for every positive integer $n$ there exists an integer $i$ with $0\le i\le l-1$ such that $\binom{n}{i}+{(-1)}^l\binom{n}{i+l}+{(-1)}^{2l}\binom{n}{i+2l}+\dots$ is not divisible by $k$?

Call four points to be in general position if they are pairwise distinct, no three of them are collinear, and no two of the six lines determined by them are parallel. Let $A$, $B$, $C$, $D$ be points in general position lying on a circle. Let $E$ be the intersection point of lines $AB$ and $CD$, let $F$ be the intersection point of lines $AC$ and $BD$, and let $G$ be the intersection point of lines $AD$ and $BC$. In triangle $EFG$, denote by $P$, $Q$, and $R$ the feet of the altitudes corresponding to the vertices $E$, $F$, and $G$, respectively. a) Show that the triples of lines $AR$-$BQ$-$CP$, $AQ$-$BR$-$DP$, $AP$-$CR$-$DQ$, and $BP$-$CQ$-$DR$ are either concurrent or parallel. b) Suppose that each of the four triples of lines is concurrent, and denote their points of concurrency by $X$, $Y$, $Z$, and $W$, respectively. Assume further that these points are in general position. Prove that the Miquel points of quadrilaterals $XWYZ$, $XYZW$, and $XZWY$ are precisely the points $P$, $Q$, and $R$.

Let $A=\{[\frac{3+\sqrt{5}}{2}\cdot n]\colon n=1,2,3,\ldots\}=\{2,5,7,10,13,15,\ldots\}.$ Two players, First and Second, play a game with a pile of $N$ coins according to the following rules. The players move alternately, with First moving first. On each move, the player on turn chooses a number $k\in A$ and removes $k$ coins from the pile. A player who is unable to make a legal move loses the game. Determine, in terms of $N$, which player has a winning strategy, and describe such a winning strategy.

Let $ABCDEF$ be a bicentric hexagon, that is a hexagon which is both cyclic and tangential. Assume that there exists a point $P$ such that line $AP$ is perpendicular to line $BF$, $CP$ is perpendicular to $BD$, and $EP$ is perpendicular to $DF$. Prove that there are two opposite sides of the hexagon whose sum is equal to the circumdiameter.

Let $a_0=0<a_1<a_2<\ldots<a_n$ be integers such that the sequence $b_k=\frac{a_{k+1}-a_k}{2k+1}$ ($k=0$, 1, $\ldots$, $n-1$) is non-decreasing. Suppose that $c_1$, $c_2$, $\ldots$, $c_n$ are real numbers such that the polynomial $1+\sum_{k=1}^n c_kx^{a_k}$ is divisible by the polynomial $(x+1)^n$. Show that $2>|c_1|>|c_2|>\ldots>|c_n|$.

Let $\mathcal{P}$ be a set of ten points in the plane. A subset $\mathcal{Q}\subseteq \mathcal{P}$ is said to be isolable if there exists a closed disk of positive integer radius that contains all points of $\mathcal{Q}$ but none of the points of $\mathcal{P}\setminus\mathcal{Q}$. Prove that at least one third of the five-element subsets of $\mathcal{P}$ are not isolable.

Does there exist a tetrahedron such that the incenters of its faces are coplanar?

Prove that for any integers $k>10$ and $n>k^4$, among the integers $({n+1^2})$, $({n+2^2})$, $({n+3^2})$, $\ldots$, $({n+k^2})$, there is at least one that has a prime divisor greater than $k$.

Let $ABC$ be an acute triangle with orthocenter $H$. Points $D$ and $E$ lie on the lines $AC$ and $AB$, respectively, such that the points $B$, $C$, $D$, $E$ are concyclic, and the line $DE$ bisects the side $BC$. Suppose that the lines $BD$ and $CE$ meet at $M$. Prove that the line $HM$ is perpendicular to the symmedian through $A$.

Let $a$, $b$, $c$ be real numbers in the interval $(1,4)$. Consider the following three inequalities: ${bx(x+y-z)}+{cx(x+z-y)}\geq {a(2yz+x)}$, ${cy(y+z-x)}+{ay(y+x-z)} \geq {b(2zx+y)}$, ${az(z+x-y)}+{bz(z+y-x)} \geq {c(2xy+z)}$. a) Determine, in terms of $a$, $b$, $c$, the maximum number of these inequalities that can hold simultaneously for positive real numbers $x$, $y$, $z$ satisfying $x+y+z=1$. b) Find an explicit formula for $x$, $y$, $z$ in terms of $a$, $b$, $c$, without using case distinctions, such that $x$, $y$, $z>0$, $x+y+z=1$, and none of the three inequalities holds.

Let $\mathcal{T}$ denote the set of countable, undirected trees. A subset $X$ of the real numbers is called bariton if every nonempty subset of $X$ has a least element. a) Prove that for every bariton $X$ there exists a function $f\colon X\to \mathcal{T}$ such that for all $y$, $z\in X$ we have $y\leq z$ if and only if $f(y)$ is isomorphic to a subgraph of $f(z)$. b) Does the statement remain true if we additionally require that for every $x\in X$, all vertices of $f(x)$ have finite degree? (A graph with vertex set $V$ is called countable if there exists an injective map $V\to\mathbb{N}$.)

A criminal starts at the origin and escapes by moving on the integer lattice points of the plane. At each step, they move to a neighboring lattice point, with the restriction that they never move in the same direction twice consecutively. It is known that their first move is upward, and that their path visits every lattice point exactly once. An agent also starts at the origin but cannot see the criminal. Fortunately, the headquarters can track the criminal's movement, but they can communicate with the agent only in the following way: a predetermined infinite set $S \subseteq \mathbb{N}$ is given, and for each $s \in S$, after the $s$-th step of the criminal, a message consisting of either $1$ or $2$ is sent to the agent. The agent moves only when a message is received, and always makes exactly as many steps as the criminal has taken since the last message. What is the value of $I=\inf_{S} \left(\lim_{n\rightarrow \infty} \frac{|S \cap \{1,2,3,\ldots,n\}|}{n} \right)$ taken over all sets $S$ for which the agent can uniquely reconstruct the movement of the criminal? Both the agent and the headquarters are aware of the rules restricting the criminal's movement.

Given an acute, scalene triangle $ABC$ in the plane. Denote its symmedian point by $K$ and the center of its Feuerbach circle by $N$. Construct, using a compass and straightedge, distinct points $X$, $X^*$, $Y$, $Y^*$ such that $\bullet$ the lines $XY$ and $X^*Y^*$ intersect at $K$; $\bullet$ the lines $XX^*$ and $YY^*$ intersect at $N$; $\bullet$ the points $X$ and $X^*$, as well as $Y$ and $Y^*$, are isogonal conjugates with respect to the triangle.

Let $P \in \mathbb{C}[x_1,\ldots,x_n]$ be an irreducible complex polynomial of degree at least 2. Assume that there exists an integer $M>1$ such that $P(x_1,\dots,x_n)\mid P(x_1^M,\dots,x_n^M)$. Prove that there exist a nonzero complex number $c$, a complex root of unity $\zeta$ and non-negative integers $a_1$, $\ldots$, $a_n$, $b_1$, $\dots$, $b_n$ such that $P(x_1,\dots,x_n)=c\left(x_1^{a_1}\cdots x_n^{a_n}-\zeta x_1^{b_1}\cdots x_n^{b_n}\right)$.

Prove that all divisors of the positive integer $1\underbrace{000\ldots00}_{2026}\underbrace{999\ldots99}_{2026}$ (written in base 10) end in a digit 1 or 9 (also in base 10).

In the acute and scalene triangle $ABC$ let $A_1$, $B_1$ and $C_1$ denote the feet of altitudes from $A$, $B$ and $C$, respectively. Let $A_2$, $B_2$ and $C_2$ denote the intersection of the internal angle bisectors from $A$, $B$ and $C$, respectively, with the opposite sides. Let $X_1$ denote the point of intersection of lines $A_1B_1$ and $A_2C_2$, and let $X_2$ denote the point of intersection of lines $A_1C_1$ and $A_2B_2$. Similarly, let $Y_1$ denote the point of intersection of lines $B_1C_1$ and $B_2A_2$ and $Y_2$ denote the point of intersection of lines $B_1A_1$ and $B_2C_2$, let $Z_1$ denote the point of intersection of lines $C_1A_1$ and $C_2B_2$, and finally let $Z_2$ denote the point of intersection of lines $C_1B_1$ and $C_2A_2$. Assume that all these points exist and are distinct. Prove that points $A$, $X_1$, $X_2$, points $B$, $Y_1$, $Y_2$ and points $C$, $Z_1$ and $Z_2$ are collinear, and these three lines are concurrent.

Let $p$ be a prime number and let $k$ be an integer with $1\le k\le p$. Suppose that $f(x)$ is a polynomial with integer coefficients such that its value at every integer point gives a remainder of $0$ or $1$ when divided by $p^k$, and both cases occur. Show that the degree of $f$ is at least $k(p-1)$.

The axes of three hyperboloids of revolution are pairwise perpendicular, and there is a point that lies on exactly two hyperboloids. Show that the common points of the three hyperboloids lie on a sphere or a plane.

Let $ABCD$ be a cyclic quadrilateral without right angles. Let $E$ denote the midpoint of diagonal $AC$, and let $F$ denote the midpoint of diagonal $BD$. Suppose that the circles $AEF$, $BEF$, $CEF$, $DEF$ do not degenerate into lines, and they intersect the circumcircle of $ABCD$ for the second time at $A'$, $B'$, $C'$, $D'$, respectively. Let $E'$ be the midpoint of segment $A'C'$, and let $F'$ be the midpoint of segment $B'D'$. Prove that points $E$, $F$, $E'$, $F'$ lie on the same circle.

Let $a_n$, $b_n$ and $c_n$ be sequences of real numbers such that $\vert a_n \vert$, $\vert b_n \vert$, $\vert c_n \vert \leq d$. Is there a real number $c>0$ such that there exists a sequence $s_n$ of signs (i.e. $s_n\in \{1, -1\}$ for all $n$) satisfying $\left\vert\sum_{n=1}^{N}s_na_n \right\vert, \left\vert\sum_{n=1}^{N}s_nb_n\right\vert, \left\vert\sum_{n=1}^{N}s_nc_n\right\vert \leq c$ for every positive integer $N$?

Let $A$ be a set of real numbers with $n$ elements. Prove that at least $4n-3$ numbers can be written in the form $a-2b+c$ such that $a$, $b$, $c\in A$ not necessarily distinct.

Let $P$ and $Q$ be the third vertices of the equilateral triangles on side $BC$ of triangle $ABC$. Prove that $AP^2+AQ^2=AB^2+BC^2+CA^2$.

Let $a$, $b$ and $c$ be real numbers such that not all of them are equal. Prove that $\frac{a+b+c}{3}>\sqrt[3]{abc}$ holds if and only if $\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}>0$.

Let the diagonal $AC$ and $BD$ of cyclic quadrilateral $ABCD$ intersect each other in point $P$. Let $K$ be the circumcenter of triangle $APB$, and let $M$ be the orthocenter of triangle $CPD$. Prove that points $K$, $M$ and $P$ are collinear.

Solve the following equation on the set of positive integers: $x^5-xy^2+y^2=1$.

Let us choose two different integers, $a$ and $b$ from interval $(n^2,n^2+n)$, for a given $n>2$ positive integer. Prove that it is not possible to find an integer number different from $a$ and $b$ that divides product $ab$.

Let $ABCD$ be a convex quadrilateral. a) Prove that if $\tan\angle BAC\cdot\tan\angle DCA =\tan\angle CAD\cdot\tan\angle ACB$, then $\tan\angle CBD\cdot\tan\angle ADB=\tan\angle DBA\cdot\tan\angle BDC$. b) Prove that if $\tan\frac{\angle BAC}{2}\cdot\tan\frac{DCA}{2}={\tan\frac{\angle CAD}{2}\cdot\tan\frac{\angle ACB}{2}}$, then ${\tan\frac{\angle CBD}{2}\cdot\tan\frac{\angle ADB}{2}}={\tan\frac{\angle DBA}{2}\cdot\tan\frac{\angle BDC}{2}}$.

Let $V=\{v_1,v_2,\ldots,v_n\}$ be the set of vertices of a graph on $n$ vertices, and let $d_i$ denote the degree of vertex $v_i$. We call function $f$ Verőcese , if it takes non-negative real values and for every edge $v_iv_j$ of the graph $f(v_i)+f(v_j)\geq d_i+d_j$ holds. Find the largest real number $\lambda$ such that for every $n$, every simple graph on $n$ vertices and every Verőcese function $f$ $f(v_1)+f(v_2)+\ldots+f(v_n)\geq\lambda(d_1+d_2+\ldots+d_n)$ holds true.

We write real numbers on the faces of a cube such that the sum of the numbers on opposite faces is always 7. Subsequently at each vertex we take the product of the numbers written on the faces containing the given vertex. What is the largest possible value of the sum of these eight numbers?

Prove that if a chord intersects the diameter of the unit circle in an angle of 45 degrees, then the sum of the squares of the lengths of the two line segments created on the chord by the diameter equals 2.

Prove that $\frac{1}{\sqrt[n]{k}}+\frac{1}{\sqrt[k]{n}}\geq 1+\frac1{kn}$ is true for every positive integer $k$ and $n$.

Inside regular hexagon $ABCDEF$ point $P$ is given such that the areas of triangles $ABP$, $BCP$ and $CDP$ are $25$, $31$ and $32$ units, respectively. Find the areas of triangles $DEP$, $EFP$ and $FAP$.

The sides of triangle $ABC$ satisfy $a \geq b \geq c$ (with the usual notations). Let $P$ be the endpoint of the angle bisector from point $C$ on side $AB$. Prove that the diagonals of the cyclic trapezoid with side lengths $CP$, $BP$, $CP$, $PA$ equals $\sqrt{ab}$.

Is it possible to write a digit on each of the vertices of a regular 1000-gon such that numbers 000, 001, 002, $\ldots$, 998, 999 can all be read from some three consecutive vertices, moving in the clockwise direction?

A right triangle is inscribed in a parabola such that the foot of the altitude corresponding to the hypotenuse is the focus of the parabola. What can be the size of the angles of the triangle?

For which prime numbers $p$ does there exist a permutation $a_1$, $a_2$, $\dots$, $a_{p-1}$ of numbers 1, 2, $\ldots$, $p-1$ such that numbers $a_1^1$, $a_2^2$, $\ldots$, $a_{p-1}^{p-1}$ have pairwise different remainders modulo $p$?

Let $n$ and $k$ be positive integers. How many ways are there to partition the lattice trapezoid with vertices $(0,0)$, $(n,0)$, $(k,1)$ and $(0,1)$ into lattice triangles of area $1/2$? (A polygon is called a lattice polygon if both coordinates of all its vertices are integers.)

Viktor has drawn a parallelogram of unit area. Bálint measured one of its sides and one of its diagonals. What is the smallest possible value of the sum of these two measurements?

Two circles are given in the plane, lying outside each other. Two line segments with lengths $d_1$ and $d_2$ are also given. How many lines are there that intersect the first circle in a chord of length $d_1$ and the second circle in a chord of length $d_2$?

Positive integer $k$ is given. Find all positive integers $n$ such that in the prime factorization of $\frac{(2^kn)!}{(n!)^{2^k}}$ the exponent of 2 is exactly $2^k-1$.

Does there exist a sequence of positive integers $a_1$, $a_2$, $\ldots$ and positive integer $N$ such that for every $n\geq N$, $a_n<a_{n+1}<{a_1+a_2+\ldots+a_n}$ and $a_{n+1}\mid a_1+a_2+\ldots+a_n$?

Find the largest possible number of distinct integer roots of polynomial $p(x)=a_{12}x^{12}+a_{11}x^{11}+\ldots+a_2x^2+a_1x+a_0$ with integer coefficients if $a_0+a_2+a_4+a_6+a_8+a_{10}+a_{12}=0$ and $a_1+a_3+a_5+a_7+a_9+a_{11}=12$.

In triangle $ABC$, $F$ is the midpoint of side $BC$. A line intersects line segments $AB$, $AC$, $AF$, in points $X$, $Y$, $Z$, respectively. Prove that $AB\cdot AX+AC\cdot AY>2AF\cdot AZ$.

For edge $e$ in tetrahedron $T$ let us define quantity $r(e)$ as the ratio of the distance of the two heights of the tetrahedron from the endpoints of e and the length of edge $e$. Let $R(T)$ be defined as the sum of $r(e)$ for all the edges of tetrahedron $T$. Show an example for a tetrahedron $T$ for which $R(T)>5.999$.

Let $P$ be an interior point of circle $k$ with center $O$, different from $O$. Let $O'$ be the reflection of point $O$ across point $P$. Let $X$ be one of the intersection points of the circle with center $O'$ and radius $O'P$ and circle $k$. Line $XP$ intersects circle $k$ at point $Y$ for the second time. Prove that $P$ trisects line segment $XY$.

The sides of acute triangle $ABC$ are $a$, $b$, $c$, and its inradius is $r$. Prove that equality $2r^2=(c-a)(c-b)$ (also featured in problem B.5495. ) holds if and only if $a+b=3c$.

Prove that no matter how we color the natural numbers using 100 colors, it is always possible to find numbers $a<b<c<d$ with the same color satisfying $a+d=b+c$.

How many of the perfect powers $6^2$, $6^3$, $6^4$, $\ldots$, $6^{2026}$ have a first digit that is smaller than $6$?

Five different positive integers satisfy the following property: no matter how we choose some of them, their geometric mean is always an integer. Find the smallest possible value of the biggest of these numbers.

The incenter of triangle $ABC$ is $I$, the circumcenters of triangles $ABI$, $BCI$ and $CAI$ are $D$, $E$ and $F$, respectively. Prove that the area of triangle $CDE$ is at least as large as the area of triangle $ABC$.

An opera house has 8 boxes. With a box subscription, one can reserve the same box for at most 7 different evenings of the 2027/28 season, chosen by the buyer. There is one performance each evening. A box can be rented by only one person for a given evening, but multiple subscriptions may refer to the same box as long as the sets of selected evenings are disjoint. What is the minimum number of different performances in the repertoire such that the opera house can always arrange the program (knowing the subscribers' chosen dates) so that every box subscriber sees only different performances?

In acute triangle $ABC$ let $A_0$ be the foot of the altitude from $A$, and let $B_0$ be the foot of the altitude from $B$. On circle $A_0B_0C$ let $X$ and $Y$ be the two points for which circles $ABX$ and $ABY$ are tangent to circle $A_0B_0C$. Prove that line $XY$ bisects side $AB$.

Solve the equation $\lbrace x \rbrace + \left\lbrace \dfrac{1}{x}\right\rbrace=1$ on the set of real numbers, where $\lbrace x \rbrace$ denotes the fractional part of $x$.

$a$, $b$ and $c$ are the sides and $r$ is the inradius of a scalene triangle. Let $s_a$ be the length of the median corresponding to side $a$, and let $t_a$ denote the distance of the incenter from the line of median $s_a$. Prove that $2t_as_a=|b-c|\cdot r.$

We have written six non-negative integers with a sum of 2027 at the vertices of a regular hexagon. In a step we replace one of the six numbers with the absolute value of the difference of the numbers at the adjacent vertices. Is it possible that after a sequence of such steps there will be 0 at every vertex?

In circle $k$ chord $CD$ bisects chord $AB$. The tangents of $k$ at $C$ and $D$, respectively intersect line $AB$ at $X$ and $Y$, respectively. Prove that $XA=YB$.

Solve the following equation on the set of real numbers: $\frac{1}{x^2-5x+9}+\frac{1}{y^2-5y+9}+\frac{1}{z^2-5z+9}=\frac{1}{\sqrt{6x-9}}+\frac{1}{\sqrt{6y-9}}+\frac{1}{\sqrt{6z-9}}$.

For any positive integer $n$ prove that $\dfrac{n!\cdot\big[\tfrac{n}{30}\big]!}{\big[\tfrac{n}{2}\big]!\cdot\big[\tfrac{n}{3}\big]!\cdot\big[\tfrac{n}{5}\big]!}$ is an integer, and divides the least common multiple of integers $1$, $2$, $\ldots$, $n$.

Find the smallest and the largest possible area of the convex polygon $K$ if its orthogonal projections on the $x$-axis, the $y$-axis and the line $x=y$ is a line segment of unit length.

Let $A$, $B$ and $C$ be three points of a parabola with focus $F$, in this order. Let the tangent of the parabola at $B$ intersect the tangent at $P$ in point $A$, and the tangent at $C$ at point $Q$. Prove that the point of intersection of circles $FAP$ and $FCQ$ different from $F$ is on line $AC$.

A group of 13 people wants to get to a city that is 50 km away. Unfortunately, public transport is not available nearby, but they can walk at a speed of 5 km/h, and they also have a five-seat car that can travel at a speed of 80 km/h. Is it possible for the group to get to the other city in 2 and a half hours time?

Let $e$ and $f$ be the opposite side of a rectangular pool table, and let $A$ and $B$ be two interior points. Prove that if a billiard ball travels from $A$ to be $B$ such that it bounces twice on side $e$ and bounces twice on side $f$ (and does not bounce on the other two sides), then it will travel through the midpoint of line segment $AB$.

The sides of an acute triangle have lengths $a$, $b$ and $c$. The lengths of the line segments connecting the triangle's isogonal point to its vertices are $x$, $y$ and $z$. Prove that $a^2$, $b^2$ and $c^2$ form an arithmetic sequence (in some order) if and only if $x$, $y$ and $z$ form an arithmetic sequence in some order.

For positive integers $a$, $b$, $c$ the values of expressions $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $\frac{a}{c}+\frac{c}{b}+\frac{b}{a}$ are also integers. Prove that $a=b=c$.

Prove that if the side lengths of a non-equilateral triangle satisfy $b^2+c^2=2a^2$, the foot of the perpendicular from vertex $A$ to the Euler line bisects the orthogonal projection of $BC$ onto the Euler line.

Solve the following equation on the set of real numbers: $\bigl(\frac{x}{\sqrt{2y-3}}\bigr)^3+\bigl(\frac{y}{\sqrt{2x-3}}\bigr)^3=6\sqrt{3}$.

Scalene triangle $ABC$ is given. Let $O$ denote the circumcenter of the triangle. Let $P$ denote the intersection point, other than $O$, of the circle with diameter $AO$ and circle $BCO$. Similarly, let $Q$ denote the second intersection point of the circle with diameter $BO$ and circle $CAO$. Finally, let $R$ denote the second intersection point of the circle with diameter $CO$ and circle $ABO$. Prove that points $P$, $Q$, $R$ and $O$ are concyclic.

Is it possible to find positive integers $a$, $b$, $c$ satisfying $\mathop{\rm lcm}(a,b)+\mathop{\rm lcm}(a,c)+\mathop{\rm lcm}(b,c)=\mathop{\rm lcm}(a,b,c)?$ ($\mathop{\rm lcm}(x_1,x_2,\ldots,x_n)$ denotes the smallest common multiple of positive integers $x_1$, $x_2$, $\ldots$, $x_n$.)

Alma and Barni play the following game on a unit square: first Alma chooses a unit line segment that is parallel to a side of the square and has no points outside the square. Subsequently, Barni chooses a point on the line segment. They repeat this two more times, resulting in three chosen points. Alma wins, if the area of the triangle created from the three points is at least $0.01$, and Barni wins if the area is less than $0.01$. Which player has a winning strategy?

How many quadruples $(a,b,c,d)$ are there such that $a-b+c-d$ is divisible by $1893$, where $a$, $b$, $c$, $d \in \{1,2,3,\ldots ,1893\}$ and $a$, $b$, $c$ and $d$ are distinct from each other?

In triangle $ABC$ let $a<b<c$, $m_a$, $m_b$, $m_c$ and $r_a$, $r_b$, $r_c$ be the sides, the heights and the exradii (using the usual notations). Find all possible orders by magnitude of $m_a$, $m_b$, $m_c$, $r_a$, $r_b$, $r_c$.

The lengths of the sides of a parallelogram are integers, and its perimeter is $84$. The area of the shape defined by the points of intersection of its internal angle bisectors is $t$, and the area of the parallelogram is $T$. Find the sides of the parallelogram if $\frac{T}{t}$ is an integer number.

We colour the points of the plane using two colours. Prove that there exists a triangle such that its vertices have the same colour and its angles are $48^\circ$, $60^\circ$, $72^\circ$.

Let $ABCD$ be a trapezoid, and let circle $k_1$ through points $A$ and $D$ be tangent to side $BC$ at an inner point. Let circle $k_2$ through points $B$ and $C$ be tangent to side $AD$ at an inner point. Prove that the line through points of intersections of the two circles passes through the intersection of the diagonals.

A polynomial with integer coefficients has exactly two real roots, both being a root of multiplicity one. Find the smallest possible degree of the polynomial if the difference of the two roots is an irrational number, and the ratio of the two roots is a positive rational number.

Find all natural numbers $n$ satisfying $n^3+25n\ge10n^2+16$.

30 students in a class did a test in mathematics. The teacher marked the tests, and sent a table of the marks to the students such that the marks appeared in a column: 15 of them were four, and 15 of them were five. Prove that it is always possible to find 14 consecutive rows such that the sum of the marks contained in them equals 63.

We inscribe reagular hexagon $ABCDEF$ in equilateral triangle $PQR$ such that points $B$, $D$ and $F$ are midpoints of sides $PQ$, $QR$ and $RP$. Find the area of triangle $QPR$ if the area of pentagon $ABQRF$ equals one unit.

The internal angles of hexagon $ABCDEF$ are equal. Prove that the area of triangles $ACE$ and $BDF$ are equal.

How many numbers are there with a base nine representation of $\overline{abcabc}_9$ and exactly 40 positive divisors?

Anna has written all the three-digit numbers on the blackboard, each appearing exactly once. Boglárka has erased Anna's numbers one by one, replacing them with the number obtained by subtracting the last digit from the sum of the first and the second digit. For example, 225 is replaced by $2+2-5=-1$ and 973 is replaced by $9+7-3=13$. Find the sum of numbers written on the blackboard by Boglárka.

Numbers 1 and $\sqrt{5}$ are marked on the number line, and nothing else. Mark 0 on the line by construction. (The basic constructions, e.g., bisecting an angle or reflecting across a line don't have to be detailed.)

Prove that the irrational number $\sqrt[3]{a}$ cannot be written as $b+c\sqrt{d}$, where $a$, $b$, $c$ and $d$ are non-negative integer numbers.

Let $I$ denote the incenter of triangle $ABC$, and let line $AI$ intersect the circumcircle of the triangle at point $D$. Let the circumcircle of triangle $CDI$ intersect line segment $BC$ at point $E$. Prove that $BE=IE$.

Prove that $\binom{\binom{n}{2}}{3}=15\binom{n}{6}+30\binom{n}{5}+16\binom{n}{4}+\binom{n}{3}$ is true for every positive integer $n$.

How many rectangular prisms are there with a volume of $2560~\mathrm{cm}^3$ where every edge has an integer length measured in centimeters.

Prove that if $n$ is an integer, then the value of $\frac{n^5}{5}+\frac{n^3}{3}+\frac{7n}{15}$ is also an integer.

The surface of a cube shaped small planet is covered in grass, its edges have a length of 1 km. A goat is tied to one of the vertives with a rope of length $\sqrt{2}$ km. What ratio of the surface of the planet can be consumed by the goat?

Let $E$ be the center of square $ABCD$. A quarter circle with center $D$ and radius $DA$ inside the square intersects the semicircle with diameter $AB$ inside the square at point $F$. Line $EF$ intersects line $AB$ at $M$, and the semicircle with diameter $DA$ outside the circle at point $N$. Prove that $F$ is the midpoint of line segment $MN$.

We choose three vertices of a regular $n$-gon such that the chance of picking any of the triples is the same. Prove that if n is even, then the probability of the triangle formed by the three vertices being obtuse is three times as big as the probability of it being acute.

We would like to select some of the nine lattice points of a $2\times 2$ square lattice such that no three of them form a right triangle. a) Prove that it's not possible to select five points with the property above. b) How many ways are there to select four points satisfying the above property? (Two selections are considered different if there is a point that is selected in one of them and not selected in the other.)

2227 is a special number for two different reasons. The first is that this is the next year when Pluto will be the closer to the Sun than Neptun, and the second is that three of its digits can be chosen such that their product is bigger by one than its remaining digit. How many four-digit numbers are there having the second property?

Real numbers $a$ and $b$ satisfy $|a-1|+|b-1|=|a|+|b|=|a+1|+|b+1|$. Find the smallest possible value of $|a-b|$.

On the smaller arc $AB$ of the circumcircle of square $ABCD$ we pick internal point $E$. Let $E'$ be the reflection of $E$ across the center of the square. Let $F$ and $G$ be the feet of the perpendiculars from $E$ to diagonals $AC$ and $BD$, respectively. We get similarly points $H$ and $I$ from point $E'$. a) Prove that quadrilateral $FGHI$ is a diamond. b) Find the maximum of the area of quadrilateral $FGHI$.

The 17 enthusiastic members of a math club would like to get together for a group math session during the summer break, so they collect everyone's availability: from each person, they receive a time interval consisting of at least two days. (For example, from July 3 to July 6.) After collecting the data, it turned out that among any three students there are two who have marked a common day. Is it guaranteed that there will be a day such that every student will be available? Is it guaranteed that there will be two days such that every student will be available on at least one of the two days?

$AB$ is the hypotenuse of right triangle $ABC$. Let $D$ be the foot of the altitude from $C$. Let $E$ be the foot of the perpendicular from $D$ to $AC$, and let $F$ be the foot of the perpendicular from $E$ to $AB$. Given that $F$ is the midpoint of the hypotenuse, find the lengths of the legs of the triangle.

Let $a$, $b$, $c$ and $d$ be distinct positive integers. Among the four statement below two are true and two are fale: (i) $a<b<c<d$, (ii) $a+b=c+d$, (iii) $a^2+b^2=c^2+d^2$, (iv) $a^3-b^3=c^3-d^3$. Find the smallest possible value of $d$.

We wrote down all the fractions between $0$ and $1$ that have a denonimator of at most $99$ in increasing order. Find the fraction right after ${\frac{11}{21}}$.

2026 is a four-digit, positive number such that all its digits are even and exactly two of them are equal. How many such four-digit positive integer numbers are there?

Let $D$ denote the midpoint of arc $BC$ not containing point $A$ on the circumcircle of equilateral triangle $ABC$. Let $E$ denote the midpoint of line segment $AC$, and let $F$ denote the midpoint of line segment $BC$. a) Find the magnitude of angle $\angle EBD$ b) Let $F'$ denote the reflection of $F$ across $E$. Prove the circle with center $F'$ and radius $F'C$ passes through the midpoint of side $AB$.

Bogi likes to play with pebbles which she calls gems. Currently she has 4052 gems, and she labeled them with numbers 1, 2, 3, ..., 4052. She wants to place her gems in a red and a blue box such that for every $k=1$, 2, 3, ..., 2026 gems with label $k$ and $2k$ should be in different boxes. How many ways can Bogi distribute her gems?

Let $F$ denote the midpoint of the side $BC$ of triangle $ABC$. We construct isosceles right triangles $BAD$ and $ACE$ on sides $AB$ and $AC$ pointing outwards such that $AB$ and $AC$ are the hypotenuses. Prove that $EDF$ is an isosceles right triangle.

Let $a_1, a_2, \ldots a_{10}$ be positive, not necessarily distinct numbers such that their sum is 95. Prove that $a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2+a_7^2+a_8^2+a_9^2+a_{10}^2 \geq 905$. When does the equality hold?

Juci and Joci, exhausted, are looking back on their school assessments from the past five days. There were at most six assessments in a single day,'' Juci begins. But at least two every day,'' Joci sighs. ``And 16 in total,'' they say at the same time, their voices cracking. In how many ways could the weekly schedule of assessments have turned out? (Two cases are considered distinct if and only if there is a day when a different number of assessments occurred.)

Solve equation $ab+2=a^3+2b$ for positive integers $a$ and $b$.

In acute triangle $ABC$ the orthocenter is $M$. We draw the tangents from midpoint $F$ of line segment $AB$ to the circle with diameter $CM$, the points of tangency being $D$ and $E$. Prove that the circumcircle of triangle $DEF$ is half as big as the circumcircle of triangle $ABC$.

The sides of a triangle are consecutive natural numbers. The inradius of the triangle is $4$ distance units. Find the circumradius of the triangle.

We want to mark $k$ squares of a $2026\times 2026$ board with an $\times$ such that no two $\times$'s are in the same row or the same column. For which value of $k$ do we have the largest number of possibilities to do this? (The cases which can be reflected or rotated into each other are also considered to be different.)