KöMaL 2023

$n$ people live in a town, and they are members of some clubs (residents can be members of more than one club). No matter how we choose some (but at least one) clubs, there is a resident of the town who is the member of an odd number of the chosen clubs. Prove that the number of clubs is at most $n$.

Let $N$ be the set of those positive integers $n$ for which $n\mid k^k-1$ implies $n\mid k-1$ for every positive integer $k$. Prove that if $n_1,n_2\in N$, then their greatest common divisor is also in $N$.

The inscribed circle of triangle $ABC$ is tangent to sides $BC$, $AC$ and $AB$ at points $D$, $E$ and $F$, respectively. Let $E'$ be the reflection of point $E$ across line $DF$, and $F'$ be the reflection of point $F$ across line $DE$. Let line $EF$ intersect the circumcircle of triangle $AE'F'$ at points $X$ and $Y$. Prove that $DX=DY$.

The incircle of triangle $ABC$ is tangent to sides $BC$, $AC$, and $AB$ at points $D$, $E$, and $F$, respectively. Let $A'$ denote the point of the incircle for which circle $(A'BC)$ is tangent to the incircle. Define points $B'$ and $C'$ similarly. Prove that lines $A'D$, $B'E$ and $C'F$ are concurrent.

Let $n$ be a positive integer and let vectors $v_1,v_2,\ldots,v_n$ be given in the plain. A flea originally sitting in the origin moves according to the following rule: in the $i^{\text{th}}$ minute (for $i=1,2,\ldots,n$) it will stay where it is with probability $1/2$, moves with vector $v_i$ with probability $1/4$, and moves with vector $-v_i$ with probability $1/4$. Prove that after the $n^{\text{th}}$ minute there exists no point which is occupied by the flea with greater probability than the origin.

Let $A$ be a given finite set with some of its subsets called pretty. Let a subset be called small, if it's a subset of a pretty set. Let a subset be called big, if it has a pretty subset. (A set can be small and big simultaneously, and a set can be neither small nor big.) Let $a$ denote the number of elements of $A$, and let $p$, $s$ and $b$ denote the number of pretty, small and big sets, respectively. Prove that $2^a\cdot p\le s \cdot b$.

Let $G$ be a planar graph, which is also bipartite. Is it always possible to assign a vertex to each face of the graph such that no two faces have the same vertex assigned to them?

For real number $r$ let $f(r)$ denote the integer that is the closest to $r$ (if the fractional part of $r$ is 1/2, let $f(r)$ be $r-1/2$). Let $a>b>c$ rational numbers such that for all integers $n$ the following is true: $f(na)+f(nb)+f(nc)=n$. What can be the values of $a$, $b$ and $c$?

Prove that there exists a positive real number $N$ such that for arbitrary real numbers $a,b>N$ it is possible to cover the perimeter of a rectangle with side lengths $a$ and $b$ using non-overlapping unit disks (the unit disks can be tangent to each other).

Let $k$, $l$ and $m$ be positive integers. Let $ABCDEF$ be a hexagon that has a center of symmetry whose angles are all $120^{\circ}$ and let its sidelengths be $AB=k$, $BC=l$ and $CD=m$. Let $f(k,l,m)$ denote the number of ways we can partition hexagon $ABCDEF$ into rhombi with unit sides and an angle of $120^{\circ}$. Prove that by fixing $l$ and $m$, there exists polynomial $g_{l,m}$ such that $f(k,l,m)=g_{l,m}(k)$ for every positive integer $k$, and find the degree of $g_{l,m}$ in terms of $l$ and $m$.

Let $(a_i,b_i)$ be pairwise distinct pairs of positive integers for $1 \le i \le n$. Prove that $(a_1+a_2+\dots+a_n) (b_1+b_2+\dots+b_n)>\frac{2}{9} n^3,$ and show that the statement is sharp, i.e. for an arbitrary $c>\frac{2}{9}$ it is possible that $(a_1+a_2+\dots+a_n) (b_1+b_2+\dots+b_n)<cn^3.$

Let points $A$, $B$, $C$, $A'$, $B'$, $C'$ be chosen in the plane such that no three of them are collinear, and let lines $AA'$, $BB'$, $CC'$ be tangent to a given equilateral hyperbola at points $A$, $B$ and $C$, respectively. Assume that the circumcircle of $A'B'C'$ is the same as the nine-point circle of triangle $ABC$. Let $s(A')$ be the Simson line of point $A'$ with respect to the pedal triangle of $ABC$. Let $A^*$ be the intersection of line $B'C'$ and the perpendicular of $s(A')$ through point $A$. Points $B^*$ and $C^*$ are defined in a similar manner. Prove that points $A^*$, $B^*$ and $C^*$ are collinear.

Prove that $\sum_{k=0}^n \frac{2^{2^k}\cdot 2^{k+1}}{2^{2^k}+3^{2^k}}<4$ holds for all positive integers $n$.

In scalene triangle $ABC$ the shortest side is $BC$. Let points $M$ and $N$ be chosen on sides $AB$ and $AC$, respectively, such that $BM=CN=BC$. Let $I$ and $O$ denote the incenter and circumcentre of triangle $ABC$, and let $D$ and $E$ denote the incenter and circumcenter of triangle $AMN$. Prove that lines $IO$ and $DE$ intersect each other on the circumcircle of triangle $ABC$.

In a rock-paper-scissors round robin tournament any two contestants play against each other ten times in a row. Each contestant has a favourite strategy, which is a fixed sequence of ten hands (for example, RRSPPRSPPS), which they play against all other contestants. At the end of the tournament it turned out that every player won at least one hand (out of the ten) against any other player. Prove that at most $1024$ contestants participated in the tournament.

Let $ABC$ be a given acute triangle, in which $BC$ is the longest side. Let $H$ be the orthocenter of the triangle, and let $D$ and $E$ be the feet of the altitudes from $B$ and $C$, respectively. Let $F$ and $G$ be the midpoints of sides $AB$ and $AC$, respectively. $X$ is the point of intersection of lines $DF$ and $EG$. Let $O_1$ and $O_2$ be the circumcenters of triangles $EFX$ and $DGX$, respectively. Finally, $M$ is the midpoint of line segment $O_1O_2$. Prove that points $X$, $H$ and $M$ are collinear.

Prove that the only integer solution of the following system of equations is $u=v=x=y=z=0$: $uv=x^2-5y^2$, $(u+v)(u+2v)=x^2-5z^2$.

Path graph $U$ is given, and a blindfolded player is standing on one of its vertices. The vertices of $U$ are labeled with positive integers between $1$ and $n$, not necessarily in the natural order. In each step of the game, the game master tells the player whether he is in a vertex with degree 1 or with degree 2. If he is in a vertex with degree 1, he has to move to its only neighbour, if he is in a vertex with degree 2, he can decide whether he wants to move to the adjacent vertex with the lower or with the higher number. All the information the player has during the game after $k$ steps are the $k$ degrees of the vertices he visited and his choice for each step. Is there a strategy for the player with which he can decide in finitely many steps how many vertices the path has?

A 0$\,$-$\,$1 sequence of length $2^k$ is given. Alice can pick a member from the sequence, and reveal it (its place and its value) to Bob. Find the largest number $s$ for which Bob can always pick $s$ members of the sequence, and guess all their values correctly. Alice and Bob can discuss a strategy before the game with the aim of maximizing the number of correct guesses of Bob. The only information Bob has is the length of the sequence and the member of the sequence picked by Alice.

Let $f(x)=x^2-2$. Let $f^{(n)}(x)$ denote the $n^{\text{th}}$ iterate of $f$ (i.e. $f^{(1)}(x)=f(x)$ and $f^{(k+1)}(x)=f\big(f^{(k)}(x)\big)$). Let $H=\big\{x\colon f^{(100)}(x)\le -1\big\}$. Find the length of $H$ (the sum of the lengths of the intervals in $H$). We expect the solution as a closed-form expression.

Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\omega$. Let $F_A$, $F_B$, $F_C$ and $F_D$ be the midpoints of arcs $AB$, $BC$, $CD$ and $DA$ of $\omega$. Let $I_A$, $I_B$, $I_C$ and $I_D$ be the incenters of triangles $DAB$, $ABC$, $BCD$ and $CDA$, respectively. Let $\omega_A$ denote the circle that is tangent to $\omega$ at $F_A$ and also tangent to line segment $CD$. Similarly, let $\omega_C$ denote the circle that is tangent to $\omega$ at $F_C$ and tangent to line segment $AB$. Finally, let $T_B$ denote the second intersection of $\omega$ and circle $F_BI_BI_C$ different from $F_B$, and let $T_D$ denote the second intersection of $\omega$ and circle $F_DI_DI_A$. Prove that the radical axis of circles $\omega_A$ and $\omega_C$ passes through points $T_B$ and $T_D$.

Let $n\ge 2$ be a given integer. Find the greatest value of $N$, for which the following is true: there are infinitely many ways to find $N$ consecutive integers such that none of them has a divisor greater than 1 that is a perfect $n^{\mathrm{th}}$ power.

Let $ABC$ be a triangle and $O$ be its circumcenter. Let $D$, $E$ and $F$ be the respective tangent points of the incircle of $\triangle ABC$, and sides $BC$, $CA$ and $AB$. Let $M$ and $N$ be the respective midpoints of sides $AB$ and $AC$. Let $M'$ and $N'$ be the respective reflections of points $M$ and $N$ across lines $DE$ and $DF$. Let lines $CM'$ and $BN'$ intersect lines $DE$ and $DF$ at points $H$ and $J$, respectively. Prove that the points $H$, $J$ and $O$ are collinear.

A crossword is a grid of black and white cells such that every white cell belongs to some $2 \times 2$ square of white cells. A word in the crossword is a contiguous sequence of two or more white cells in the same row or column, delimited on each side by either a black cell or the boundary of the grid. Show that the total number of words in an $n\times n$ crossword cannot exceed $\frac{{(n+1)}^2}{2}$.

A graph is called 2-connected, if after deleting any point (and the edges adjacent to it) from the graph it remains still connected. Is it true that in any 2-connected graph with a countably infinite number of vertices it's always possible to find a trail that is infinite in one direction (i.e. a sequence of not necessarily distinct points $v_1, v_2,\ldots$ such that $v_i$ and $v_{i+1}$ is always connected with an edge, and all edges $(v_i,v_{i+1})$ are distinct from each other)?

Let $p(x)$ be a monic integer polynomial (polynomial with integer coefficients and leading coefficient 1) of degree $n$ that has $n$ real roots, $\alpha_1, \alpha_2,\ldots, \alpha_n$. Let $q(x)$ be an arbitrary integer polynomial that is relatively prime to polynomial $p(x)$ (i.e. it's not possible to find an integer polynomial different from constant 1 and $-1$ that divides both $p(x)$ and $q(x)$). Prove that $\sum\limits_{i=1}^{n} \big|q(\alpha_i)\big|\ge n$.

A set of points in the plane is called disharmonic, if the ratio of any two distances between the points is between $100/101$ and $101/100$, or at least 100 or at most $1/100$. Is it true that for any distinct points $A_1, A_2,\ldots, A_n$ in the plane it is always possible to find distinct points $A'_1, A'_2,\ldots,A'_n$ that form a disharmonic set of points, and moreover $A_i$, $A_j$ and $A_k$ are collinear in this order if and only if $A'_i$, $A'_j$ and $A'_k$ are collinear in this order (for all distinct $1\le i,j,k\le n$)?

What is the smallest positive integer $n$ for which the number $\underbrace{11\ldots1}_{n}$ (in decimal notation) is divisible by the number $\underbrace{33\ldots3}_{100}$?

Two circles touch each other externally. The line passing through the centres of the circles intersects the circles again at points $A$ and $B$. The points of tangency on one of the common external tangents of the circles are $P$ and $Q$, respectively. Prove that the lines $AP$ and $BQ$ intersect on the common internal tangent of the circles.

Two players are playing the following game on a $8 \times 8$ chessboard. They take turns in selecting one side of a field of the chessboard, separating two adjacent fields of the board, and colouring it yellow. The player who is first forced to create a closed yellow polygon will lose the game. Which player has a winning strategy?

Let $a$, $b$, $c$ and $d$ denote non-negative real numbers such that $a+b+c+d=1$. Prove that $\frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}+\frac{1}{d^2+1} \ge \frac72.$

Solve the following equation over the set of positive integers: $3^n+4^n+\dots+{(n+2)}^n= {(n+3)}^n.$

The centre of the inscribed circle of a triangle $ABC$ is $I$, and the centre of its circumscribed circle is $O$. Given that $\angle OIA =90^\circ$, $AI=89$ and $BC=160$, what is the area of the triangle?

Given a circle $k$ and points $P$ and $Q$ in its interior, construct (write down the steps of the construction - basic construction steps like bisecting an angle, reflecting over a line, etc. do not need to be described in detail - and explain why the method is correct; it is not required to perform the construction on paper) a circle through points $P$ and $Q$ that intersects the circle $k$ at two diametrically opposite points. Depending on the position of points $P$ and $Q$, how many solutions will the problem have?

Let $p$ denote a prime number. What is the largest number of polynomials with integer coefficients such that there are no pairs of polynomials such that the value of their difference is divisible by $p^2$ at every integer?

Two altitudes of an acute-angled triangle $ABC$ are $AT_A$ and $BT_B$. The midpoint of $AB$ is $F$, and the midpoint of $T_AT_B$ is $G$. Prove that $FG$ is perpendicular to $T_AT_B$.

Find the largest integer $k$ for which $1722$ divided by $k$ leaves a remainder of $2m$ and $2179$ divided by $k$ leaves a remainder of $3m$ (for some appropriate natural number $0 \le m < k/3$).

A rook is moving from the bottom left corner of a $8\times 8$ chessboard to the top right corner. It starts by moving to the right, then up, and so on, right and up alternatingly. How many different sequences of moves are there?

In a triangle $ABC$, $\angle BAC =2\angle CBA$. Let $A'$, $B'$, $C'$ denote interior points of sides $CA$, $AB$ and $BC$, respectively, such that triangle $A'B'C'$ is similar to triangle $ABC$. Show that the angle bisectors of angles $BAC$ and $B'A'C'$ intersect each other on the line segment $B'C'$.

Solve the following system of equations over the set of real numbers: $$\begin{align*} y+yx^2-2x & =0,\\ z+zy^2-2y & =0,\\ x+xz^2-2z & =0. \end{align*}$$

There is a flea sitting at each of points $1,2,3$ of the number line. If one flea is at point $a$ and another is at point $b$, then the flea at $a$ may jump to the point $2b-a$. Is it possible for the fleas to end up at points $2^{100},3^{100},2^{100}+3^{100}$ with a finite sequence of such steps?

Given that the sum of the reciprocals of ten distinct positive integers is 1, prove that each of them is smaller than $10^{1000}$.

Either $+1$ or $-1$ is written in every field of an $8 \times 8$ table so that the sum of all entries is 0. The sum of the numbers is calculated for each row and column. What is the maximum possible number of positive sums out of the 16 sums obtained in this way?

The isosceles right-angled triangle $ABC$ has its right angle at $C$. $D$ is an interior point of side $BC$ such that the angle $CDA$ is $75^\circ$. Given that triangle $ADC$ has unit area, prove that $BD = 2$.

$a)$ Are there positive integers $a$ and $b$ such that $a+b \mid a^2+b^2, \text{but} a+b \nmid a^4 + b^4?$ $b)$ Are there positive integers $a$ and $b$ such that $a+b \mid a^4+b^4, \text{but} a+b \nmid a^2+b^2?$

Let $A_1$ and $A_2$, respectively, denote the points lying closer to $B$ and closer to $C$ that divide side $BC$ of a triangle $ABC$ in a 1:2 ratio. Define points $B_1$ and $B_2$ on side $CA$, and points $C_1$ and $C_2$ on side $AB$ in an analogous way. Prove that the centroid of triangle $ABC$ lies on the line joining the common points of the circumscribed circles of triangles $A_1B_1C_1$ and $A_2B_2C_2$.

We have a weighted (six-sided) die and a weighted coin. On one side of the coin there is one dot, and on the other side there are two dots. The expected value of the number of dots appearing on top is the same for the die and the coin. Show that if the die and the coin are thrown simultaneously then the probability of getting more dots on the coin than on the die is greater than the probability of getting more dots on the die than on the coin.

The area of an acute-angled triangle is $T$, its inradius is $r$, and its circumradius is $R$. Show that $\sqrt{3}T \le {(r+R)}^2.$

Let $a_n$ denote the least common multiple of the positive integers $n+1,n+2,\ldots,n+10$. Find the greatest real number $\lambda$ for which $\lambda a_{n}\le a_{n+1}$ is always true.

Given the axis and two points of a parabola, construct its focus and directrix.

A game of strategy is played by four teams on a map represented by an $n \times n$ square grid ($n \ge 3$). Every square field is either sea or land. The bases of the four teams are in the four corners of the map, on land. Given that there is one large connected region of sea on the map and no two bases are connected by a path on land, determine the minimum possible number of fields that represent the sea. (A path consists of adjacent fields. Two fields are adjacent if they have an edge in common. The region of sea is connected in this sense: any two sea squares can be connected by a path consisting of sea squares.)

Is it true that if the sine of each angle of a triangle is rational then the cosine of each angle is rational, too?

Let $F_k$ denote the $k$th Fibonacci number ($F_1=F_2=1$, $F_{k+1}=F_k+F_{k-1}$). Prove that $2\sum_{k=1}^nF_k^2F_{k+1}=F_nF_{n+1}F_{n+2}$ for all positive integers $n$.

Triangle $ABC$ is acute angled, and $AC<AB<BC$. The centre of the circumscribed circle is $O$, the orthocentre is $M$. The perpendicular bisector of side $AB$ intersects line $AM$ at point $P$, and the circle $OMP$ intersects line $BM$ again at a point $Q$, different from $M$. Prove that line $BC$ is tangent to the circle $ABQ$.

Let $S$ be an $n$-element set, and let $1\le k\le n$ be an odd integer. What is the largest number of subsets of $S$ that can be selected so that the symmetric difference of no pair of subsets should have exactly $k$ elements?

Consider a triangle $ABC$. Let $B'$ be a point on the extension of side $AB$ beyond $B$, and let $C'$ be the point on the extension of side $AC$ beyond $C$ such that $BB'=CC'$. Let $k$ and $k'$ denote the circumscribed circles of triangles $ABC'$ and $AB'C$, respectively. Prove that the common chord of $k$ and $k'$ lies on the angle bisector drawn from $A$.

Prove that if $0<a,b<1$ then $(a+b-ab)\big(a^b+b^a\big) > a+b.$

An ellipse lies in the closed positive orthant, its foci are $(x_1;y_1)$ and $(x_2;y_2)$, and it touches the coordinate axes at the points of abscissa $p$, and ordinate $q$, respectively. Show that the point $(p;q)$ is collinear with the origin and the centre of the ellipse, and calculate the numerical eccentricity of the ellipse.

All positive divisors of a positive integer are written down on a sheet of paper. There are two numbers on the sheet that leave a remainder of 2 when divided by 8, and there are four numbers that leave a remainder of 4. How many numbers may there be on the sheet that leave a remainder of 6 when divided by 8?

Is it true that every acute-angled triangle has at least one altitude whose foot lies in the middle third of the side?

Given that $\frac{a_{n+3}}{a_{n+1}} + \frac{a_n}{a_{n+2}} = 2$ for each term of a sequence $a_n$, and the first three terms are $a_1=1$, $a_2=4$ and $a_3=2$, prove that $\frac{2^{2021}}{a_{2023}}$ is an integer.

Show that the sum of the squares of the medians of a triangle is smaller than one and a half times the square of the semi-perimeter.

Prove that if $\frac{\cos\alpha}{s-b}-\frac{\cos\beta}{s-a}=\frac{\cos\alpha-\cos\beta}{s-c}$ in a triangle, with conventional notations, then the triangle is right-angled or isosceles. (As usual, $s$ denotes the semi-perimeter of the triangle.)

We are playing the following game: arbitrary real numbers from the interval $[0, 100]$ are written on 2023 cards. Then the cards are dropped in an urn, and one card is pulled out at random. If the number on the card equals 2/3 of the mean of the numbers on all the cards then we will win as much money as the number on the card pulled out. Otherwise we do not win anything. What numbers should be written on the cards so that the expected value of the money gained is a maximum?

Determine all bounded convex polyhedra such that the planes of the faces divide the space into $c+e+\ell+1$ regions, where $c$, $e$ and $\ell$, respectively, stand for the number of vertices, edges and faces.

At the end of a meeting with English and Hungarian participants, everyone bid farewell to everyone else, one by one. The English participants said Goodbye'', and the Hungarian participants said Viszlát'' to everyone else. There were 198 occurrences of Goodbye'' and 308 occurrences of Viszlát''. How many participants of each nationality were there?

The heights of triangle $ABC$ are $m_a$, $m_b$ and $m_c$. Suppose that one can construct a triangle having sides $m_a$, $m_b$ and $m_c$, and the heights of that triangle are $x$, $y$ and $z$. Show that one can again construct a triangle having sides $x$, $y$ and $z$.

The number 2023 is written on the first page of a notebook. On every following page, the positive divisors of the numbers on the previous page are written (each divisor is written down $n$ times if it divides $n$ numbers on the previous page). How many numbers will there be on page 4?

A fair die is rolled. The game terminates when we decide to stop or a number 1 is rolled. The award is the value of the last roll. Is there a strategy to achieve an expected award of at least 4?

Suppose that $a$, $b$, $c$ is a primitive Pythagorean triple, i.e., $a$, $b$ and $c$ are relatively prime positive integers satisfying $a^2+b^2=c^2$. Show an axially symmetrical polygon that can be decomposed into $c$ right-angled triangles with sides $a$, $b$, $c$.

Prove that it is possible to cover a regular tetrahedron of unit length with two spheres of unit diameter.

What may be the positive integer $n$ if any sequence of $2^n$ consecutive positive integers contains a term that can be represented as a sum of the $n$th powers of at most $n$ non-negative integers?

The foot of the altitude drawn from vertex $A$ of an acute-angled triangle $ABC$ is $T_A$. The ray drawn from vertex $A$ through the centre $O$ of the circumcircle intersects side $BC$ at point $R_A$. Denote the midpoint of line segment $AR_A$ by $F_A$. Starting from vertices $B$ and $C$, obtain the points $T_B$, $R_B$, $F_B$, $T_C$, $R_C$, $F_C$ in the same way. Show that the lines $T_AF_A$, $T_BF_B$ and $T_CF_C$ are concurrent.

What is the smallest $n$ for which every convex $n$-gon has two adjacent obtuse angles?

The product of the positive numbers $x$, $y$, $z$ is 1. What may be the value of the expression $\left(x+\frac{1}{x}\right)^{2} +\left(y+\frac{1}{y}\right)^{2} +\left(z+\frac{1}{z}\right)^{2} -\left(x+\frac{1}{x}\right) \left(y+\frac{1}{y}\right) \left(z+\frac{1}{z}\right)?$

The organizers of a school event bought four kinds of sweets to use them as awards for correct answers. The prices paid for them were $1300$ forints (HUF, Hungarian currency), $3000$ forints, $3300$ forints and (for questions for the audience) a large number of sweets for $50$ forints a piece. The total was $41 300$ forints, and the mean price per item was exactly $100$ forints. How many of each kind did they buy?

A regular triangle is drawn over each side of a regular $n$-gon, on the outside. The third vertices of the triangles form a larger $n$-gon. What may the number $n$ be if the ratio of the areas of the two polygons is an integer?

There are 10 numbered seats in a particular row of an auditorium. The seats are accessible from each end of the row. The 10 spectators holding tickets for that row arrive in a random order and they take their seats. Since the spectators do not like passing others already seated, they will approach their seats from the direction where that can be avoided, if possible. Determine the probability that at least one of the 10 ticket holders will not be able to access their seat without passing a seated spectator.

A circle $k_1$ touches another circle $k_2$ from the inside at point $P$. Let $M$ denote an arbitrary point on the circumference of $k_1$, and let the tangent drawn to $k_1$ at $M$ intersect $k_2$ at points $A$ and $B$. Show that $PM$ bisects the angle $APB$.

Let $n$ be a positive integer, and let $f$, $g$ and $h$ denote polynomials with real coefficients whose degrees are at most $n$. What is the largest possible number of real numbers $x$ for which $f(x)$, $g(x)$, $h(x)$, in some order, are consecutive terms of a non-constant arithmetic sequence, assuming that there are only a finite number of such $x$?

The centroid of a regular tetrahedron $ABCD$ is $S$, and an arbitrary interior point is $P$. Reflect point $P$ in the planes of the four faces of the tetrahedron to obtain tetrahedron $XYZW$. Show that the centroid of $XYZW$ is the point that divides line segment $PS$ in the ratio $2:1$, with the shorter piece next to $S$.

Let us pick four consecutive integers of the same parity, and take the products of all possible pairs. Prove that the sum of these products cannot be a perfect square.

Decide which one of numbers $$\begin{align*} A & = 1! - 2! + 3! - 4! + \ldots + 2021! - 2022! + 2023! \text{and}\\ B & =(1-2+3-4+\ldots+2021-2022+2023)! \end{align*}$$ is the bigger.

Anti and Bandi would like to get from Balatonmáriafürdő to Balatonlelle, which is 30 kilometers away, partly on foot and partly by bicycle. They start at the same time, and they only have a single bicycle. Anti's speed by bicycle is 30 km/h, and he can run with a speed of 15 km/h. Bandi's speed by bicycle is 20 km/h, and he can run with a speed of 12 km/h. Find the least amount of time they need to get to their destination in minutes. (During their trip they can swap the bicycle with each other as many times as they want, and the bicycle can be safely left at the side of the road.)

Prove that if the incircle and the nine-point circle of a triangle are concentric, then the triangle is equilateral.

For which $n$ is it possible to find an equilateral (all sides have the same length) $n$-gon such that each of its sides is parallel to exactly two other sides?

Prove that if $r$ is a positive rational number, and $r^r$ is also rational, then $r$ is an integer.

Let $b$, $c$ and $n$ be non-negative integers satisfying $0\le c\le b-2n$. Prove that $\sum_{a=0}^n \binom{2a}{a}\binom{b-2a}{n-a} = \sum_{a=0}^n \binom{2a+c}{a}\binom{b-c-2a}{n-a}.$

The edges of parallelepiped $P$ all have lengths that are at most $1$. Let $X$ be an arbitrary point of $P$. Show that we can find a vertex of $P$ such that its distance from $X$ is at most $\sqrt 3 /2$.

$a)$ Is it possible to find positive integers $a$, $b$, $c$ and $d$ such that the arithmetic mean of $a$ and $b$ is greater than the quadratic mean of $c$ and $d$, however, the geometric mean of $a$ and $b$ is smaller than the harmonic mean of $c$ and $d$? $b)$ Is it possible to find positive integers $a$, $b$, $c$ and $d$ such that the geometric mean of $a$ and $b$ is greater than the quadratic mean of $c$ and $d$, however, the arithmetic mean of $a$ and $b$ is smaller than the harmonic mean of $c$ and $d$?

Let $P$ be an arbitrary point inside the equilateral triangle $ABC$. The line parallel to $AB$ through $P$ intersects sides $BC$ and $AC$ in points $C_1$ and $C_2$, respectively. Similarly, the parallel through point $P$ with side $BC$ intersects sides $AC$ and $AB$ in points $A_1$ and $A_2$, respectively. Finally, let the parallel through point $P$ with side $AC$ intersect sides $AB$ and $BC$ in points $B_1$ and $B_2$, respectively. Prove that the areas of triangles $A_1B_1C_1$ and $A_2B_2C_2$ are equal.

For which positive integers $n>3$ is it possible to find $n$ lines in the plane such that any three of them form an isosceles triangle?

Prove the following equality for every positive integer $n>1$: $\sum\limits_{i=1}^n \sum\limits_{j=1}^n|i-j|=\frac{n(n^2-1)}{3}.$

Prove that the Euler line of a scalene triangle is parallel to one of the interior angle bisectors if and only if the bisected angle is $120^{\circ}$.

We colored $n$ squares red in a square lattice. We numbered the red squares from $1$ to $n$, and calculated the sum of the numbers for each pair of squares that share a common side. Is it true that for all arrangements of the $n$ squares it is possible to number the red squares in a way that we will get a different sum for all pairs that share a common side?

Prove the following inequality for every integer $n\geq 2$ and non-negative real numbers $x_{1},\ldots,x_{n}$: $\sqrt[\scriptstyle n]{\prod\limits_{i=1}^{n} (1+x_{i})}\ge 1+\sqrt[\scriptstyle n]{\prod\limits_{i=1}^{n}x_{i}}.$

Let $ABC$ be a triangle. The tangents at $B$ and $C$ to the circumcircle of $ABC$ intersect at $P$. Let $PB\cap AC=D$ and $PC\cap AB=E$. The perpendicular bisector of the line segment $BC$ intersects $AC$ and $AB$ at $F$ and $G$, respectively. The circumcircles of the triangles $PDF$ and $PEG$ intersect at $M$ for the second time. Let $A'$ be the reflection of $A$ over $FG$ and let $O$ be the circumcenter of the triangle $AFG$. Prove that the line $OA'$ passes through one of the intersections of the circles $MFG$ and $ADE$.

Show that the shortest side of a cyclic quadrilateral inscribed in a unit circle cannot be longer than $\sqrt{2}\,$.

Calculate the exact value of $\sqrt[\scriptsize 3]{K}$, where $K$ denotes the product of all positive factors of $2025$.

$M$ and $N$ are the common points of circles $k_1$, centred at $O_1$ and $k_2$, centred at $O_2$. A secant drawn through point $M$ intersects circle $k_1$ at point $A$ and circle $k_2$ at point $B$ such that $A$ lies outside circle $k_2$ and $B$ lies outside circle $k_1$. Lines $AO_1$ and $BO_2$ intersect at point $P$. Points $N$ and $P$ lie in the same half plane determined by line $O_1O_2$. Show that $P$ lies on the circumscribed circle of triangle $O_1NO_2$.

Let $a$ and $b$ denote positive real numbers such that $a^2+b^2=\frac{2}{9}$. Prove that $\frac{1}{2-3a}+\frac{1}{2-3b} \ge 2.$

There are six people waiting in a line. It is taking very long, so they decide to play. They select a permutation of the six positions at random, and perform it three times successively. (A permutation is a rule of obtaining a new order by assigning a (possibly) new position to everyone. For example they may chose the following rule: the person at position 1 goes to position 3, the one at position 2 goes to position 1, the third goes to position 2, the fourth goes to position 6, the fifth stays in place, and the sixth goes to position 4.) What is the probability that at least one of them will be standing in their initial position at the end?

A long strip of paper is divided into squares. The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 are written in the first ten squares, then the same numbers are written in the next ten squares, too, and so on. There are exactly 2030 squares numbered in this way. A token is placed in square $1$. In each move, the token is moved as many squares ahead as the number written in the square it is standing on. What is the number in the square where the token is when its next move would get out off the paper strip of length $2030$?

Squares $ABCD, BEFC$ and $EGHF$ are drawn next to each other in a plane. The foot of the perpendicular drawn from $B$ to $DE$ is $K$. Show that the points $A$, $K$, $H$ are collinear.

What integers $a$, $b$ and $c$ satisfy the equality $a^2+b^2-8c=6$?

Solve the equation $4\cdot \cos\big(\pi\cdot \sin {(\pi \cdot x)}\big)=-5x^2+15x-\frac{61}{4}$ over the set of real numbers.

Johnny got a robot for Christmas, and he is trying it on a square carpet. The length of the sides of the square is $4$ metres, and its vertices are $A$, $B$, $C$ and $D$, in this order. Let $P$ denote the interior point of square $ABCD$ which is exactly $1$ metre away from both of sides $AB$ and $BC$. The robot is initially standing at point $P$, and starts to move in a random direction. When it has covered $2$ metres, it will stop. What is the probability that the robot will get off the carpet?

Find all positive integers for which the sum of the number and its factorial equals the cube of the number.

An equilateral triangle is cut with a line parallel to one of the sides. The circumscribed circles of the resulting triangle and trapezium are drawn. Is it possible that the ratio of the circumradius of the trapezium to that of the triangle is $\frac{\sqrt{3}}{3}$?

Is there a positive prime number $p$ for which $\log_{p-2}(4p-11)=m,$ where the parameter $m$ denotes one of the digits of $2023$?

Prove that the number $4^{52}+52^{2023}+2023^{52}$ is divisible by $15$.

Solve the simultaneous equations $$\begin{align*} x(2x+6)(3x+5y) & =64;\\ 2x^2+9x+5y & =16 \end{align*}$$ where $x$, $y$ are positive real numbers.

The base of a regular four-sided pyramid $ABCDE$ is the square $ABCD$, and each edge of the pyramid is $32$ units long. A snail starts at vertex $E$ and crawls to vertex $A$ as follows: first moves along edge $EA$ to the point $P$ for which $EP=2$. Then continues to cross the face $ABE$ and arrives at point $Q$ of edge $EB$, where $EQ=4$. Then crosses the face $BCE$ to reach that point $R$ of edge $EC$ for which $ER=8$, and continues along face $CDE$ to point $S$ on edge $ED$, with $ES=16$. Finally, it crawls on the surface of face $DAE$, from $S$ to point $A$. What is the minimum distance covered by the snail altogether?

Show that in every triangle (with conventional notations), $\sqrt{a\sin{\alpha}}+\sqrt{b\sin{\beta}}+\sqrt{c\sin{\gamma}}=\sqrt{(a+b+c)(\sin{\alpha}+\sin{\beta}+\sin{\gamma})}.$

Given are $2$ coins of $7$, $3$ coins of $17$, $5$ coins of $119$, $7$ coins of $289$, $11$ coins of $2023$ and $n$ coins of $1$. Two coins are selected at random, and their values are multiplied together, and a result of $2023$ is obtained. Find the value of $n$, given that the probability of obtaining that result is $\frac{12}{55}$.