KöMaL 2024

Let $A$ and $B$ be given real numbers. Let the sum of real numbers $0\le x_1\le x_2\le\dots\le x_n$ be $A$, and let the sum of real numbers $0\le y_1\le y_2\le\dots\le y_n$ be $B$. Find the largest possible value of ${\sum_{i=1}^n (x_i-y_i)^2}$.

We label every edge of a simple graph with the difference of the degrees of its endpoints. If the number of vertices is $N$, what can be the largest value of the sum of the labels on the edges?

Let $ABC$ be an obtuse triangle, and let $H$ denote its orthocenter. Let $\omega_A$ denote the circle with center $A$ and radius $AH$. Let $\omega_B$ and $\omega_C$ be defined in a similar way. For all points $X$ in the plane of triangle $ABC$ let circle $\Omega(X)$ be defined in the following way (if possible): take the polars of point $X$ with respect to circles $\omega_A$, $\omega_B$ and $\omega_C$, and let $\Omega(X)$ be the circumcircle of the triangle defined by these three lines. With a possible exception of finitely many points find the locus of points $X$ for which point $X$ lies on circle $\Omega(X)$.

For every positive integer $k$ let $a_{k,1},a_{k,2},\ldots$ be a sequence of positive integers. For every positive integer $k$ let sequence $\{a_{k+1, i}\}$ be the difference sequence of $\{a_{k, i}\}$, i.e. for all positive integers $k$ and $i$ the following holds: $a_{k, i+1}-a_{k, i}=a_{k+1, i}$. Is it possible that every positive integer appears exactly once among numbers $a_{k,i}$?

Let $ABCD$ be a convex cyclic quadrilateral satisfying $AB\cdot CD=AD\cdot BC$. Let the inscribed circle $\omega$ of triangle $ABC$ be tangent to sides $BC$, $CA$ and $AB$ at points $A'$, $B'$ and $C'$, respectively. Let point $K$ be the intersection of line $ID$ and the nine-point-circle of triangle $A'B'C'$ that is inside line segment $ID$. Let $S$ denote the centroid of triangle $A'B'C'$. Prove that lines $SK$ and $BB'$ intersect each other on circle $\omega$.

Nyihaha and Bruhaha are two neighbouring islands, both having $n$ inhabitants. On island Nyihaha every inhabitant is either a Knight or a Knave. Knights always tell the truth and Knaves always lie. The inhabitants of island Bruhaha are normal people, who can choose to say whatever they want. When a visitor arrives on any of the two islands, the following ritual is performed: every inhabitant points randomly to another inhabitant (independently from each other with uniform distribution), and says He is a Knight'' or He is a Knave''. On island Nyihaha , Knights have to tell the truth and Knaves have to lie. On island Bruhaha every inhabitant says either He is a Knight'' or He is a Knave'' with probability $1/2$ independently from each other. Sinbad arrives on island Bruhaha , but he does not know whether he is on island Nyihaha or island Bruhaha . Let $p_n$ denote the probability that after observing the ritual he can rule out being on island Nyihaha . Is it true that $p_n \to 1$ if $n \to \infty$?

$a)$ Two players play a cooperative game. They can discuss a strategy prior to the game, however, they cannot communicate and have no information about the other player during the game. The game master chooses one of the players in each round. The player on turn has to guess the number of the current round. Players keep note of the number of rounds they were chosen, however, they have no information about the other player's rounds. If the player's guess is correct, the players are awarded a point. Player's are not notified whether they've scored or not. The players win the game upon collecting 100 points. Does there exist a strategy with which they can surely win the game in a finite number of rounds? $b)$ How does this game change, if in each round the player on turn has two guesses instead of one, and they are awarded a point if one of the guesses is correct (while keeping all the other rules of the game the same)?

Find all non-negative integers $a$ and $b$ satisfying $5^a+6=31^b$.

Convex quadrilateral $ABCD$ is circumscribed about circle $\omega$. A tangent to $\omega$ parallel to diagonal $AC$ meets diagonal $BD$ at point $P$ outside of $\omega$. The second tangent from $P$ to $\omega$ touches $\omega$ at point $T$. Prove that $\omega$ and the circumcircle of triangle $ATC$ are tangent.

Let point $A$ be one of the intersections of circles $c$ and $k$. Let $X_1$ and $X_2$ be arbitrary points on circle $c$. Let $Y_i$ denote the intersection of line $AX_i$ and circle $k$ for $i=1,2$. Let $P_1$, $P_2$ and $P_3$ be arbitrary points on circle $k$, and let $O$ denote the center of circle $k$. Let $K_{ij}$ denote the center of circle $(X_iY_iP_j)$ for $i=1,2$ and $j=1,2,3$. Let $L_j$ denote the center of circle $(OK_{1j}K_{2j})$ for $j=1,2,3$. Prove that points $L_1$, $L_2$ and $L_3$ are collinear.

An integer $k>2$ is given. Xavier and Yvette play the following game: a number $n>k$ is initially written on the blackboard. The two players take turns, Xavier starts. In each turn the integer $m$ on the blackboard is replaced by integer $m'$ satisfying $k\le m'<m$ and $\gcd (m,m')=1$. The player who cannot make a legal move loses the game. We say that integer $n>k$ is good if Yvette has a winning strategy. Prove that if $n$, $n'>k$ are two integers satisfying the condition that every prime $p\le k$ divides $n$ if and only if it divides $n'$, then $n$ is good if and only if $n'$ is good.

Find all triples $(a,b,c)$ of real numbers for which there exists a function $f\colon \mathbb{Z}^+ \to \mathbb{Z}^+$ satisfying $af(n)+bf(n+1)+cf(n+2)<0$ for every $n \in \mathbb{Z}^+$ ($\mathbb{Z}^{+}$ denotes the set of positive integers).

We visit all squares exactly once on a $n\times n$ chessboard (colored in the usual way) with a king. Find the smallest number of times we had to switch colors during our walk.

Let $H_1$, $H_2$, $\ldots$, $H_m$ be non-empty subsets of the positive integers, and let $S$ denote their union. Prove that $\sum_{i=1}^m \sum_{a, b\in H_i} \gcd(a,b) \ge \frac{1}{m} \sum_{a, b\in S} \gcd(a, b)$. (The $\sum_{(a,b)\in X}$ notation means that the summation is over ordered pairs $(a,b)$ where $a\in X$ and $b\in X$.)

Let $J \subsetneq I \subseteq \mathbb{R}$ be non-empty open intervals, and let $f_1$, $f_2$, $\ldots$ be real polynomials satisfying the following conditions:

• $f_i(x) \geq 0$ for all $i \geq 1$ and $x \in I$,
• $\sum_{i=1}^\infty f_i(x)$ is finite for all $x \in I$,
• $\sum_{i=1}^\infty f_i(x)=1$ for all $x \in J$. Do these conditions imply that $\sum_{i=1}^\infty f_i(x)=1$ also for all $x \in I$?

We fill in an $n\times n$ table with real numbers such that the sum of the numbers in each row and each coloumn equals 1. For which values of $K$ is the following statement true: if the sum of the absolute values of the negative entries in the table is at most $K$, then it's always possible to choose $n$ positive entries of the table such that each row and each coloumn contains exactly one of the chosen entries.

Let triangle $ABC$ be a given acute scalene triangle with altitudes $BE$ and $CF$. Let $D$ be the point where the incircle of $\triangle ABC$ touches side $B C$. The circumcircle of $\triangle B D E$ meets line $A B$ again at point $K$, the circumcircle of $\triangle C D F$ meets line $A C$ again at point $L$. The circumcircle of $\triangle B D E$ and $\triangle CDF$ meet line $KL$ again at $X$ and $Y$, respectively. Prove that the incenter of $\triangle DXY$ lies on the incircle of $\triangle ABC$.

Let $k$ and $n$ be two given distinct positive integers greater than 1. There are finitely many (not necessarily distinct) integers written on the blackboard. Kázmér is allowed to erase $k$ consecutive elements of an arithmetic sequence with a difference not divisible by $k$. Similarly, Nándor is allowed to erase $n$ consecutive elements of an arithmetic sequence with a difference that is not divisible by $n$. The initial numbers on the blackboard have the property that both Kázmér and Nándor can erase all of them (independently from each other) in a finite number of steps. Prove that the difference of biggest and the smallest number on the blackboard is at least $\varphi(n)+\varphi(k)$, where $\varphi$ denotes Euler's totient function, i.e., $\varphi(n)$ is the number of positive integers not exceeding $n$ that are coprime to $n$.

A non self-intersecting polygon is given in a Cartesian coordinate system such that its perimeter contains no lattice points, and its vertices have no integer coordinates. A point is called semi-integer if exactly one of its coordinates is an integer. Let $P_1$, $P_2$, $\ldots$, $P_k$ denote the semi-integer points on the perimeter of the polygon. Let $n_i$ denote the floor of the non-integer coordinate of $P_i$. Prove that integers $n_1, n_2, \ldots , n_k$ can be divided into two groups with the same sum.

Let $n$ be a given positive integer. Find the smallest positive integer $k$ for which the following statement is true: for any given simple connected graph $G$ and minimal cuts $V_1$, $V_2$, $\ldots$, $V_n$, at most $k$ vertices can be chosen with the property that picking any two of the chosen vertices there exists an integer $1\leq i\leq n$ such that $V_i$ separates the two vertices. A partition of the vertices of $G$ into two disjoint non-empty sets is called a minimal cut if the number of edges crossing the partition is minimal.

Let $W, A, B$ be fixed real numbers with $W>0$. Prove that the following statements are equivalent.

• For all $x$, $y$, $z\geq0$ satisfying $x+y\leq z+W$, $x+z\leq y+W$, $y+z\leq x+W$ we have $Axyz+B\geq x^{2}+y^{2}+z^{2}$.
• $B\geq W^{2}$ and $AW^{3}+B\geq3W^{2}$.

Bart, Lisa and Maggie play the following game: Bart colors finitely many points red or blue on a circle such that no four colored points can be chosen on the circle such that their colors are blue-red-blue-red (the four points do not have to be consecutive). Lisa chooses finitely many of the colored points. Now Bart gives the circle (possibly rotated) to Maggie with Lisa's chosen points, however, without their colors. Finally, Maggie colors all the points of the circle to red or blue. Lisa and Maggie wins the game, if Maggie correctly guessed the colors of Bart's points. A strategy of Lisa and Maggie is called a winning strategy, if they can win the game for all possible colorings by Bart. Prove that Lisa and Maggie have a winning strategy, where Lisa chooses at most $c$ points in all possible cases, and find the smallest possible value of $c$.

Let $ABC$ be an acute triangle. Points $B'$ and $C'$ are located on the interior of sides $AB$ and $AC$, respectively. Let $M$ denote the second intersection of the circumcircles of triangles $ABC$ and $AB'C'$, while let $N$ denote the second intersection of the circumcircles of triangles $ABC'$ and $AB'C$. Reflect $M$ across lines $AB$ and $AC$, and let $l$ denote the line through the reflections. a) Prove that the line through $M$ perpendicular to line $AM$, the line $AK$, and $l$ are either concurrent or all parallel. b) Show that if the three lines are concurrent at $S$, then triangles $SBC'$ and $SCB'$ have equal areas.

Given two integers, $k$ and $d$ such that $d$ divides $k^3-2$. Show that there exists integers $a$, $b$, $c$ satisfying $d=a^3+2b^3+4c^3-6abc$.

In a text editor program, initially there is a footprint symbol (L) that we want to multiply. Unfortunately, our computer has been the victim of a hacker attack, and only two functions are working: Copy and Paste, each costing $1$ Dürer dollar to use. Using the Copy function, we can select one or more consecutive symbols from the existing ones, and the computer memorizes their number. When using the Paste function, the computer adds as many new footprint symbols to the sequence as were selected in the last Copy. If no Copy has been done yet, Paste cannot be used. Let $D(n)$ denote the minimum number of Dürer dollars required to obtain exactly $n$ footprint symbols. Prove that for any positive integer $k$, there exists a positive integer $N$ such that $D(N)={D(N+1)}+1={D(N+2)}={D(N+3)}+1={D(N+4)}=\ldots ={D(N+2k-1)}+1={D(N+2k)}$.

In convex polyhedron $ABCDE$ line segment $DE$ intersects the plane of triangle $ABC$ inside the triangle. Rotate the point $D$ outward into the plane of triangle $ABC$ around the lines $AB$, $BC$, $CA$; let the resulting points be $D_1$, $D_2$, and $D_3$. Similarly, rotate the point $E$ outward into the plane of triangle $ABC$ around the lines $AB$, $BC$, $CA$; let the resulting points be $E_1$, $E_2$, and $E_3$. Show that if the polyhedron has an inscribed sphere, then the circumcircles of $D_1D_2D_3$ and $E_1E_2E_3$ are concentric.

Let's call a function $f\colon\mathbb{R}\to\mathbb{R}$ weakly periodic if it is continuous and satisfies ${f(x+1)}=f(f(x))+1$ for all $x\in \mathbb{R}$. a) Does there exist a weakly periodic function such that $f(x)>x$ for all $x\in\mathbb{R}$? b) Does there exist a weakly periodic function such that $f(x)<x$ for all $x\in\mathbb{R}$?

Find the largest number of distinct integers that can be chosen in a way that the sum of any two chosen integers is a perfect power of 2 (with a non-negative exponent).

$a)$ Does there exist a triangle such that all of its sides have integer length, and its area is less than $1/10$? $b)$ Does there exist a quadrilateral such that all of its sides have integer length, and its area is less than $1/10$?

The largest angle of scalene triangle $ABC$ is at vertex $C$. Let $T$ denote the foot of altitude from vertex $C$, and let the angle bisector from vertex $C$ intersect side $AB$ at point $F$. Prove that $\frac{AT}{TB}=\bigg(\frac{AF}{FB}\bigg)^2$ if and only if $\angle ACB =90^\circ$.

Let $a_n$ denote the number of ways $2n$ can be obtained as a sum of two positive prime numbers. Is it true that sequence $(a_n)$ will be strictly increasing from somewhere?

In triangle $ABC$ it is given that $\angle BAC=50^\circ$ and $\angle ABC=70^\circ$. On sides $AC$ and $BC$ points $E$ and $D$ are chosen satisfying the following properties: $\angle ABE =\angle BAD = 30^\circ$. Let $M$ denote the intersection point of line segments $AD$ and $BE$. Find the magnitude of angles $\angle ACM$ and $\angle BED$.

Square $ABCD$ is the base of a right square pyramid, and point $E$ is its apex. Let $F$ be the midpoint of lateral edge $CE$, and let point $H$ be the trisection point of lateral edge $BE$ closer to point $B$. Find the ratio of the volumes of the two parts created by plane $AHF$ in pyramid $ABCDE$.

We label the two sides of a coin with integers 1 and 2, and the faces of a cube with integers 1, 2, 3, 4, 5 and 6. Find all weightings of the coin and the cube in a way that tossing both will give the same probabilities for sums 2, $\ldots$, 8 as two regular tetrahedrons labeled with 1, 2, 3 and 4. (Tossing a tetrahedron will result in the number appearing on the face that rests on the table; and weighting means that the weight distribution of an object can be altered in a way that it will land on its sides with different probabilities.)

Find the smallest real number $\alpha$ for which it is possible to find infinitely many positive integers $n$ with the property that the difference of $\sqrt{13}\cdot n$ and the nearest integer is less than $\alpha /n$.

Is it possible to find a composite number $n>1$ with the following property: if $1=d_1<d_2<\ldots<d_k=n$ denote the positive divisors of $n$, then $d_i$ is divisible by $(d_{i-1} + d_{i-2})$ for all $3 \leq i \leq k$?

$a)$ In the open unit disk we place two open line segments of length $\ell$ with no common points, perpendicularly to each other. What can be the value of $\ell$? $b)$ In the open unit ball we place three line segments of length $\ell$ such that any two are disjoint (have no common points) and perpendicular to each other. What can be the value of $\ell$?

In a table tennis championship any two contestants played with each other exactly once. In each match, the winner got 1 point, and the loser got 0 points (there is no tie in table tennis). Interestingly, one contestant has won against exactly those who scored more than him in the championship and lost to exactly those who scored less than him in the championship. Find the smallest possible number of contestants participating in the championship. (It is possible that several contestants got the same score at the end of the championship. At least two contestants participated in the championship.)

Inside an equilateral triangle $ABC$, point $P$ is chosen such that $\angle APB=150^\circ$. Prove that ${PA^2+PB^2=PC^2}$.

Let $k$ be a positive integer, and assume that real numbers $a_1$, $\ldots$, $a_k$ satisfy ${\sum_{i=1}^k (k-i+1) a_i=0}$. Prove that there exists positive integer $m\le k$ for which $2m\sum_{i=1}^{m} a_i\le \sum_{i=1}^m i a_i$.

Let point $P$ be chosen inside triangle $ABC$. Let $D$, $E$ and $F$ denote the orthogonal projections of point $P$ onto sides $BC$, $CA$ and $AB$, respectively. Prove that $\frac{PE+PF}{PA}+\frac{PF+PD}{PB}+\frac{PD+PE}{PC}\le 3$.

A sphere has been partitioned into spherical triangles and quadrilaterals by drawing some great circles. No three of the great circles pass through the same point, and at least one quadrilateral was created. Prove that there must be exactly eight spherical triangles and six spherical quadrilaterals.

Let $n$ be a positive integer. Prove that there are at least 8 odd coefficients in polynomial $a_{7n}x^{7n}+\dots+a_1x+a_0=(x^7+x^6+x^5+x^4+x^3+x^2+x+1)^n$.

Square $ABCD$ and rhombus $BAEF$ are drawn on the opposite sides of line segment $AB$. Let $K$ and $M$ denote the center of the square and the rhombus, respectively. Prove that $KM$ bisects angle $AMB$.

Solve equation $\left(m-k \right)^{2}={m+k}$ for non-negative integers $m$ and $k$.

Let us divide positive integer $n$ with all the positive integers smaller than $n$, and let $f(n)$ denote the sum of the remainders we've obtained. (For example, when we divide $n=5$ by $1$, $2$, $3$ and $4$, we get remainders $0$, $1$, $2$ and $1$, respectively, therefore $f(5)=4$.) Solve equation $f(n)=n$.

Determine the set of real numbers $p$ satisfying the following property: inequality $\sqrt{a^2+b^2-ab}+\sqrt{b^2+c^2-bc}\geq \sqrt{a^2+c^2-p\cdot ac}$ holds for all triples of positive real numbers $a$, $b$ and $c$, for which the square roots are defined.

Let $n$ and $k$ be positive integers. Prove that if $n \leq k^{11}$, then $n$ can be obtained as the product of ten positive integers, among which no composite number greater than $k^2$ appears.

Let $ABC$ be a triangle with a right angle at $C$. On the hypotenuse $AB$, let $H$ and $D$ be the feet of the altitude and the angle bisector starting from $C$, respectively. The bisector of angle $AHC$ meets $AC$ at point $E$, and the bisector of angle $CHB$ meets $BC$ at $F$. Mark a point $M$ on line segment $HE$ and a point $N$ on line segment $HF$ that satisfy $HM:HE=HN:HF$. Show that lines $CD$, $AM$ and $BN$ are concurrent.

Find the smallest possible degree of polynomial $f$ of one variable, if it satisfies the following property: the range of $f$ and $f\circ f$ are different, while the range of $f \circ f$ and $f\circ f\circ f$ are the same. (The symbol $\circ$ denotes function composition.)

Do there exist prime numbers $2<p<q$ for which more than one third of the elements of set $\{ p+1, p+2, \ldots, q-1 \}$ is a prime number?

Cyclic quadrilateral $ABCD$ has the following properties: $\angle BAD = 90^{\circ}$, $BC = CD$ and $AC = 1$. Find the area of $ABCD$.

Prove that if $a$, $b$, $c>0$ and $a^2+b^2+c^2=abc$, then $2(a+b+c)+\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab} \le abc$.

Let $F$ denote the center of the nine-point circle of acute triangle $ABC$. Prove that $AF^2-BF^2=Rc\sin(\beta-\alpha)$, where $R$ denotes the radius of the circumcircle, $c$ denotes the length of side $AB$, and $\alpha$ and $\beta$ denote the interior angles at $A$ and $B$.

Anna and Balázs play the following game: Anna tosses a fair coin 101 times, and Balázs tosses a fair coin 10 times. Anna wins, if she receives more than ten times as many heads as Balázs, otherwise Balázs wins. Find the player to whom this game is favorable.

We have colored finitely many points on the plane to red, blue or green such that no three points of the same color are collinear, however, any line segment connecting two points with the same color contains a third colored point (with a different color). Find the maximum number of the colored points.

Prove that any $2n$ consecutive positive integers can be divided into $n$ pairs in at least $n!$ different ways such that in each pair the product of the two numbers is not a perfect square.

Let $I$ denote the incenter of acute triangle $ABC$. Let the incircle touch sides $BC$ and $AC$ at points $D$ and $E$, respectively. Let $H$ denote the orthocenter of triangle $ABC$. Prove that if $H$ is on line segment $DE$, then line $HI$ bisects side $AB$.

Is it possible to find even integers $a_0$, $a_1$, $\ldots$, $a_{n-1}$ for which polynomial $x^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0$ is divisible by $x^2+x+1$?

Let $C$ be a point on a circle with unit diameter $AB$. Let points $D$ and $E$ be chosen on line segment $AB$ such that $BD = BC$ and $AE = AC$. Find the smallest possible value of $AD^2 + DE^2 + EB^2$.

Let us consider a trapezoid whose area is equal to the product of the lengths of its bases. Prove that this trapezoid is tangential if and only if it is right.

Let $f$ be a real function (that is, a real-valued function of a real variable) satisfying $\left|f(x+y+z)+\sin x+\sin y+\sin z \right|\le 3$, for all $x$, $y$, $z\in \mathbb{R}$. Prove that $|f(x)-\sin x|\le 1$ for all $x\in \mathbb{R}$.

Let $O$ be the center of square $ABCD$, and let $X$ be an arbitrary point on its circumcircle. Let $T$ denote the orthogonal projection of $X$ onto $BC$. Let $E$ denote the intersection of lines $XB$ and $AC$, and let $F$ denote the intersection of lines $XC$ and $BD$. Prove the $EF$ is perpendicular to $TO$.

Let $d(k)$ denote the number of positive divisors of a positive integer $k$, and let $1<n$ be an integer. Which sum is larger: $d(2)+d(4)+\dots+d(2n)$ or $(d(1)+d(3)+\dots+d(2n-1))+(d(1)+d(2)+\dots+d(n))$?

An equilateral pentagon in the three-dimensional space has four right angles. What can be its fifth angle?

Let $G$ be a graph (possibly with multiple edges) with the following property: whenever the vertices of $G$ are partitioned into $t$ disjoint sets, there are at least $2t-2$ edges connecting different sets. Show that the edges of $G$ can be colored red or blue such that the red edges and the blue edges each form a connected graph (and reaching all the vertices).

In trapezoid $ABCD$ the bases are $AB \parallel CD$ and $\angle ADC - \angle CBA = 90^{\circ}$. Prove that the sum of the squares of the legs equals the difference of the squares of the bases.

A five-digit perfect square does not contain the digit 9. If we increase each of its digits by 1, we get another perfect square. Find all possible perfect squares with this property.

In a $3\times 3$ magic square we increased one of the nine entries by 1. Find the least number of additional entries that has to be changed to obtain a magic square again. (The $3\times 3$ magic square is a number grid in which all three numbers in each of the rows, columns, and diagonals add up to the same number.)

Find the biggest possible value of product $mn$, provided that $m$, $n$ and $\sqrt{25+\sqrt{n+\sqrt m}} + \sqrt{25-\sqrt{n+\sqrt m}}$ are all positive integers.

The side lengths $a$, $b$ and $c$ of a triangle satisfy $a^2+b^2+c^2=a^2b^2c^2$. Prove that the area of the triangle is at most $\frac34$, and if equality holds, then the triangle is equilateral.

Suppose that the edges of a simple, connected $k$-regular ($k \ge 2$) graph $G$ can be colored with $k$ colors such that the colors of the edges meeting at any given vertex are all different. Prove that if we delete any edge from $G$, the resulting graph will still be connected.

The altitudes of the acute triangle $ABC$ are $AT_A$, $BT_B$ and $CT_C$. The midpoints of the sides $BC$, $CA$ and $AB$ are $F_A$, $F_B$ and $F_C$, respectively. Let $r$ denote the radius of the inscribed circle and let $P_A$ denote the point on altitude $AT_A$ satisfying $AP_A = r$. Points $P_B$ and $P_C$ are defined similarly. Prove that line segments $F_AP_A$, $F_BP_B$ and $F_CP_C$ are concurrent.

Positive integers $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ satisfy the following property: for all indices $i < j \leq n$ the greatest common divisor of $b_i$ and $b_j$ does not divide difference $(a_i-a_j)$. Prove that $\sum_{i=1}^n \frac1{b_i} \leq 1$.

Prove that in any base-$n$ number system, equality $\sqrt{\frac{123456787654321}{1234321}}=10001$ holds, where $n\ge 9$.

Find positive integers $a$, $b$, $c$ and $d$ satisfying $\frac{b}{a}=\frac{c}{b}=\frac{d}{c}$ and $\frac{a+c}{2}=b+1$.

In a triangle the length of one of the sides is the arithmetic mean of the lengths of the other two sides. Prove that the length of the angle bisector corresponding to the middle side is $\frac{\sqrt{3}}{2}$ times the geometric mean of the lengths of the other two sides.

Each card in a deck of german suited playing cards has a suit and a value. The suit can be Hearts, Bells, Leaves or Acorns, and the value can be VII, VIII, IX, X, Unter, Ober, King or Ace. The deck contains all possible combinations of the suits and the values. We arrange the 32 cards of a deck in 4 rows and 8 coloumns. Let $A$ denote the random event that no coloumn contains two cards with the same suit, and let $B$ denote the random event that no row contains two cards with the same value. Which of the two events has the larger probability?

We draw three circles centered at each of the three vertices of a right triangle such that each two of these circles are externally tangent to each other. Determine the center and the radius of the circle that is internally tangent to all three of these circles.

Prove the following statement for every $n\ge 2$ positive integer: $\sum_{k=1}^{n^2-1} \lfloor \sqrt{k} \rfloor]^2=\frac{(n-1)n(3n^2-n-1)}{6}$. $\lfloor x \rfloor$ denotes the floor of the real number $x$.

A fugitive at a vertex of a finite connected graph is being chased by three policemen at three other vertices. First the fugitive makes a move along an edge, then each of the three policemen can also make a move along an edge. The game continues by alternating turns between the fugitive and the policemen. Can the policemen always catch the fugitive in a finite number of steps, i.e., can they always guarantee that one of them will be on the same vertex as the fugitive after a finite number of steps?

Let $M$ be the orthocenter, $S$ be the centroid and $I$ be the incenter of the non-equilateral triangle $ABC$. Prove that $\angle MIS>90^\circ$.

Let $ABCD$ be a rectangle. Points $P$ and $Q$ are chosen such that the circumcenter of triangle $ABP$ is $Q$ and the circumcenter of triangle $BCQ$ is $P$. Find the magnitude of angle $PDQ$.

Each of the three kids, Beni, Lili and Domi completes three laps on the athletics track. The referee keeps track of the names of each kid completing a lap, and thus gets a list of nine names. Find the number of different lists the referee can get supposing that no two kids complete a lap at the same time and the speed of each kid remains constant for all three laps.

Real numbers $x$, $y$ and $z$ satisfy $x+y+z=8$ and $xy+yz+zx=5$. Find the biggest possible value of $z$.

Which number is bigger, $\left(2^{1000}\right)!$ or $2^{1000!}?$

Let $a$, $b$ and $c$ denote the side lengths of acute triangle $ABC$, and let $R$ denote its circumradius. Prove that $\frac{1}{-a^2+ b^2+c^2}+\frac{1}{a^2-b^2+c^2}+\frac{1}{a^2+b^2-c^2}\ge \frac{1}{R^2},$ and find those triangles where equality holds.

Let $q(n)$ denote the greatest odd divisor of positive integer $n$. Let $P(n)=q(1)+q(2)+\ldots+q(n)$ and $S(n)=1+2+\ldots+n$. Prove that the ratio $P(n)/S(n)$ is smaller than $2/3$ for infinitely many values of $n$, and also greater than $2/3$ for infinitely many values of $n$.

Ádám, the famous conman signed up for the following game of luck. There is a rotating table with a shape of a regular 13-gon, and at each vertex there is a black or a white cap. (Caps of the same colour are indistinguishable from each other.) Under one of the caps 1000 dollars are hidden, and there is nothing under the other caps. The host rotates the table, and then Ádám chooses a cap, and take what is underneath. Ádám's accomplice, Béla is working at the company behind this game. Béla is responsible for the placement of the 1000 dollars under the caps, however, the colors of the caps are chosen by a different collegaue. After placing the money under a cap, Béla a) has to change the color of the cap, b) is allowed to change the color of the cap, but he is not allowed to touch any other cap. Can Ádám and Béla find a strategy in part a) and in part b) , respectively, so that Ádám can surely find the money? (After entering the casino, Béla cannot communicate with Ádám, and he also cannot influence his colleague choosing the colors of the caps on the table.)

The incenter and the inradius of the acute triangle $ABC$ is $I$ and $r$, respectively. The excenter and exradius relative to vertex $A$ is $I_a$ and $r_a$, respectively. Let $R$ denote the circumradius. Prove that if $II_a=r_a+R-r$, then $\angle BAC = 60^{\circ}$.

Two natural numbers are relatives if they differ in at most one digit. For example, 135 and 175 are relatives, and so are 101 and 1 (that is, 001), but 135 and 513 are not. Does there exist a number whose relatives are all composite?

$\{x\}$ denotes the fractional part of $x$. Does there exist a positive integer $n$ for which $\bigl\{\sqrt{2} n\bigr\}\cdot\bigl\{\frac{n}{\sqrt{2}}\bigr\}$ is rational?

Let $ABCD$ be a cyclic quadrilateral, let $E$ be the midpoint of $AC$ and $O$ be the centre of the circumcircle such that $O$ and $E$ are distinct. Prove that if $OEBD$ is cyclic then $EC$ bisects angle $DEB$.

Jumpy, the grasshopper is jumping around on the positive integers of the numberline, visiting each exactly once. Is it possible that the lengths of his jumps produce every positive integer exactly once?

Let $P$ be an internal point in triangle $ABC$. Let lines $AP$, $BP$ and $CP$ intersect sides $BC$, $CA$ and $AB$ in points $L$, $M$ and $N$, respectively. Prove that $P$ is the centroid of triangle $ABC$ if and only if $P$ is the centroid of triangle $LMN$.

Solve the equation $5^{a}+12^{b}=13^{c}$ on non-negative integers.

For every points $X$, $Y$, $Z$, denote by $[XYZ]$ the area of triangle $XYZ$. Show that if points $A$, $B$, $C$, $D$, $E$, $F$ lie on a non-degenerate conic section then ${[ABC]\cdot[CDE]\cdot[EFA]\cdot[BDF]=[BCD]\cdot[DEF]\cdot[FAB]\cdot[ACE]}$.

Function $f$ defined on the real numbers satisfies the following two conditions (for all $x \in \mathbb{R}$): $(1)$ $f(x)=f(147-x),$ $(2)$ $f(x+100)=f(46-x).$ Find the value of $f(200)+f(201)+f(202)$, if we also know that $f(50)+f(51)+f(52)+f(53)=2024$.

Circles $k_1$ and $k_2$ are externally tangent to each other at point $A$, circles $k_2$ and $k_3$ are externally tangent to each other at point $B$, and finally circles $k_1$ and $k_3$ are externally tangent to each other at point $C$. Line $AB$ intersect circle $k_3$ at point $D$ (different from $B$). Prove that line segments $AC$ and $CD$ are perpendicular to each other.

Let $p$ be a real parameter. Find the number of solutions of the following equation as a function of $p$: $|x^2-6x+5|=px^2-6px+9p+4$.

A regular octahedron with edges of unit length is placed on the plane of one of its triangular face. Find the distance of this plane and the vertices of the octahedron that are not contained in this plane.

Find the largest integer value of $x$ for which $x>2$ and the values $\log_2 (x)$, $\log_4 (2x)$ and $\log_8 (3x)$ can be the side lengths of a triangle.

Let parameter $p$ be a given integer. Find all integer solutions of the equation $\big(p+\frac{1}{p}\big)\cdot\big(x-\frac{1}{x}\big)+\big(p-\frac{1}{p}\big)\cdot\big(x+\frac{1}{x}\big)=4px+5+\frac{1}{p}$.

Prove that if $n$ is a natural number, then the interval $\left[\sqrt{16n+21},\sqrt{16n+24}\right]$ contains no integers.

Let sequence $a_n$ be defined as $a_1=2$ and $a_n=a_{n-1}+2n$. Find the sum of the reciprocals of the first 2024 terms of the sequence (i.e. find the value of $\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+ \dots +\frac{1}{a_{2024}}$).

In regular hexagon $ABCDEF$ point $M$ is chosen on diagonal $AC$ and point $N$ is chosen on diagonal $DE$ satisfying $\frac{AM}{AC}=\frac{CN}{CE}=k$. For which values of $k$ will points $B$, $M$ and $N$ be collinear?

Determine the number of triples of positive integers with greatest common divisor $4$ and least common multiple 2024.

In triangle $ABC$ let $D$, $E$ and $F$ denote the midpoints of the sides $BC$, $CA$ and $AB$, respectively. Let $K_a$, $K_b$ and $K_c$ denote the centers of the incircles of triangles $AFE$, $BDF$ and $CED$, respectively. Prove that the sum of the areas of quadrilaterals $K_aFDE$, $K_bDEF$ and $K_cEFD$ equals the area of triangle $ABC$.

Solve equation ${\frac{6x-3}{3x}-\Big(3y^2-14xy+8x\Big)^2=x}$ for positive real numbers $x$ and $y$.

In triangle $ABC$ angles satisfy $2\beta = 3\gamma$. Let points $D$ and $E$ be chosen on side $AC$ such that $BD$ and $BE$ trisect angle $\beta$, and point $D$ is located between points $A$ and $E$. Further, let $F$ denote the intersection of side $AB$ and the angle bisector of $\gamma$. Prove that $BE$ and $DF$ are parallel.