KöMaL 2022

Let $q$ be a monic polynomial with integer coefficients. Prove that there exists a constant $C$ depending only on polynomial $q$ such that for an arbitrary prime number $p$ and an arbitrary positive integer $N\le p$ the congruence $n! \equiv q(n) \pmod{p}$ has at most $CN^{2/3}$ solutions among any $N$ consecutive integers.

Peter has 2022 pieces of magnetic railroad cars, which are of two types: some has the front with north and the rear with south magnetic polarity, and some has the rear with north and the rear with south magnetic polarity (on these railroad cars the front and the rear can be distinguished). Peter wants to decide whether there are the same number of both type of cars. He can try to fit together two cars in one try. What is the least number of tries needed?

Let $ABC$ be a triangle. Let $T$ be the point of tangency of the circumcircle of triangle $ABC$ and the $A$-mixtilinear incircle (the circle which is tangent to sides $AB$, $AC$, and internally tangent to the circumcircle of triangle $ABC$). The incircle of triangle $ABC$ has center $I$ and touches sides $BC$, $CA$ and $AB$ at points $D$, $E$ and $F$, respectively. Let $N$ be the midpoint of line segment $DF$. Prove that the circumcircle of triangle $BTN$, line $TI$ and the perpendicular from $D$ to $EF$ are concurrent.

Find all pairs of positive integers $m$, $n$ such that $9^{|m-n|}+3^{|m-n|}+1$ is divisible by $m$ and $n$ simultaneously.

Let $G$ be an arbitrarily chosen finite simple graph. We write non-negative integers on the vertices of the graph such that for each vertex $v$ in $G$ the number written on $v$ is equal to the number of vertices adjacent to $v$ where an even number is written. Prove that the number of ways to achieve this is a power of 2.

Let $ABC$ be an arbitrary triangle. Let the excircle tangent to side $a$ be tangent to lines $AB$, $BC$ and $CA$ at points $C_a$, $A_a$ and $B_a$, respectively. Similarly, let the excircle tangent to side $b$ be tangent to lines $AB$, $BC$ and $CA$ at points $B_c$, $B_a$ and $B_b$, respectively. Finally, let the excircle tangent to side $c$ be tangent to lines $AB$, $BC$ and $CA$ at points $C_c$, $A_c$ and $B_c$, respectively. Let $A'$ be the intersection of lines $A_bC_b$ and $A_cB_c$. Similarly, let $B'$ be the intersection of lines $B_aC_a$ and $A_cB_c$, and let $C'$ be the intersection of lines $B_aC_a$ and $A_bC_b$. Finally, let the incircle be tangent to sides $a$, $b$ and $c$ at points $T_a$, $T_b$ and $T_c$, respectively. $a)$ Prove that lines $A'A_a$, $B'B_b$ and $C'C_c$ are concurrent. $b)$ Prove that lines $A'T_a$ and $B'T_b$ and $C'T_c$ are also concurrent, and their point of intersection is on the line defined by the orthocentre and the incentre of triangle $ABC$.

$a)$ Is it possible to find a function $f\colon \mathbb{N}^2\to \mathbb{N}$ such that for every function $g\colon \mathbb{N}\to \mathbb{N}$ and positive integer $m$ there exists $n \in \mathbb{N}$ such that set $\big\{k\in \mathbb{N} : f(n,k)=g(k)\big\}$ has at least $m$ elements? $b)$ Is it possible to find a function $f\colon \mathbb{N}^2\to \mathbb{N}$ such that for every function $g\colon \mathbb{N}\to \mathbb{N}$ there exists $n \in \mathbb{N}$ such that set $\big\{k\in \mathbb{N} \colon f(n,k)=g(k)\big\}$ has an infinite number of elements?

Is it possible to find rational numbers $p$, $q$ and $r$ such that $p+q+r=0$ and $pqr=1$?

For positive integers $n$ consider the lattice points $S_n=\big\{(x,y,z)\colon 1\le x\le n$, $1\le y\le n$, $1\le z\le n$, $x,y,z\in \mathbb{N}\big\}$. Is it possible to find a positive integer $n$ for which it is possible to choose more than $n\sqrt{n}$ lattice points from $S_n$ such that for any two chosen lattice points at least two of the coordinates of one is strictly greater than the corresponding coordinates of the other?

An infinite set $S$ of positive numbers is called thick, if in every interval of the form $\big[1/(n+1), 1/n\big]$ (where $n$ is an arbitrary positive integer) there is a number which is the difference of two elements from $S$. Does there exist a thick set such that the sum of its elements is finite?

Find all functions $f\colon \mathbb{Z}^+\to \mathbb{R}^+$ that satisfy $f(nk^2)=f(n)f^2(k)$ for all positive integers $n$ and $k$, furthermore $\lim\limits_{n\to \infty} \frac{f(n+1)}{f(n)}=1$.

An antelope is a chess piece which moves similarly to the knight: two cells $(x_1, y_1)$ and $(x_2, y_2)$ are joined by an antelope move if and only if $\big\{|x_1-x_2|, |y_1-y_2|\big\} = \{3, 4\}.$ The numbers from $1$ to $10^{12}$ are placed in the cells of a $10^6 \times 10^6$ grid. Let $D$ be the set of all absolute differences of the form $|a-b|$, where $a$ and $b$ are joined by an antelope move in the arrangement. How many arrangements are there such that $D$ contains exactly four elements?

Let $n>1$ be a given integer. In a deck of cards the cards are of $n$ different suites and $n$ different values, and for each pair of a suite and a value there is exactly one such card. We shuffle the deck and distribute the cards among $n$ players giving each player $n$ cards. The players' goal is to choose a way to sit down around a round table so that they will be able to do the following: the first player puts down an arbitrary card, and then each consecutive player puts down a card that has a different suite and different value compared to the previous card that was put down on the table. For which $n$ is it possible that the cards were distributed in such a way that the players cannot achieve their goal? (The players work together, and they can see each other's cards.)

Triangle $ABC$ has incenter $I$ and excircles $\Omega_A$, $\Omega_B$, and $\Omega_C$. Let $\ell_A$ be the line through the feet of the tangents from $I$ to $\Omega_A$, and define lines $\ell_B$ and $\ell_C$ similarly. Prove that the orthocenter of the triangle formed by lines $\ell_A$, $\ell_B$, and $\ell_C$ coincides with the Nagel point of triangle $ABC$. (The Nagel point of triangle $ABC$ is the intersection of segments $AT_A$, $BT_B$, and $CT_C$, where $T_A$ is the tangency point of $\Omega_A$ with side $BC$, and points $T_B$ and $T_C$ are defined similarly.)

Let $G$ be a simple graph on $n$ vertices with at least one edge, and let us consider those $S\colon V(G)\to \mathbb{R}^{\ge 0}$ weighings of the vertices of the graph for which $\sum\limits_{v\in V(G)}S(v)=1$. Furthermore define $f(G)=\max_S \min_{(v,w)\in E(G)} S(v)S(w),$ where $S$ runs through all possible weighings. Prove that $f(G)=\frac{1}{n^2}$ if and only if the vertices of $G$ can be covered with a disjoint union of edges and odd cycles. ($V(G)$ denotes the vertices of graph $G$, $E(G)$ denotes the edges of graph $G$.)

For $H\subset \mathbb{Z}$ and $n\in \mathbb{Z}$, let $h_n$ denote the number of finite subsets of $H$ in which the sum of the elements is $n$. Does there exist $H\subset \mathbb{Z}$, for which $0\notin H$, and $h_n$ is a (finite) even number for every $n\in \mathbb{Z}$? (The sum of the elements of the empty set is 0.)

In triangle $ABC$ let $F$ denote the midpoint of side $BC$. Let the circle passing through point $A$ and tangent to side $BC$ at point $F$ intersect sides $AB$ and $AC$ at points $M$ and $N$, respectively. Let line segments $CM$ and $BN$ intersect in point $X$. Let $P$ be the second point of intersection of the circumcircles of triangles $BMX$ and $CNX$. Prove that points $A$, $F$ and $P$ are collinear.

Let us assume that the number of offsprings for every man can be $0, 1,\dots$ or $n$ with probabilities $p_0,p_1,\dots,p_n$ independently from each other, where $p_0+p_1+\dots+p_n=1$ and $p_n \ne 0$. (This is the so called Galton-Watson process.) Which positive integer $n$ and probabilities $p_0,p_1,\dots,p_n$ will maximize the probability that the offsprings of a given man go extinct in exactly the tenth generation?

Some lattice points in the Cartesian coordinate system are colored red, the rest of the lattice points are colored blue. Such a coloring is called finitely universal, if for any finite, non-empty $A \subset \mathbb{Z}$ there exists $k\in \mathbb{Z}$ such that the point $(x,k)$ is colored red if and only if $x\in A$. $a)$ Does there exist a finitely universal coloring such that each row has finitely many lattice points colored red, each row is colored differently, and the set of lattice points colored red is connected? $b)$ Does there exist a finitely universal coloring such that each row has a finite number of lattice points colored red, and both the set of lattice points colored red and the set of lattice points colored blue are connected? A set $H$ of lattice points is called connected if, for any $x,y\in H$, there exists a path along the grid lines that passes only through lattice points in $H$ and connects $x$ to $y$.

Let $A_1A_2\ldots A_8$ be a convex cyclic octagon, and for $i=1,2\ldots,8$ let $B_i=A_iA_{i+3}\cap A_{i+1}A_{i+4}$ (indices are meant modulo 8). Prove that points $B_1,\ldots,B_8$ lie on the same conic section.

A. 835. Let $f^{(n)}(x)$ denote the $n^\text{th}$ iterate of function $f$, i.e $f^{(1)}(x)=f(x)$, $f^{(n+1)}(x)=f\Big(f^{(n)}(x)\Big)$. Let $p(n)$ be a given polynomial with integer coefficients, which maps the positive integers into the positive integers. Is it possible that the functional equation $f^{(n)}(n)=p(n)$ has exactly one solution $f$ that maps the positive integers into the positive integers?

For every $i \in \mathbb{N}$ let $A_i$, $B_i$ and $C_i$ be three finite and pairwise disjoint subsets of $\mathbb{N}$. Suppose that for every partition of $\mathbb{N}$ consisting of sets $A$, $B$ and $C$ there exists $i\in \mathbb{N}$ such that $A_i \subset A$, $B_i \subset B$ and $C_i \subset C$. Prove that there also exists a finite $S\subset \mathbb{N}$ such that for every partition of $\mathbb{N}$ consisting of sets $A$, $B$ and $C$ there exists $i\in S$ such that $A_i \subset A$, $B_i \subset B$ and $C_i \subset C$.

Let all the edges of tetrahedron $A_1A_2A_3A_4$ be tangent to sphere $S$. Let $a_i$ denote the length of the tangent from $A_i$ to $S$. Prove that $\bigg(\sum_{i=1}^4 \frac 1{a_i}\bigg)^{\! 2}> 2\bigg(\sum_{i=1}^4 \frac1{a_i^2}\bigg).$

Sets $X\subset \mathbb{Z}^{+}$ and $Y\subset \mathbb{Z}^{+}$ are called comradely, if every positive integer $n$ can be written as $n=xy$ for some $x\in X$ and $y\in Y$. Let $X(n)$ and $Y(n)$ denote the number of elements of $X$ and $Y$, respectively, among the first $n$ positive integers. Let $f\colon \mathbb{Z}^{+}\to \mathbb{R}^{+}$ be an arbitrary function that goes to infinity. Prove that one can find comradely sets $X$ and $Y$ such that $\frac{X(n)}{n}$ and $\frac{Y(n)}{n}$ goes to $0$, and for all $\varepsilon>0$ exists $n \in \mathbb{Z}^+$ such that $\frac{\min\big\{X(n), Y(n)\big\}}{f(n)}<\varepsilon.$

We are given a finite, simple, non-directed graph. Ann writes positive real numbers on each edge of the graph such that for all vertices the following is true: the sum of the numbers written on the edges incident to a given vertex is less than one. Bob wants to write non-negative real numbers on the vertices in the following way: if the number written at vertex $v$ is $v_0$, and Ann's numbers on the edges incident to $v$ are $e_1,e_2, \dots, e_k$, and the numbers on the other endpoints of these edges are $v_1, v_2, \dots, v_k$, then $v_0=\sum\limits_{i=1}^{k}e_iv_i+2022$. Prove that Bob can always number the vertices in this way regardless of the graph and the numbers chosen by Ann.

The incircle of triangle $ABC$ touches the sides in $X$, $Y$ and $Z$. In triangle $XYZ$ the feet of the altitude from $X$ and $Y$ are $X'$ and $Y'$, respectively. Let line $X'Y'$ intersect the circumcircle of triangle $ABC$ at $P$ and $Q$. Prove that points $X$, $Y$, $P$ and $Q$ are concyclic.

Find all non-negative integer solutions of the equation $2^a+p^b=n^{p-1}$, where $p$ is a prime number.

The sequence of digits 110 represents an even integer, whatever positive integer greater than 1 is the base of notation. Is there a sequence of digits 1 and 0 such that it represents a multiple of 3, whatever positive integer greater than 1 is the base of notation?

Find all positive real numbers $x$ for which $x + \frac1{x}$ is an integer, and $x^3 + \frac1{x^3}$ is a prime number.

The tangents drawn to the circumscribed circle of a right triangle $ABC$ at the right-angled vertex $C$ and at another vertex $A$ intersect at $D$. Prove that the line $BD$ bisects the altitude drawn from vertex $C$.

A new triangle is constructed out of the line segments obtained by multiplying the medians of a triangle by $\frac{2}{\sqrt3}$. The procedure is repeated with the triangle obtained. Show that the triangle obtained in the second step is congruent to the original triangle.

What is the largest number of elements that can be selected out of the first $2022$ positive integers such that the difference of any two selected numbers is not a prime?

Prove that $\frac{|a+b+c|}{1+|a+b+c|}\le \frac{|a|}{1+|a|}+\frac{|b|}{1+|b|}+\frac{|c|}{1+|c|}.$ for all real numbers $a$, $b$, $c$. On what condition will equality occur?

Let $n$ be a positive integer. Prove that it is possible to select $n$ perfect squares from the numbers $1$ to $2^{n+2}$ such that the sums obtained by adding an arbitrary subset of the selected numbers (including sums of single terms and the sum of all the numbers) are all distinct.

In an acute-angled triangle $ABC$, the points of tangency of the inscribed circle on sides $BC$, $CA$, $AB$ are $D$, $E$, and $F$, respectively. The circumscribed circle of the triangle intersects circle $AEF$ at a point $P$ different from $A$, intersects circle $BFD$ at a point $Q$ different from $B$, and intersects circle $CDE$ at a point $R$ different from $C$. Show that the lines $DP$, $EQ$ and $FR$ are concurrent.

Let $A$ denote the set of even positive integers for which the sum of the digits decreases by 2 if the number is halved. Let $B$ denote the set of positive integers for which the sum of the digits increases by 5 if the number is multiplied by 5. What is the number of elements in the set $A\cap B$ and in the set $B\setminus A$?

Define the sequence $\{a_n\}$ as follows: $a_1=-3, a_{n+1}=4+a_n+4\sqrt{a_n+4}\,.$ Determine the value of $a_{2022}$.

$P$ is a point on side $BC$ of a unit square $ABCD$, and $Q$ is a point on side $CD$, such that $\angle PAQ =45^{\circ}$. For which positions of points $P$ and $Q$ will the sum $BP+PQ+QD$ be minimal?

The inscribed circle of triangle $ABC$ is centred at $I$ and has a radius $\varrho$, the radius of the circumscribed circle is $R$. Prove that if $\overline{AI}=R$, then the area of the triangle $ABC$ is $\frac{a\cdot R}{4}+\varrho \cdot a$, where $a$ denotes the length of the side opposite to vertex $A$.

The length of each side of a triangle is at most 2 units. Each pair of vertices is joined with an arc of a unit circle, not longer than a semicircle. Prove that $a'+b'>2c'/3,$ where $a'$, $b'$, $c'$ denote the lengths of the arcs.

Give an example of a positive integer $k$, along with a finite tree graph $F$ of at least $k$ vertices in which the degree of each vertex is at most 3, and $F$ will fall apart into at least $2022$ components if an arbitrary connected subgraph of $k$ vertices is deleted from $F$.

A parabola intersects side $AB$ of a triangle $ABC$ at interior points $C_1$ and $C_2$, side $BC$ at interior points $A_1$ and $A_2$, and it intersects side $CA$ at interior points $B_1$ and $B_2$. Prove that if $AC_1 = C_2B$ and $BA_1 = A_2C$ then $CB_1=B_2A$.

Let $a\ne 0$ be a real number, and let $f\colon \mathbb{R} \to \mathbb{R}$ be a function, such that $f\big(x+f(y)\big) = f(x) + f(y) + ay$ for all $x,y\in \mathbb{R}$. Prove that $f$ is additive, that is, $f(x+y) = f(x) + f(y)$ for all $x,y\in \mathbb{R}$.

Points $C$ and $D$ lie on a semicircular arc of diameter $AB$. Let $A'$ and $B'$ denote the feet of the perpendiculars dropped to the line $CD$ from the points $A$ and $B$, respectively. Prove that the line segments $A'C$ and $B'D$ are equal in length.

Prove that $\sum_{k=1}^n k \cdot 2^{k-1} = \sum_{k=1}^n 2^{n-k} \cdot (2^k-1).$ for all positive integers $n$.

$P$ is an interior point lying on the median drawn from vertex $C$ of an acute-angled triangle $ABC$ such that $\angle APB =180^{\circ}-\angle ACB$. Show that the line $AB$ is tangent to the circle $APC$.

The vertices of a regular hexagon are labelled $1, 2, \ldots, 6$ in a random order. Then the absolute value of the difference of the labels of the adjacent endpoints is written on each side of the hexagon. Find the expected value of the sum of the six numbers written on the sides.

A positive integer $n$ is defined to be a mythical number if each of its divisors is 2 smaller than a prime number. For example, 15 is a mythical number. What is the largest possible number of divisors that a mythical number may have? Find all mythical numbers that have the maximum number of divisors.

Show that all prime numbers greater than $3$ that occur in the Fibonacci sequence are of the form $4k+1$.

Let $a$, $b$, $c$ denote positive real numbers such that $abc=1$. Show that $a+a^2+a^3 + b + b^2 + b^3 + c+ c^2 + c^3 \le {(a^2+b^2+c^2)}^2.$

In a triangle, $r$ denotes the inradius, $R$ is the circumradius, and $s$ denotes the semiperimeter. Prove that if $r+2R=s$ then the triangle is right angled.

Solve the following equation over the set of positive integers: $(k+n)!=k^3+n^3+(k+n)(3kn-1).$

The sides $a$, $b$ and $c$ of a triangle, in this order, form an arithmetic sequence. Show that the centre of the inscribed circle divides the angle bisector drawn to side $b$ in a $1:2$ ratio.

Show that every positive integer $n$ has a multiple in which the sum of the digits is $n$.

In a triangle $ABC$, the centre of the circumscribed circle is $O$, and $\angle ABC > 90^\circ$. The tangent drawn to the circumscribed circle at $C$ intersects line $AB$ at point $P$, and the perpendicular drawn from $P$ to $BC$ intersects line $OC$ at $Q$. Prove that $AB$ is perpendicular to $AQ$.

Let $m$ and $n$ denote arbitrary positive integers. Consider those lattice points $(x;y)$ in the Cartesian coordinate plane for which $1\le x\le m$ and $1\le y\le n$. What is the maximum possible number of points that can be selected out of these $mn$ lattice points such that no four points selected should form a non-degenerate parallelogram?

In a triangle $ABC$, $\angle CAB =48^{\circ}$ and $\angle ABC =54^{\circ}$. $D$ is an interior point of the triangle such that $\angle CDB =132^{\circ}$ and $\angle BCD =30^{\circ}$. Prove that the line segments forming the polygon $ACDB$ cannot form a triangle.

Determine all integers $n > 4$ such that $(k,n) > 1$ for all composite numbers $k$ less than $n$.

$a)$ Prove that there exist infinitely many, pairwise non-similar triangles in which the lengths of the sides are integers, and one angle is 3 times as large as another. $b)$ Is there a triangle with the above property in which the lengths of the sides are all at most 10?

There are 14 people sitting around a table. Each of them is wearing either a blue shirt or a yellow shirt. What is the maximum possible number of people who have adjacent neighbors with shirts of different color?

The ends of a rope are fixed to the ground at two points separated by a distance shorter than the length of the rope. The rope will become taut if its midpoint is raised to a height of 150 cm. The rope will also become taut if a point of the rope 90 cm from one end is raised to a height of 90 cm. How long is the rope?

Solve the following simultaneous equations over the set of real numbers: $$\begin{align*} \frac{x^2}{y}+\frac{y^2}{x}+x+y & =\frac{8}{xy},\\ x(x+1)+y(y+1) & =6. \end{align*}$$

Let $T_0$ denote the area of the triangle formed by the points of tangency of the inscribed circle of triangle $ABC$ on the sides, and let $T_1$ denote the area of the triangle formed by the centres of the escribed circles. Show that the geometric mean of $T_0$ and $T_1$ equals the area of triangle $ABC$.

Prove that for all non-negative integers $n$, $2^{2^{n}(n-2)+n+2}\le (2^n)!\le 2^{2^{n}(n-1)+1}.$

The vertices of a rectangle $ABCD$ in the coordinate plane are $A(0,0)$, $B(2022,0)$, $C(2022,2)$, $D(0,2)$. Consider those triangles of unit area that have all three vertices at lattice points lying on the longer sides of the rectangle. These triangles are to be coloured so that no triangles of the same colour have an interior point in common. What is the minimum number of colours needed?

A polyhedron $ABCA_1B_1C_1$ has six vertices. Faces $ABC$ and $A_1B_1C_1$ are triangles. The edges $AA_1$, $BB_1$ and $CC_1$ are parallel. Faces $AA_1B_1B$, $BB_1C_1C$ and $CC_1A_1A$ are trapeziums in which the diagonals intersect at points $P$, $Q$ and $R$, respectively. Show that the volumes of polyhedra $ABCPQR$ and $A_1B_1C_1PQR$ are equal.

Is it true that if $\binom{n}{k}$ is even, then the $k$-element subsets of an $n$-element set $S$ can be paired up so that the symmetric difference of every pair should have exactly 2 elements?

Prove that the difference of the squares of any two odd numbers not divisible by 3 is divisible by 24.

Vertex $A$ of a triangle $ABC$ is reflected about vertex $B$, $B$ is reflected about $C$, and $C$ is reflected about $A$. The reflections are points $C_1$, $A_1$ and $B_1$, respectively. Show that there exists a triangle with sides of lengths $AA_1$, $BB_1$ and $CC_1$.

In the lottery game, five numbers are drawn out of ninety every week. Andrew fills out a single lottery ticket with the same five numbers in each of the 52 weeks of the year. Belle uses a different scheme. She plays only once a year with 52 tickets simultaneously: she fills them out in pairwise different ways. Is it true that both of them have the same chance of having a ticket with five correct numbers?

In an acute-angled triangle $ABC$, the heights are $AA_1$, $BB_1$, $CC_1$, and the midpoint of side $AB$ is $F$. A circle $k$ passes through the points $F$ and $C_1$, and intersects the extensions of line segments $A_1C_1$ and $B_1C_1$ beyond $C_1$ at points $P$ and $Q$, respectively. Prove that $A_1P=B_1Q$.

Is it true that every positive integer has a positive multiple in which the sum of the digits in decimal notation is at most 2022?

$G$ and $H$ are points of chord $AB$ of a circle $k$ such that $AG=GH=HB=1$. Let $F$ denote the midpoint of one of the arcs $AB$. The secants $FH$ and $FG$ intersect the circle again at points $C$ and $D$, respectively. Show that $CD=BC^2$.

Starting Player and Second Player are playing a game on the edges of a complete graph of 100 vertices. They take turns in colouring an edge of the graph that has not been coloured before. In each step, Starting Player colours his edge red, and Second Player colours his edge blue. The game terminates and Starting Player wins if there is a set of four vertices such that all the six connecting edges are red. The game terminates and Second Player wins if there is a set of four vertices such that all the six connecting edges are blue. The game terminates with a draw if there is no such set of four vertices but there remain no further vertices to colour. Who has a winning strategy?

Louisa wrote down a natural number, not containing 0 but containing at least two different digits. Then she also listed all the numbers which can be formed by permuting the digits of the original number. What is the maximum of the greatest common divisor of all the numbers (including the original one)?

Prove that the sum of the squares of the medians of a triangle is not less than the square of the semiperimeter of the triangle.

First Player and Second Player play the following game. First Player starts and prescribes arbitrarily many (even infinitely many) terms of a binary sequence (i.e., any term is $0$ or $1$) in a way that infinitely many terms can still be determined. Then Second Player sets the value of the first digit which has not been prescribed yet. They then repeat this procedure forever by taking turns. First Player wins if the binary sequence is periodic from a certain term, otherwise, Second Player wins. Is there a winning strategy, and if yes, who has it?

Enlarge the incircle of a right-angled triangle by a scale factor of 2, where the center of enlargement is the vertex at the right angle. Show that this circle touches the circumcircle of the triangle.

Some football players are on holiday together. Altogether they are from $k$ clubs and from $n$ nations where $k<n$. Show that there are at least $n-k+1$ players having more club fellows than compatriots.

We are given two line segments of length $p$ and $q$, and a triangle $ABC$ determined by the lines $a$, $b$ and $c$ (where the usual convention is used: points $B$ and $C$ lie on line $a$, and so on). Construct the point $P$ on the circumcircle of the triangle for which the point $P_a$ divides the line segment $P_bP_c$ in a ratio $p: q$, where $P_x$ is the orthogonal projection of the point $P$ onto the line $x$.

Let $I$ denote the incenter of the triangle $ABC$. Let $P$ denote an arbitrary interior point of the triangle on the circle $ABI$. The reflection of the line $AP$ about the line $AI$ intersects the circle $ABI$ at a point $Q$ different from the point $A$. Prove that $CP=CQ$.

Let $p \ge 19$ be an odd integer, and color the numbers $0,1,\dots,p-1$ with two colors. For $1\le i\le p$, let $x_i$ denote a random element of the set $\{0,1,\dots,p-1\}$ (the choices are independent, and have uniform distribution). Prove that the probability of the event that $x_1,\dots,x_p$ have the same color and $p$ divides $x_1+\cdots + x_p$ is at least $3/(2^pp)$.

$ABC$ is an isosceles right angled triangle with the right angle lying at vertex $C$. $A'$, $B'$ and $C'$ are interior points of sides $AB$, $BC$ and $CA$, respectively, such that triangle $A'B'C'$ is similar to $ABC$. Show that the midpoint of the side $AB$, the midpoint of line segment $A'B'$, and point $C$ are collinear.

A flea starts out from point $(a,b)$ of the coordinate plane, where $a$, $b$ are positive integers. With each jump, the flea will move one unit to the left or downwards. It keeps jumping until it reaches either the $x$ axis or the $y$ axis. What fraction of the possible sequences of jumps terminate on the $x$ axis?

$D$ is a point on side $AB$ of an equilateral triangle $ABC$, and $E$ is a point on side $BC$ such that $\angle BCD =45^\circ$ and $\angle CDE =30^\circ$. Show that $BE=2AD$.

The product of the positive integers $a<b$ is a perfect square. Show that there is a positive integer $x$ such that $a\le x^2\le b$.

Is there an irrational number $a$ such that $a^{\sqrt{3}}$ is rational?

Prove that there exist infinitely many positive integers $k$ for which the sum of the digits of $2^k$ is $a)$ smaller; $b)$ greater than the sum of the digits of $2^{k+1}$.

The centre of the inscribed circle of triangle $ABC$ is $I$. The midpoint of the circular arc $BCA$ is $F$, and line $FI$ intersects the circumscribed circle again at point $M$. Show that line $CM$ passes through the external centre of similitude of the inscribed circle and the circumscribed circle.

In Noname School of Nowhere, the graduating class consists of four groups, each specializing in a different subject: math, physics, chem and bio. On day, they are all sitting around a large round table in the school cafeteria: everyone has someone sitting directly opposite them, and everyone has neighbours sitting right next to them on either side. However a student is selected, it is observed that this student, the two neighbours, and the one sitting straight across the table all belong to different groups. How many students may there be in the graduating class if there are less than $20$ of them?

Two circles are drawn inside a right angle. One circle touches one of the arms at point $A$, and the other circle touches the other arm at point $B$. The circles also touch each other at point $C$. What is the measure of $\angle ACB$?

Let $a>2$, $b$ and $c$ be real numbers. Consider the three statements below. (1) The equation $ax^2 + bx + c = 0$ has no real solution. (2) The equation $(a-1)x^2 + (b-1)x + (c-1) = 0$ has 1 real solution. (3) The equation $(a-2) x^2 + (b-2)x + (c-2) = 0$ has 2 real solutions. $a)$ Given that statements (1) and (2) are true, can we conclude that statement (3) is also true? $b)$ Given that statements (2) and (3) are true, can we conclude that statement (1) is also true?

Let $d > 1$ denote a positive integer. Prove that there exists a positive integer that has the same number of factors divisible by $d$ as factors not divisible by $d$.

In an acute-angled triangle $ABC$, the foot of the altitude drawn from vertex $A$ is $T$. The projection of $T$ on side $AB$ is $D$, and its projection on side $AC$ is $E$. Let $F$ be the intersection of side $BC$ and the circle $ABE$ that is different from $B$. Analogously, let $G$ be the intersection of side $BC$ and circle $ACD$, different from $C$. Show that $TF=TG$.

A convex quadrilateral $N$ contains a circular disc of radius $r$. Show that the perimeter of $N$ is at least $8r$.

Let $n>2$. Alan has selected an edge of the complete graph of $2n$ vertices. Paula wants to find out which. If she pays 1 forint (Hungarian currency, HUF) she may name any pairing of all vertices and ask whether the selected edge is contained in it. What is the minimum number of forints that Paula needs to have in her pockets in order to be certain that she can find out the selected edge by asking the appropriate questions?

In an acute-angled triangle $ABC$, $AB=AC$. Points $A'$, $B'$ and $C'$ are moving along the circumscribed circle of the triangle, so that triangle $A'B'C'$ always remain congruent to triangle $ABC$, and have the same orientation. Let $P$ be the intersection of lines $BB'$ and $CC'$. Show that the lines $A'P$ all pass through a certain point.

In the expansion of the product $(x+1)\cdot(x^2+1)\cdot(x^3+1)\cdot \ldots \cdot(x^{12}+1)$, what is the coefficient of the term in $x^{14}$?

In a circle of centre $O$, $A$ is an interior point different from $O$. For a point $B$ on the circumference of the circle, $\angle OAB =\alpha$. Let $C$ be a point on the circumference such that $2\alpha+\beta=180^{\circ}$, where $\angle BAC =\beta$, and the angles $\angle BAO$ and $\angle BAC$ have no common points apart from the ray $AB$. Prove that the points $O$, $A$, $B$, $C$ are concyclic. \

What is the sum of all integers $x$ for which $\sqrt{2x^2-6x-20}<-x+5?$

Vertex $A$ of a quadrilateral $ABCD$ lies on the plane $S$, its diagonal $BD$ is parallel to the plane $S$, and its vertex $C$ is at a distance of $8$ units from the plane $S$. Given that the orthogonal projection of the quadrilateral onto $S$ is a square with a diagonal $6$ units long, prove that quadrilateral $ABCD$ is a rhombus, and calculate the length of its sides.

The natural numbers $a$ and $b$ each have only digits of $1$ in decimal notation. Prove that if $a$ and $b$ are not relatively prime then the sums of their digits, $S(a)$ and $S(b)$ are not relatively prime either.

For which real numbers $a$ will the minimum of the function $f(x)=4x^2-4ax+a^2-2a+2$ defined on the segment $[0;2]$ be equal to 3?

Given that a certain quadrilateral is a kite, it is cyclic and its sides are 42 and 56 units long, what is the distance between the centres of the inscribed and circumscribed circles?

Prove that in every set of 2022 positive integers there exist two numbers such that their difference or sum is divisible by 4040.

In a triangle $ABC$ (with the usual notations) $b=6$, $a=2$ and they enclose an angle of $\gamma=120^{\circ}$. Find the exact length of the interior angle bisector of angle $\gamma$.

Solve the following equation over the set of real number pairs: $\log_2^2(x+y)+\log_2^2(xy)+1=2\log_2(x+y).$

The integers $a$ and $b$ are factors of 720, but $ab$ is not a factor of 720. How many such ordered pairs $(a;b)$ are there?

Solve the equation $\sqrt{x-1801}+\sqrt{y-1860}=2-\frac{1}{\sqrt{x-1801}},$ where $x$ and $y$ denote real numbers.

The sides of a pentagon are all equal in length, and two of its angles are right angles. What may be the measures of the other three angles?

Let $x$ and $a$ denote real numbers such that $x+\frac{1}{x}=a$. Determine the value of $x^{13}+\frac{1}{x^{13}}$ as a function of $a$.

The integers 1 to 22 are written on a blackboard. In each move, a pair of numbers is selected, erased and replaced with the absolute value of their difference. Prove that the last number added to the board is odd.

In factorial representation, the place values of the digits are not the powers of a base: the $n$th place value is $n$ factorial. Thus the digit in the first place is to be multiplied by 1, the digit in the second place is multiplied by 2, that in the third place is multiplied by 6, and so on. For example, the number $3310_!$ in factorial representation corresponds to the number $3\cdot4!+3\cdot3!+1\cdot2!=92$ in decimal notation. (If there is a number of more than one decimal digits in a certain place then it is indicated by using brackets.) (It can be shown that the representation is unique, that is, every positive integer has a single factorial representation. See the Informatics problems I. 553. of the January issue.) It is observed that one third of $111_!$ is $11_!$, one third of $111 111_!$ is $22 011_!$, and one third of $111 111 111_!$ is $33 022 011_!$. Determine the factorial representation of one third of the number that consists of $3n$ digits of 1, also given in factorial representation.