KöMaL 2021

A lightbulb is given that emits red, green or blue light and an infinite set $S$ of switches, each with three positions labeled red, green and blue. We know the following: $i)$ For every combination of the switches the lighbulb emits a given color. $ii)$ If all switches are in a position with a given color, the lightbulb emits the same color. $iii)$ If there are two combinations of the switches where each switch is in a different position, the lightbulb emits a different color for the two combinations. We create the following set $U$ containing some of the subsets of $S$: for each combination of the switches let us observe the color of the lightbulb, and put the set of those switches in $U$ which are in the same position as the color of the lightbulb. Prove that $U$ is an ultrafilter on $S$. ($U$ is an ultrafilter on $S$ if it satisfies the following: $a)$ The empty set is not in $U$. $b)$ If two sets are in $U$, their intersection is also in $U$. $c)$ If a set is in $U$, every subset of $S$ containing it are also in $U$. $d)$ Considering a set and its complement in $S$, exactly one of these sets is contained in $U$.) See also problem N.$\,$35. from the May issue of 1994 (in Hungarian). ( Problem and solution .)

Let $p\ge 3$ be a prime number and $0\le r\le p-3$. Let $x_1, x_2, \ldots, x_{p-1+r}$ be integer numbers satisfying $\sum_{j=1}^{p-1+r}x_j^k\equiv r~ \textrm{(mod}\ p\textrm{)}$ for all $1\le k\le p-2$. What are the possible remainders of numbers $x_1,x_2,\ldots,x_{p-1+r}$ modulo $p$?

In the 43 dimension Euclidean space the convex hull of finite set $S$ contains polyhedron $P$. We know that $P$ has 47 vertices. Prove that it is possible to choose at most 2021 points in $S$ such that the convex hull of these points also contain $P$, and this is sharp.

A polyomino $P$ occupies $n$ cells of an infinite grid of unit squares. In each move, we lift $P$ off the grid and then we place it back into a new position, possibly rotated and reflected, so that the preceding and the new position have $n-1$ cells in common. We say that $P$ is a caterpillar of area $n$ if, by means of a series of moves, we can free up all cells initially occupied by $P$. How many caterpillars of area $10^6 + 1$ are there?

The following game is played with a group of $n$ people: $n+1$ hats are numbered from $1$ to $n+1$. The people are blindfolded, and each of them is getting one of the $n+1$ hats on his head (the remaining hat is hidden). Now a line is formed from the $n$ people, and their eyes are uncovered: each of them can see the numbers on the hats of the people standing in front of him. Now starting from the last person (who can see all the other players) the players take turns to guess the number of the hat on their head, but no two players can guess the same number (each player hears all the guesses from the other players). What is the highest number of guaranteed correct guesses, if the $n$ people can discuss a common strategy after learning about the game?

Let $ABCD$ be a cyclic quadrilateral. Let lines $AB$ and $CD$ intersect in $P$, and lines $BC$ and $DA$ intersect in $Q$. The feet of the perpendiculars from $P$ to $BC$ and $DA$ are $K$ and $L$, and the feet of the perpendiculars from $Q$ to $AB$ and $CD$ are $M$ and $N$. The midpoint of diagonal $AC$ is $F$. Prove that the circumcircles of triangles $FKN$ and $FLM$, and the line $PQ$ are concurrent.

We call a system of non-empty sets $H$ entwined, if for every disjoint pair of sets $A$ and $B$ in $H$ there exists $b\in B$ such that $A\cup\{b\}$ is in $H$ or there exists $a\in A$ such that $B\cup\{a\}$ is in $H$. Let $H$ be an entwined system of sets containing the following $n$ one-element sets: $\{1\}, \{2\},\dots,\{n\}$. Prove that if $n>k(k+1)/2$, then $H$ contains a set with at least $k+1$ elements, and this is sharp for every $k$, i.e. if $n=k(k+1)$, it is possible that every set in $H$ have at most $k$ elements.

Let $0<p<1$ be given. Initially we have $n$ coins, all of which has probability $p$ of landing on heads, and probability $1-p$ landing on tails (the results of the tosses are independent from each other). In each round we toss our coins and remove those that result in heads. We keep repeating this until all our coins are removed. Let $k_n$ denote the expected number of rounds that was needed to get rid of all the coins. Prove that there exists $c>0$ for which the following inequality holds for all positive integers $n$: $c\left(1+\frac12+\cdots+\frac1{n}\right)<k_n<1+c\left(1+\frac12+\cdots+\frac1{n}\right).$

For a given quadrilateral $A_1A_2B_1B_2$ point $P$ is called phenomenal, if line segments $A_1A_2$ and $B_1B_2$ subtend the same angle at point $P$ (i.e. triangles $PA_1A_2$ and $PB_1B_2$ which can be also also degenerate have equal inner angles at point $P$ disregarding orientation). Three non-collinear points, $A_1$, $A_2$ and $B_1$ are given on the plane. Prove that it is possible to find a disc on the plane such that for every point $B_2$ on the disc quadrilateral $A_1A_2B_1B_2$ is convex for which it is possible to construct seven distinct phenomenal points only using a right ruler. With a right ruler the following two steps are allowed: $i)$ given two points it is possible to draw the straight line connecting them; $ii)$ given a point and a straight line, it is possible to draw the straight line passing through the given point which is perpendicular to the given line.

In a finite, simple, connected graph $G$ we play the following game: initially we color all the vertices with a different color. In each step we choose a vertex randomly (with uniform distribution), and then choose one of its neighbors randomly (also with uniform distribution), and color it to the the same color as the originally chosen vertex (if the two chosen vertices already have the same color, we do nothing). The game ends when all the vertices have the same color. Knowing graph $G$ find the probability for each vertex that the game ends with all vertices having the same color as the chosen vertex.

For which values of positive integer $m$ is it possible to find polynomials $p,q\in \mathbb{C}[x]$ with degrees at least two such that $x(x+1)\cdots(x+m-1) =p\big(q(x)\big)$?

Let $P$ be a given regular 100-gon. Prove that if we take the union of two polygons that are congruent to $P$, the ratio of the perimeter and area of the resulting shape cannot be more than the ratio of the perimeter and area of $P$.

Let $\pi (n)$ denote the number of primes less than or equal to $n$. A subset of $S=\{1,2,\dots,n\}$ is called primitive if there are no two elements in it with one of them dividing the other. Prove that for $n\ge 5$ and $1\le k<\frac{\pi (n)}{2}$ the number of primitive subsets of $S$ with $k+1$ elements is greater or equal to the number of primitive subsets of $S$ with $k$ elements.

There is a city with $n$ citizens. The city wants to buy sceptervirus tests with which it is possible to analyze the samples of several people at the same time. The result of a test can be the following: $\bullet$ Virus positive: there is at least one currently infected person among the people whose samples were analyzed, and none of them were healed from an earlier infection. $\bullet$ Antibody positive: there is at least one person who was healed from an earlier infection among the people whose samples were analyzed, and non of them are infected now. $\bullet$ Neutral: either all of the people whose samples were analyzed are not infected, or there is at least one currently infected person and one person who was healed from an earlier infection. (Viruses and antibodies in samples completely neutralize each other.) What is the smallest number of tests to buy if we would like to know if the sceptervirus is present now or it has been present earlier? (The samples are taken from the people at the same time. The people are either infected now, have been infected earlier, or haven't contacted the virus yet.)

In acute triangle $ABC$ the feet of the altitudes are $A_1$, $B_1$ and $C_1$ (with the usual notations on sides $BC$, $CA$ and $AB$, respectively). The circumcircles of triangles $AB_1C_1$ and $BC_1A_1$ intersect the circumcircle of triangle $ABC$ at points $P\ne A$ and $Q\ne B$, respectively. Prove that lines $AQ$, $BP$ and the Euler line of triangle $ABC$ are either concurrent or parallel to each other.

Four distinct lines are given in the plane, which are not concurrent and no three of which are parallel. Prove that it is possible to find four points in the plane, $A$, $B$, $C$ and $D$ with the following properties: $(i)$ $A$, $B$, $C$ and $D$ are collinear in this order; $(ii)$ $AB=BC=CD$; $(iii)$ with an appropriate order of the four given lines $A$ is on the first, $B$ is on the second, $C$ is on the third and $D$ is on the fourth line.

Let $n \ge 2$ be a given integer. Let $G$ be a finite simple graph with the property that each of its edges is contained in at most $n$ cycles. Prove that the chromatic number of the graph is at most $n+1$.

Find all triples of positive integers $a$, $b$ and $c$ such that $a$, $b$ and $c$ are pairwise relatively prime and $a^2+3b^2c^2=7^c.$

Let the lengths of the sides of triangle $ABC$ be denoted by $a$, $b$ and $c$ using the standard notations. Let $S$ denote the centroid of triangle $ABC$. Prove that for an arbitrary point $P$ in the plane of the triangle the following inequality is true: $a\cdot PA^3+b\cdot PB^3+c\cdot PC^3\ge 3abc\cdot PS.$

For all positive integers $n$ let $r_n$ be defined as $r_n=\sum_{t=0}^{n} {(-1)}^t \binom{n}{t} \frac{1}{(t+1)!}.$ Prove that $\sum_{n=1}^{\infty} r_n=0$.

Let $A$ be a given set with $n$ elements. Let $k<n$ be a given positive integer. Find the maximum value of $m$ for which it is possible to choose sets $B_i$ and $C_i$ for $i=1,2,\dots,m$ satisfying the following conditions: $(i)$ $B_i \subset A$, $|B_i|=k$, $(ii)$ $C_i \subset B_i$ (there is no additional condition for the number of elements in $C_i$), $(iii)$ $B_i \cap C_j \ne B_j \cap C_i$ for all $i \ne j$.

Two players play the following game: there are two heaps of tokens, and they take turns to pick some tokens from them. The winner of the game is the player who takes away the last token. If the number of tokens in the two heaps are $A$ and $B$ at a given moment, the player whose turn it is can take away a number of tokens that is a multiple of $A$ or a multiple of $B$ from one of the heaps. Find those pair of integers $(k,n)$, for which the second player has a winning strategy, if the initial number of tokens is $k$ in the first heap and $n$ in the second heap.

Let $p$ be a prime number and $k$ a positive integer. Let $t=\sum_{j=0}^{\infty} \left\lfloor\frac{k}{p^j}\right\rfloor.$ $a)$ Let $f(x)$ be a polynomial of degree $k$ with integer coefficients such that its leading coefficient is 1 and its constant is divisible by $p$. Prove that there exists $n\in \mathbb{N}$ for which $p\mid f(n)$, but $p^{t+1}\nmid f(n)$. $b)$ Prove that the statement above is sharp, i.e. there exists polynomial $g(x)$ of degree $k$, integer coefficients, leading coefficient 1 and constant divisible by $p$ such that if $p\mid g(n)$ is true for a certain $n\in \mathbb{N}$, then $p^t\mid g(n)$ also holds.

There are given 666 points in the plane such that they cannot be covered with 10 lines. Prove that it is possible to choose 66 of them such that they still cannot be covered with 10 lines.

In a football championship, there are four teams in a group. Within the group, each team plays every other team once. The teams receive 3 points for winning, 1 point for a draw and 0 points for losing a game. The two teams scoring the highest qualify for the semi-finals, and the other two teams are eliminated. In the case of equal scores, the qualification is decided by chance. Determine those values of the number $p$ for which it may happen that a qualifying team and an eliminated team both have $p$ points.

Find the real solutions of the equation $16x^2+9x+117=24x\sqrt{x+13}\,$.

The area of a convex quadrilateral $ABCD$ is $t$, and an interior point is $O$. Show that $2t\le OA^2+OB^2+OC^2+OD^2.$ When will equality occur?

Show that there are $\binom{n+1}{2k+1}$ different strings of zeros and ones of length $n$ in which it occurs exactly $k$ times that a $0$ is followed by a $1$.

$T$ is a cuboid of unit volume, and $M$ is a point in its interior. Point $M$ is reflected in the planes of the faces. Let $D$ be the convex hull of the $6$ images obtained. Determine the volume of the solid $T\cap D$.

Let $k>1$ be a positive integer. Is there $a)$ a finite subset (of any size) $b)$ an infinite subset of the set of positive integers in which the greatest common divisor of any $k$ elements is greater than 1 but the greatest common divisor of any $k+1$ elements is equal to 1?

A triangle $ABC$ is right angled at $C$. The inscribed circle touches the leg $BC$ at point $D$, and the leg $AC$ at point $E$. The escribed circle of side $BC$ touches line segment $BC$ at point $G$; and the escribed circle of side $AC$ touches line segment $AC$ at point $H$. The intersection of line segments $DH$ and $EG$ is $M$. Show that the other intersection of the circumscribed circles of triangles $DGM$ and $EHM$ lies on the inscribed circle.

In how many different ways is it possible to fill in a $6\times 6$ table with the numbers $1,2,\dots,36$ so that however 6 fields are selected, all lying in different rows and in different columns, the sum of the numbers in 6 such fields should always be the same?

Prove that there are only a finite number of positive integers that cannot be obtained by adding one of the digits of a smaller number to that number. What is the largest of these finite number of integers?

Prove that if $a^2=b^2+ac=c^2+ab$, then two of the numbers $a$, $b$, $c$ are equal.

Determine those positive integers greater than $1$ whose positive divisors can all be written on the circumference of a circle so that the ratio of every adjacent pair should be a prime.

Find all functions $f$ taking on positive integer values, and defined on the set of positive integers such that $f\big(f(n)\big)=2n$ and $f(4n-3)=4n-1$ for all positive integers $n$.

The convex quadrilateral $ABCD$ has no parallel sides, and the intersection of lines $AB$ and $CD$ is $M$. Point $X$ is moving along the interior of side $AB$, and point $Y$ is moving along the interior of side $CD$ so that the equality $AX:XB=DY:YC$ remains true. Show that the circles $MXY$ all have another common point, different from $M$.

Let $K$ be a convex polygon of $2n$ vertices where all sides have unit length and the opposite sides are parallel. Show that $K$ can be dissected into a finite number of rhombuses of unit sides. How many rhombuses may there be?

There are some integers on the blackboard. Each of three students (independently of each other) selects a number from the board at random, and writes it down in their notebook. Prove that the probability that the sum of the three numbers written down is divisible by 3 is at least $1/4$.

Let $A$, $B$, $C$ and $D$ be points in the plane such that $AB<CB$ and $CD<AD$. Prove that the line segments $AB$ and $CD$ do not intersect each other.

Solve the equation $\left[\frac{2020-x}{x-1}\right]+\left[\frac{2021+x}{x+1}\right]=82$ over the set of integers, where $[c]$ denotes the greatest integer not greater than $c$.

What may be the value of $x+y+z$ if $\sqrt{x-1}+2\sqrt{y-4}+3\sqrt{z-9}=\frac{x+y+z}{2}?$

The sides of triangle $ABC$ are 9, 10 and 17 units long. What is the area of the triangle formed by the exterior angle bisectors of triangle $ABC$?

Triangle $ABC$ is right-angled at $C$. The right angle is divided $2:1$ by a line such that the smaller part is closer to the shorter leg. This line intersects hypotenuse $AB$ at $T$, and the circumscribed circle at $D$. What are the measures of the acute angles of the triangle if the feet of the perpendiculars dropped from $D$ onto the lines of the legs are collinear with $T$?

Two players continue playing rock-paper-scissors games until one of them wins the third time. Assume that each player in each game selects rock, paper or scissors at random (independently of each other and of previous games), with probabilities of $\frac13 : \frac13 : \frac13$. Determine the expected value of the number of games needed.

Given a positive integer $k$, is there a function $f \colon \mathbb{N} \to \mathbb{N}$, such that $f(x) + f\big(f(x)\big) = x + k$ for all $x \in \mathbb{N}$?

Are there prime numbers $p$, $r$ greater than $3$ such that the sum of the digits of $2p^2+7r^2+2021$ should be a perfect square?

Consider two circles in the plane that have common interior tangents. Show that the circle passing through the points of contact of the internal tangents bisects the line segment connecting the centres of the two original circles.

Each of the integers 1 to 100 is written on a piece of paper. 16 pieces of paper are selected out of the 100 pieces. Is it certain that there will always be four pieces of paper among the selected ones such that the sum of the numbers on two of them equals the sum of the numbers on the other two?

Find the real solutions of the equation $\sqrt[3]{2x+11}+\sqrt[3]{3x+4}=\sqrt[3]{x+9}+\sqrt[3]{4x+6}.$

Let $\alpha$ and $\beta$ be acute angles such that $\sin^2\alpha+\sin^2\beta=\sin{(\alpha+\beta)}$. Prove that $\alpha+\beta=90^{\circ}$.

Let $OLMN$ be a tetrahedron, and the vertices $A$, $B$ and $C$ of another tetrahedron $OABC$ lie on the rays $OL$, $OM$ and $ON$, respectively. The centre of the inscribed circle of triangle $LMN$ coincides with the centroid of triangle $ABC$. Show that the volume of tetrahedron $OLMN$ is greater than or equal to the volume of tetrahedron $OABC$. On what condition will the volumes of the two tetrahedra be equal?

Six regular dice are placed in a cup, and rolled simultaneously. Those dice that do not show a 6 are returned to the cup, and rolled again. If there are dice that still not show a 6, those dice are rolled a third time. The procedure is repeated until every dice shows a 6. What is the probability that exactly six rolls are needed?

The orthocentre of an acute-angled triangle $ABC$ is $H$, and the centre of the circumscribed circle is $O$. Let $D$ and $E$ denote interior points on the line segments $AB$ and $AC$, respectively. The orthocentre and circumcentre of triangle $ADE$ are $H'$ and $O'$, respectively. Show that lines $HH'$ and $OO'$ are parallel if and only if $BD = CE$.

Prove that $(2n)! \le {(n^2 + n)}^n$ for all positive integers $n$.

In a triangle $ABC$, $AC=BC$, $D$ is an interior point of side $AC$, and $K$ is the centre of the circle $ABD$. Show that quadrilateral $BCDK$ is cyclic.

The first $n$ positive integers need to be written on the circumference of a circle (each number exactly once), so that the sums of all sets of three adjacent numbers should form exactly two different values. Find all possible values of $n$.

In a right-angled triangle $ABC$, line segment $CD$ is the altitude drawn to the hypotenuse. The circle $k$ of diameter $CD$ intersects the legs $AC$ and $BC$ again at points $E$ and $F$, respectively. The tangent drawn to circle $k$ at point $E$ intersects the line of leg $BC$ at point $P$, and the hypotenuse $AB$ at point $M$. The tangent drawn to circle $k$ at point $F$ intersects the line of leg $AC$ at point $Q$, and the hypotenuse $AB$ at point $N$. Prove that $4\cdot MN^2=PE^2+QF^2+2\cdot EF^2.$

Let $x$ be a positive real number. Show that $\sqrt{6x+9}+\sqrt{16x+64}\le \left(\sqrt{x}+\frac{3}{\sqrt{x}}\right) \left(\sqrt{x}+\frac{8}{\sqrt{x}}\right).$

Is there a set $H$ of integers with the following property: every nonzero integer can be represented in infinitely many ways as a sum of some distinct elements of $H$, but $0$ cannot be represented at all?

The radius of the circumscribed circle of a regular heptagon $ABCDEFG$ is $r$. Prove that the circle of radius $2r$ centred at $A$ passes through the orthocentre of triangle $BCE$.

Given eight points on the plane, no three of which are collinear and no five of which are concyclic, what is the maximum possible number of circles that pass through four of the points each?

The number $612^2=374 544$ ends in two digits of $4$ in base $10$ notation. What is the maximum number of digits of $4$ at the end of a perfect square?

Side $AB$ of a triangle $ABC$ has unit length, $\angle BAC = 60^{\circ}$, $\angle ACB =100^{\circ}$ and the midpoint of side $BC$ is $F$. $D$ is a point on side $AB$ such that $DB = FB$. Find the exact value of $T_{ABC\triangle}+2T_{FBD\triangle}$, where $T_{ABC\triangle}$ denotes the area of triangle $ABC$. ($T$ means the area of the triangle of the triangle named in the index.)

Cornelia marked four non-concyclic points in the plane. Then she drew all the circles that pass equidistant from the four points. What is the maximum possible number of circles she may have drawn? (The distance between point $P$ and a circle $k$ centred at $O$ is defined as follows: let $M$ denote the point where the ray starting at $O$ and passing through $P$ intersects the circle $k$. Then the distance is the length of line segment $PM$.)

Find the real solutions of the equation $\sqrt[3]{4-x^2}+\sqrt{x^2-3}=1.$

Al and Bill are playing the following game. They agree on a fixed number $n \ge 3$, and then Al thinks of a number from the set $\{1,2,\ldots,n \}$. Now Bill can guess the number. He will only get yes or no answers. If the answer is yes, the game terminates. If the answer is no, Al will change the number: either increases or reduces it by 1, but the number must remain positive (it is allowed to go beyond $n$ though). Then Bill can guess again, trying to hit the new number. The procedure is repeated until finally Bill gets the number. Prove that Bill has a strategy to end the game with at most $(3n-5)$ guesses.

A subset of the set $S=\{1,2,\dots,n\}$ is called primitive, if it does not contain two elements such that one is a divisor of the other. Show that if it is not possible to add a further element of $S$ to a particular primitive set $A\subseteq S$ and keep it primitive, then either $A=\{1\}$ or the size of $A$ is greater than or equal to the number of primes up to $n$.

Prove that the height of a circumscribed trapezium cannot be greater than the geometric mean of the bases.

The base edge of a right pyramid with a regular triangular base is $a$. Let $r$ be the radius of the inscribed sphere, and let $R$ be the radius of the escribed sphere touching the base. Prove that $a^2=12rR$.

In a table of $n$ rows and $k$ columns, there is $-1$ written in each field. In each move, one row and one column is selected. Each number in the row is changed to the opposite, and then each number in the column is changed to the opposite. For what values of $n$ and $k$ is it possible to achieve a value of $+1$ in every field of the whole table?

We have a wooden set square, but the hypotenuse has been chewed by a rabbit. With the set square we can join points lying close enough, we can extend straight line segments, and we can draw a perpendicular to any line at any point. Can we construct the centre of a given circle however large its size?

Eight boys decided to play a game of football, four against four, on each of the first seven days of the autumn break. Is it possible to organize the teams so that any set of three boys would play in the same team at least once?

In an acute-angled triangle $ABC$, $\angle BCA =45^{\circ}$, the feet of the altitudes on sides $BC$, $CA$, $AB$ are $D$, $E$, $F$, respectively, and the orthocentre is $M$. Point $F$ divides line segment $AB$ in a ratio ${AF}:{FB}={2}:{3}$. $G$ is the point on side $AC$ for which $CG=BM$. Show that the centroid of triangle $ABG$ is $M$.

In a triangle $ABC$, $\angle ABC =2\angle CAB$. Side $AB$ touches the inscribed circle at point $E$, and intersects the angle bisector drawn from $C$ at point $F$. Prove that $AF=2BE$.

Prove that the inequality $x^{p}\cdot y^{1-p}<x+y$ holds for every pair of positive real numbers $(x,y)$, and all real numbers $0<p<1$.

Let $p(x)=2x+1$. $A$ is a subset of set $S=\{1,2,\dots,2021\}$ such that it contains at most one of the numbers $n$, $p(n)$, $p\big(p(n)\big)$ for every $n$, but this condition will not hold anymore if any extra element of $S$ is added to $A$. What may be the number of elements in the set $A$?

Let $\mathbb{N}$ denote the set of non-negative integers, and let $k$ be a given positive integer. Is there a monotonically increasing function $f \colon \mathbb{N} \to \mathbb{N}$ such that $f\big(f(x)\big) = f(x) + x + k$ for all $x \in \mathbb{N}$?

I have a tortoise, a cat and a dog. If the cat is standing on the floor and I place the tortoise on the table, the head of the tortoise will be 70 cm above the head of the cat. If the dog is standing on the floor and I place the cat on the table, the head of the cat will be 80 cm above the head of the dog. Finally, if the tortoise is standing on the floor and I place the dog on the table, the head of the dog will be 120 cm above the head of the tortoise. How tall is the table?

A coin is placed on each field of a chessboard, heads facing up. In each move, we can (simultaneously) turn over three adjacent coins in any row or column. Is it possible to achieve an arrangement where all coins show tails on top?

Let $1=d_1<d_2<\dots<d_k=n$ be the divisors of a positive integer $n$. Determine those composite numbers $n$ for which the numbers $d_1$, $d_1+d_2$, $d_1+d_2+d_3$, $\dots,$ $d_1+d_2+\dots+d_{k-1}$ all divide $n$.

Two rational numbers are said to be acquainted to each other if they can be represented in the forms $p/q$ and $r/s$, respectively ($p$, $q$, $r$, $s$ are integers), such that $|ps-qr|=1$. If two rational numbers are acquainted to each other, how many common acquaintances may they have?

In a triangle $ABC$, $AB>BC$, the inscribed circle touches sides $BC$, $CA$ and $AB$ at the points $A_0$, $B_0$ és $C_0$, respectively, and the escribed circle drawn to side $AC$ touches $AC$ at point $B_1$. Show that the intersection of the line segments $A_0B_1$ and $B_0C_0$ lies on the interior angle bisector drawn from vertex $B$ if and only if the angle at vertex $C$ measures $90^{\circ}$.

Let $1\le a, b, c, d\le 4$ denote real numbers. Prove that $16\le (a+b+c+d)\left(\frac{1}a+\frac{1}b+\frac{1}c+\frac{1}d\right)\le 25.$

There are four circles given in the plane: circle $k_2$ lies in the interior of circle $k_1$, circle $k_3$ lies in the interior of $k_2$, and circle $k_4$ lies in the interior of $k_3$. Also given are three lines $e_1$, $e_2$ and $e_3$ which are pairwise non-parallel and each line intersects each circle. For all $i=1,2,3$ let the intersections of line $e_i$ with the circles be $A_i$, $B_i$, $C_i$, $D_i$, $E_i$, $F_i$, $G_i$ and $H_i$, in this order. Prove that if $A_1B_1+E_1F_1=C_1D_1+G_1H_1$ and $A_2B_2+E_2F_2=C_2D_2+G_2H_2$ then $A_3B_3+E_3F_3=C_3D_3+G_3H_3$.

An $n$-digit number $\overline{a_1a_2a_3\ldots a_n}$ is called hill type if there exists an integer $1 \le k \le n$ for which the sequence $a_1,a_2,\ldots,a_k$ is strictly increasing and the sequence $a_k,a_{k+1},\ldots,a_n$ is strictly decreasing. (For example, the numbers 1, 121, 1231 are of hill type, whereas 1442 or 12313 are not.) How many hill type numbers are there?

Let $n\ge 2$ be a natural number. Prove that there exist positive integers $2\le x_1<x_2<x_3< \dots<x_n$, such that $x_1!\cdot x_2!\cdot x_3!\cdot \dots \cdot x_n!$ is a perfect square.

The lines of two perpendicular chords $AB$ and $CD$ of a circle intersect at an exterior point $P$. The length of the tangent drawn from $P$ to the circle is $e$. Show that the geometric mean of the lengths $AD$ and $BC$ is at least $\sqrt2 e$.

What is the maximum possible number of two-element subsets in a $2022$-element set of integers for which the sum of the two elements also belongs to the set?

The parabolas $\mathcal{P}_1$, $\mathcal{P}_2$ and $\mathcal{P}_3$ have the same focus, and any two of them intersect at exactly two points. Let $e_{ij}$ denote the line passing through the two intersections of the parabolas $\mathcal{P}_i$ and $\mathcal{P}_j$. Show that the lines $e_{12}$, $e_{13}$ and $e_{23}$ are concurrent.

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

Prove that there exists a positive integer which can be represented in at least 2021 different ways by taking an appropriate positive integer (in decimal notation), and adding the sum of its digits to it.

Prove that if $a$, $b$, $c$ are positive real numbers, then $c\sqrt{a^2+b^2-ab}+a\sqrt{b^2+c^2-bc}\ge b\sqrt{c^2+a^2+ca}\,.$ When will equality occur?

We have made a $10 \mathrm{cm} \times 30 \mathrm{cm}$ rectangular tin of shortbread. It has a delicious crispy edge. We want to divide the bread into pieces by using cuts running all the way parallel to the edges of the tin. How many pieces may we obtain if we would like each piece to have the same length of crispy edge?

In an acute-angled triangle, the sides are $a$, $b$, $c$, and $m_b$ is the height drawn to side $b$. The lengths of $m_b$, $a$, $b$, $c$, in this order, are consecutive positive integers. What is the area of the triangle?

Find the integer solutions of the equation ${(xy-1)}^2 = {(x + 1)}^2 + {(y + 1)}^2$.

The medians drawn to the legs of an isosceles triangle are perpendicular to each other. Let $r$ and $R$ denote the inradius and circumradius, respectively. Find the exact value of the ratio $\frac{r}{R}$.

King Arthur and Sir Lancelot are running a horse race. Sir Lancelot says, ``Since Your Majesty's speed is only $\frac{2}{3}$ of mine, I will give Your Majesty a handicap of $100$ metres, and then I would catch up within the length of the race track. Alternatively, if Your Majesty reduced speed by $2 \frac{\mathrm{m}}{\mathrm{s}}$ and I reduced mine by $5 \frac{\mathrm{m}}{\mathrm{s}}$, but I gave Your Majesty a $50$ metre handicap only, I would also be able to catch up within the length of the track. The sum of the two time intervals required to catch up is exactly $75$ seconds.'' Determine the speeds of King Arthur and of Sir Lancelot.

The diagonals of a cyclic quadrilateral intersect each other at right angles at point $M$. The diagonals divide the quadrilateral into triangles. Prove that the altitude of any triangle drawn from point $M$ is collinear with the median drawn from point $M$ in the opposite triangle.

Prove the inequality $\log_{ab}c\le \frac{\log_ac + \log_bc}{4}$ for $a, b, c>1$.

The terms of a number sequence are generated as follows: the first term is $895$, and the following term is always obtained by multiplying the sum of the digits of the previous term by $61$. Determine the $2021$st term of the sequence, and the sum of the first $2021$ terms.

The shorter leg of each of two right-angled triangles has unit length. In each triangle, the right-angled vertex is at a unit distance from a point dividing the hypotenuse in a $2:1$ ratio: in one case it is the point closer to the right-angled vertex, and in the other case it is the point farther away. Prove that it is possible to select three out of the non-unit sides of the two triangles such that the three lengths form a right-angled triangle.

How many solutions does the inequality $|x|+|y|<2021$ have over the set of pairs of integers?

Find the radius of each circle that is tangent to the graphs of the functions $f(x)=\frac{3x-6}{4}$ and $g(x)=\frac{28-4x}{3}$, and also touches the $x$-axis.

Solve the equation $2{(x+y-1831)}^2=(2x-1802)(2y-1860)$ over the set of pairs of real numbers.

Three consecutive terms of an arithmetic sequence are prime numbers greater than 3. Show that the common difference of the sequence is divisible by 3.

$BCD$ and $CAE$ are regular triangles drawn on legs $BC$ and $CA$ of a right-angled triangle $ABC$, on the outside. Prove that the midpoints of the line segments $AB$, $CD$ and $CE$ also form a regular triangle.

A circular disc is divided into six congruent sectors. A circle is inscribed in each sector. The circle touches the arc of the sector as well as the two radii. What fraction of the area of the large circle is covered by the six smaller circles?

Ray $a$ starts from point $A$ of a line segment $AB$, and encloses an angle $0^\circ<\alpha<90^{\circ}$ with it. Ray $b$ starts from point $B$, and encloses an angle $0^\circ<\beta<90^{\circ}$ with the line segment $AB$. The two rays lie in two different half planes of a plane containing line $AB$. The circle of diameter $AB$ intersects $a$ again at $A_1$, and $b$ at $B_1$. The circle of diameter $A_1B_1$ intersects the line containing $a$ again at $A_2$, and the line containing $b$ at $B_2$. What is the relationship between $\alpha$ and $\beta$ if $A_1B_1$ and $A_2B_2$ are perpendicular?

The positive integers $1$ to $61^2$ are written in the fields of a $61\times61$ chessboard, starting from the top left corner and proceeding along each row in succession. Then some changes are made as follows. In the first move, the sign of each number is changed to negative. In the second move, the signs of all even numbers are changed. In the third move, the sign of every multiple of 3 is changed, and so on, while the moves are meaningful. When all this is completed, how many $1\times2$ rectangles will there be on the chessboard in which the sum of the numbers is negative?

In a lottery game, the player bets 5 numbers out of the positive integers 1 to 90. Otto Lotter insists on increase, and he always keeps the following rules when making his bets: he marks 5 numbers such that every digit may only occur once, and if the five numbers are listed in increasing order, the digits must be ascending, too. For example, 1, 2, 3, 46, 78. How many suitable selections of five numbers are there?

For what values of the real parameter $a>0$ will the equation $x^2+a=\sqrt{x-a}$ have exactly one solution in the set of real numbers? What is the solution of the equation in that case?

The circles $k_1$ and $k_2$ touch each other externally at point $E$. Lines $f$ and $g$ pass through point $E$. One of the common external tangents touches the circle $k_1$ and $k_2$ at points $C$ and $D$, respectively. Line $h$ is obtained by dropping perpendiculars from point $C$ onto lines $f$ and $g$, and connecting the feet of the perpendiculars. Line $m$ is obtained in the same way, with perpendiculars from point $D$. Prove that $h$ and $m$ are perpendicular to each other.

Each of the diagonals $AD$, $BE$ and $CF$ of a convex hexagon $ABCDEF$ halves the area of the hexagon. Prove that these diagonals are concurrent.

Each letter of the word $\textit{KÖMAL}$ denotes a digit in decimal notation. Given the equalities below, determine the value of the five-digit number $\overline{\textit{KÖMAL}}$. $$\begin{align*} M+\textit{Ö}+L & =\overline{KA}, \tag{1}\\ \textit{Ö}+L & =\overline{KK}, \tag{2}\\ K+\textit{Ö}+M & =10, \tag{3}\\ A\cdot{L} & =42. \tag{4} \end{align*}$$

In an acute-angled triangle $ABC$, let $K$ and $D$, respectively, be the intersections of the interior angle bisector drawn from point $A$ with the interior angle bisector drawn from $B$, and with side $BC$. The perpendicular drawn to angle bisector $AD$ at point $K$ intersects side $AB$ at point $E$. $F$ is the foot of the perpendicular drawn from point $E$ to $BC$. $T$ is the foot of the perpendicular drawn from point $D$ to line $AB$. Prove that $T$ lies on the circumscribed circle of triangle $KEF$.

Let $$\begin{align*} A & ={(-1)}^1+{(-1)}^2+{(-1)}^3+\dots+{(-1)}^{2021},\\ B & ={(-2)}^1+{(-2)}^2+{(-2)}^3+\dots+{(-2)}^{2021} \end{align*}$$ and $C={(-3)}^1+{(-3)}^2+{(-3)}^3+\dots+{(-3)}^{2021}.$ Determine the last digit of the number $B+C-A$.

The midpoints of sides $AB$, $BC$, $CD$, $DA$ of a parallelogram $ABCD$ are $E$, $F$, $G$, $H$, respectively. The lines $AF$ and $AG$ intersect diagonal $BD$ at points $K$ and $L$, respectively. Show that the sum of the areas of triangles $EFK$ and $GHL$ equals the area of triangle $EKL$.

Let $N$ be $\overline{abc}$ a three-digit number in decimal notation. The value of a number $M=\overline{abc}$ represented in some non-decimal notation is $2N$. Determine the number $N$.

Given that $a$ and $b$ are integers such that $3a-2b$ is divisible by $13$, prove that $4a+19b$ and $38a+57b$ are also divisible by $13$.

Line segments $AE$, $BF$, $CG$, $DH$ are perpendicular to the plane $S$ of parallelogram $ABCD$, in the same half space formed by plane $S$. $T$ and $t$ denote the areas of quadrilaterals $CGEA$ and $DHFB$, respectively. Prove that if $\frac{T}{t}=\frac{AC}{BD},$ then the points $E$, $F$, $G$, $H$ are coplanar.

Find all number pairs $p$, $r$ such that $p$, $r$ and $\frac{p+r}{p-r}$ are all positive and prime.