KöMaL 2020

In an $n\times n$ array all the fields are colored with a different color. In one move one can choose a row, move all the fields one place to the right, and move the last field (from the right) to the leftmost field of the row; or one can choose a column, move all the fields one place downwards, and move the field at the bottom of the column to the top field of the same column. For what values of $n$ is it possible to reach any arrangement of the $n$ fields using these kinds of steps?

Let $S$ be a shape in the plane which is obtained as a union of finitely many unit squares. Prove that the ratio of the perimeter and the area of $S$ is at most 8.

Find all triples $(a,b,c)$ of distinct positive integers so that there exists a subset $S$ of the positive integers for which for all positive integers $n$ exactly one element of the triple $(an,bn,cn)$ is in $S$.

Find all positive integers $n$ such that $n!$ can be written as the product of two Fibonacci numbers.

Let $\omega$ denote the incircle of triangle $ABC$, which is tangent to side $BC$ at point $D$. Let $G$ denote the second intersection of line $AD$ and circle $\omega$. The tangent to $\omega$ at point $G$ intersects sides $AB$ and $AC$ at points $E$ and $F$. The circumscribed circle of $DEF$ intersects $\omega$ at points $D$ and $M$. The circumscribed circle of $BCG$ intersects $\omega$ at point $G$ and $N$. Prove that lines $AD$ and $MN$ are parallel.

Each of $N$ people chooses a random integer number between 1 and 19 (including 1 and 19, and not necessarily with the same distribution). The random numbers chosen by the people are independent from each other, and it is true that each person chooses each of the 19 numbers with probability at most $99\%$. They add up the $N$ chosen numbers, and take the remainder of the sum divided by 19. Prove that the distribution of the result tends to the uniform distribution exponentially, i.e. there exists a number $0<c<1$ such that the mod 19 remainder of the sum of the $N$ chosen numbers equals each of the mod 19 remainders with probability between $1/19-c^N$ and $1/19+c^N$.

Let $b \ge 3$ be a positive integer and let $\sigma$ be a nonidentity permutation of the set $\{0, 1, \ldots, b - 1\}$ such that $\sigma(0) = 0$. The substitution cipher $C_\sigma$ encrypts every positive integer $n$ by replacing each digit $a$ in the representation of $n$ in base $b$ with $\sigma(a)$. Let $d$ be any positive integer such that $b$ does not divide $d$. We say that $C_\sigma$ complies with $d$ if $C_\sigma$ maps every multiple of $d$ onto a multiple of $d$, and we say that $d$ is cryptic if there is some $C_\sigma$ such that $C_\sigma$ complies with $d$. Let $k$ be any positive integer, and let $p = 2^k + 1$. $a)$ Find the greatest power of $2$ that is cryptic in base $2p$, and prove that there is only one substitution cipher that complies with it. $b)$ Find the greatest power of $p$ that is cryptic in base $2p$, and prove that there is only one substitution cipher that complies with it. $c)$ Suppose, furthermore, that $p$ is a prime number. Find the greatest cryptic positive integer in base $2p$, and prove that there is only one substitution cipher that complies with it.

Let $O$ be the circumcenter of triangle $ABC$, and $D$ be an arbitrary point on the circumcircle of $ABC$. Let points $X$, $Y$ and $Z$ be the orthogonal projections of point $D$ onto lines $OA$, $OB$ and $OC$, respectively. Prove that the incenter of triangle $XYZ$ is on the Simson-Wallace line of triangle $ABC$ corresponding to point $D$.

Let $H \subseteq \mathbb{R}^3$ such that if we reflect any point in $H$ across another point of $H$, the resulting point is also in $H$. Prove that either $H$ is dense in $\mathbb{R}^3$ or one can find equidistant parallel planes which cover $H$.

Let$k>1$ be a fixed odd number, and for non-negative integers $n$ let $f_n=\sum_{\substack{0\le i\le n\\ k\mid n-2i}}\binom{n}{i}.$ Prove that $f_n$ satisfy the following recursion: $f_n^2=\sum_{i=0}^{n}\binom{n}{i}f_if_{n-i}.$

Finite graph $G(V,E)$ on $n$ points is drawn in the plane. For an edge $e$ of the graph let $x(e)$ denote the number of edges that cross over edge $e$. Prove that $\sum\limits_{e\in E} \frac{1}{x(e)+1}\le 3n-6.$

Find all square-free integers $d$ for which there exist positive integers $x$, $y$ and $n$ satisfying $x^2+dy^2 = 2^n$.

Two circles are given in the plane, $\Omega$ and inside it $\omega$. The center of $\omega$ is $I$. $P$ is a point moving on $\Omega$. The second intersection of the tangents from $P$ to $\omega$ and circle $\Omega$ are $Q$ and $R$. The second intersection of circle $IQR$ and lines $PI$, $PQ$ and $PR$ are $J$, $S$ and $T$, respectively. The reflection of point $J$ across line $ST$ is $K$. Prove that lines $PK$ are concurrent.

We colored the $n^2$ unit squares of an $n\times n$ square lattice such that in each $2\times 2$ part at least two of the four unit squares has the same color. What is the largest number of colors we could have used?

We want to construct an isosceles triangle using a compass and a straightedge. We are given two of the following four data: the length of the base of the triangle ($a$), the length of the leg of the triangle ($b$), the radius of the inscribed circle ($r$) and the radius of the circumscribed circle ($R$). In which of the six possible cases will we definitely be able to construct the triangle?

Prove that the edges of a simple planar graph can always be oriented such that the outdegree of all vertices is at most three.

A polyomino is a figure which consists of unit squares joined together by their sides. (A polyomino may contain holes.) Let $n \ge 3$ be a positive integer. Consider a grid of unit square cells which extends to infinity in all directions. Find, in terms of $n$, the greatest positive integer $C$ which satisfies the following condition: For every colouring of the cells of the grid in $n$ colours, there is some polyomino within the grid which contains at most $n - 1$ colours and whose area is at least $C$.

Let $n$, $s$, $t$ be positive integers and $0<\lambda<1$. A simple graph on $n$ vertices with at least $\lambda n^2$ edges is given. We say that $(x_1,\ldots,x_s,y_1,\ldots y_t)$ is a good insertion, if letters $x_i$ and $y_j$ denote not necessarily distinct vertices and every $x_iy_j$ is an edge of the graph ($1~\le i \le s$, $1\le j\le t$). Prove that the number of good insertions is at least $\lambda^{st}n^{s+t}$.

Let $k\ge t\ge 2$ positive integers. For integers $n\ge k$ let $p_n$ be the probability that if we choose $k$ from the first $n$ positive integers randomly, any $t$ of the $k$ chosen integers have greatest common divisor 1. Let $q_n$ be the probability that if we choose $k-t+1$ from the first $n$ positive integers the product is not divisible by a perfect $t$-th power that is greater then 1. Prove that sequences $p_n$ and $q_n$ converge to the same value.

In a convex set $S$ that contains the origin it is possible to draw $n$ disjoint unit circles such that viewing from the origin non of the unit circles blocks out a part of another (or a complete) unit circle. Prove that the area of $S$ is at least $n^2/100$.

Let $p_n$ denote the $n^{\text{th}}$ prime number and define $a_n=\lfloor p_n \nu \rfloor$, where $\nu$ is a positive irrational number. Is it possible that there exist only finitely many $k$ such that $\binom{2a_k}{a_k}$ is divisible by $p_i^{10}$ for all $i=1,2,\ldots, 2020$?

Solve the following system of equations: $x+\frac1{x^3}=2y, y+\frac1{y^3}=2z, z+\frac1{z^3}=2w, w+\frac1{w^3}=2x.$

Let $p(x)=a_{21}x^{21}+a_{20}x^{20}+\cdots +a_1x+1$ be a polynomial with integer coefficients and real roots such that the absolute value of all of its roots are less than $1/3$, and all the coefficients of $p(x)$ are lying in the interval $[-2019a,2019a]$ for some positive integer $a$. Prove that if this polynomial is reducible in $\mathbb{Z}[x]$, then the coefficients of one its factors are less than $a$.

Andrew and Barry plays the following game: there are two heaps with $a$ and $b$ pebbles, respectively. In the first round Barry chooses a positive integer $k$, and Andrew takes away $k$ pebbles from one of the two heaps (if $k$ is bigger than the number of pebbles in the heap, he takes away the complete heap). In the second round the roles are reversed: Andrew chooses a positive integer and Barry takes away the pebbles from one the two heaps. This goes on, in each round the two players are reversing the roles. The player that takes the last pebble loses the game. Which player has a winning strategy?

There are two kinds of people living on an island: some always tell the truth, while some always lie. Ten inhabitants of the island were numbered as $1,2,\ldots,10$. Then everyone was asked the same three questions: Is your number even?'', Is your number divisible by 4?'', Is your number divisible by 5?''. The number of those saying Yes'' was three, six, and two, respectively in the three cases. Which numbers were assigned to liars?

Let us consider a triangle $ABC$. Let $A_1$ and $A_2$ denote the points that divide side $BC$ into three equal parts, with $A_1$ lying closer to vertex $B$. Analogously, let $B_1$, $B_2$ divide side $CA$ into equal parts, with $B_1$ lying closer to $C$, and let $C_1$ és $C_2$ divide $AB$ into three equal parts, with $C_1$ lying closer to $B$. Prove that the triangles $A_1B_1C_1$ and $B_2C_2A_2$ are congruent, and both of their areas are equal to the third of the area of the triangle $ABC$.

Prove that $\big[\sqrt n+\sqrt{n+3}\,\big]=\big[\sqrt{4n+5}\,\big]$ for all positive integers $n$.

The tangents drawn to the incircle of triangle $ABC$, parallel to the sides, cut off three small triangles at the corners. The radii of the incircles of the small triangles are 2, 3 and 10 units long. Show that triangle $ABC$ is right-angled.

Find all positive integers $n$ and distinct (positive) primes $p$, $q$, $r$ for which $\frac1{pq}+\frac1{pr^3}+\frac1{qr^2}=\frac1n.$

The midpoints of sides $AD$ and $BC$ of a convex quadrilateral $ABCD$ are $E$ and $F$, respectively. Line segment $EF$ intersects diagonal $AC$ at point $P$, and diagonal $BD$ at point $Q$. Prove that the circles $AEP$ and $BFQ$ intersect each other on the line $AB$.

Find the real solutions of the system $$\begin{align*} x+y+z+v & =0, \\ x^2+y^2+z^2+v^2 & = 12, \\ x^3+y^3+z^3+v^3 & = 24. \end{align*}$$

Solve the equation $\log_2 \log_3 x+\log_3\log_2 x=\log_2\frac{6}{\log_2 3}$ over the set of real numbers.

Let $D$ denote the midpoint of the base $AB$ in an isosceles triangle $ABC$. Let $H$ be the point lying closer to $C$ that divides the leg $AC$ in a $1:2$ ratio. The circle $BCH$ intersects line $CD$ at the points $C$ and $X$. Show that $CX=\frac 43r$, where $r$ is the radius of the circle $ABC$.

In a triangle, the medians drawn to sides $a$ and $b$ are perpendicular. Prove that $\frac 12<\frac ab<2$.

Prove that the geometric mean, the arithmetic mean, and the quadratic mean of the altitudes in any triangle is not greater than the geometric mean, the arithmetic mean, and the quadratic mean of the radii of the escribed circles, respectively.

Is there a polynomial $p(x)$ of degree 100 with real coefficients for which the polynomial $p\big(p(x)\big)$ has 10000 distinct real roots?

Let $n$ be a positive integer, and let $\mathcal{S}$ denote the set of $0-1-2$ sequences of length $n$. Determine which sets $\emptyset\ne A\subseteq\mathcal{S}$ have the following property: no matter how a vector $(c_1,c_2,\ldots,c_n)\in \mathcal{S}\setminus\big\{(0,0,\dots,0)\big\}$ is selected, the probabilities that the sum of products $c_1a_1+c_2a_2+\dots+c_na_n$ formed with a randomly chosen element $(a_1,a_2,\ldots,a_n)$ of set $A$ will leave a remainder of $0$, $1$, or $2$ are $1/3$ each.

Show that a regular heptagon can be dissected into a finite number of symmetrical trapezoids, all similar to each other.

Solve the equation $\big(x^3-y^2\big)^{2}= \big(x^2-y^3\big)^{2}$ over the set of pairs of integers.

The distances of an interior point $P$ of a square $ABCD$ from the vertices $A$, $B$, $D$ are $1$, $\sqrt2\,$, and $2$, respectively. Calculate the measure of the angle $APB$.

With respect to a given set $G$ of numbers, the positive integer $k>1$ is called interesting if there exist $k$ distinct elements in set $G$ such that the arithmetic mean of these elements also belongs to set $G$. Let $H=\{1;3;4;9;10;\ldots\}$ be the set of those numbers that can be represented as a sum of some different powers of 3. $a)$ What numbers $k>1$ are interesting with respect to the set $H$? $b)$ Let $c \notin H$ be an arbitrary positive integer. Prove that every number $k>1$ is interesting with respect to the set $H'=H \cup \{c\}$.

Two skew edges of a tetrahedron are perpendicular to each other, their lengths are 12 and 13, and the distance between their lines is 14 units. Determine the volume of the tetrahedron.

The inscription on one side of a fair coin is $+1$, and $-1$ is on the other side. The coin is tossed $n$ times in a row, and the $n$ results are written down in a row. Then the product of every pair of consecutive items is written below them, resulting in a new list of numbers that only consists of $(n-1)$ items. The procedure is repeated until a single number remains. What is the expected value of the sum of the $\frac{n(n+1)}{2}$ numbers written down in the resulting triangular arrangement of numbers?

In a regular dodecagon $A_1A_2\ldots A_{12}$, let $P$ denote the intersection of the diagonals $A_1A_8$ and $A_6A_{11}$, and let $R$ denote the intersection of lines $A_7A_8$ and $A_9A_{11}$. Show that the line $PR$ divides diagonal $A_1A_4$ in a $2:1$ ratio.

The sum of the elements is calculated for each subset of the set $\{0,1,\dots,n-1\}$. What may the value of $n$ be if exactly one $n$th of the resulting $2^n$ sums is divisible by $n$?

The intersection of two congruent regular pentagons is a decagon with sides of $a_1, a_2, \ldots, a_{10}$ in this order. Prove that $a_1a_3+a_3a_5+a_5a_7+a_7a_9+a_9a_1 = a_2a_4+a_4a_6+a_6a_8+a_8a_{10}+a_{10}a_2.$

Prove that if two right-angled triangles have the same perimeter and the same area, then they are congruent.

Let $a$, $b$, $c$ denote distinct nonzero integers. Prove that if the sum of the three numbers $\frac{ab}{c}$, $\frac{bc}{a}$ and $\frac{ca}{b}$ is an integer, then each of the three numbers is an integer.

In a regular triangle $ABC$ of unit sides, let $P$ be an arbitrary point on the circumference of the incircle. Let $D$, $E$, and $F$ denote the orthogonal projections of point $P$ on the sides $BC$, $AC$ and $AB$, respectively. Prove that the area of triangle $DEF$ is a constant, independent of the choice of $P$.

The product of the positive numbers $x_1,x_2,\dots,x_n$ is $1$. Prove that $x_1^4+x_2^4+\dots+x_n^4 \ge x_1^3+x_2^3+\dots+x_n^3.$

Two players, First and Second, are playing the following game: First selects a positive integer not greater than 2020, which Second is trying to find out by guessing (by naming a number as a guess). The possible answers of First are as follows: My number is smaller than that.''; You are right.''; My number is greater than that.'' If the answer is My number is smaller than that'' or You are right'', then Second pays 10 forints (Hungarian currency) to First. If the answer is My number is greater than that'' then he pays 20 forints. What is the minimum possible amount of money that Second needs to have in order to be certain that he can find out the number, and what strategy should he use? (The game terminates with the first ``You are right'' answer, even if Second already knows the number before asking the last question.)

The angle at vertex $A$ of a rhombus $ABCD$ is $60^\circ$. An ellipse is inscribed in the rhombus, with the axes lying along the diagonals of the rhombus. The points of tangency of the ellipse on sides divide the sides in a ratio $1:3$. On sides $AB$ and $AD$ it is the point closer to $A$, and on sides $BC$ and $CD$ it is the point closer to $C$. Let some point $P$ move along the ellipse. Draw lines through $P$, parallel to the midlines of the rhombus, and consider the intersections with the other midline. Let these point be $Q$ and $R$. Show that the length of the line segment $QR$ is independent of the position of $P$.

Show that it is always possible to select some numbers (at least one) out of $n$ consecutive integers such that their sum is divisible by $(1+2+\dots+n)$.

$ABCDO$ is a four-sided pyramid, and $P$ is a point in the interior of base $ABCD$. A plane not passing through $O$ cuts the lines $OA$, $OB$, $OC$, $OD$ and $OP$ at points $A'$, $B'$, $C'$, $D'$, and $P'$, respectively. Prove that $\frac{t_{PAB}\cdot t_{PCD}}{t_{PBC}\cdot t_{PDA}} = \frac{t_{P'A'B'}\cdot t_{P'C'D'}}{t_{P'B'C'}\cdot t_{P'D'A'}},$ where $t_{XYZ}$ denotes the area of triangle $XYZ$.

There are $n$ distinct points in the plane, which are not all collinear. Show that there exists a closed polygon with these vertices that does not cut through itself. (A polygon is allowed to have angles equal to $180^{\circ}$, too.)

Let $a$, $b$, $c$, $x$, $y$ and $z$ be positive numbers that satisfy the equalities $a^2+b^2=c^2$ and $x^2+y^2=z^2$. Prove that ${(a+x)}^2+{(b+y)}^2\le {(c+z)}^2$, and determine the condition for equality.

Let $A_1$, $B_1$ and $C_1$ denote the points of tangency of the incircle of triangle $ABC$ on the sides, and let $R$ and $r$ be the radii of the circumscribed and inscribed circles, respectively. Prove that the ratio of the areas of triangles $A_1B_1C_1$ and $ABC$ is $r:2R$.

Let $n$ denote a positive integer. Determine the smallest number of colours $k$ that are sufficient for colouring the edges of any directed simple graph of $n$ vertices without producing a circuit of the same colour.

The numbers $n+1, n+2, \ldots, 2n$ are written on a blackboard ($n\ge2$), and the following procedure is repeated: two numbers are selected ($x$ and $y$) from the board, erased, and replaced with the numbers $x+y+\sqrt{x^2+y^2}$ and $x+y-\sqrt{x^2+y^2}$. Prove that there will never be a number less than 1.442 written on the board.

The diagonals of a cyclic quadrilateral $ABCD$ intersect at $F$, the lines of sides $AB$ and $CD$ intersect at $E$, the midpoint of line segment $EF$ is $G$, the midpoint of line segment $BF$ is $H$, and the midpoint of side $BC$ is $I$. Show that $\angle GFD=\angle GIH$.

Let $x_1 = 2, x_2 = 7, x_{n+1} = 4 x_n - x_{n-1} (n=2,3,\ldots).$ Is there a perfect square in this sequence?

The tangents drawn to the inscribed circle of an isosceles triangle, parallel to the sides, cut off three small triangles. Prove that in the small triangles lying on the base of the large triangle, the heights drawn to the base are equal to the radius of the inscribed circle of the large triangle.

Let $a$ and $b$ be real numbers such that $a + b = 1$ and $a^2 + b^2 = 2$. Find the value of $a^8 + b^8$.

A deck of card consists of $p$ red cards and $k$ blue cards. In how many different ways is it possible to select some of the cards so that the number of red cards should be $n$ more than the number of blue cards?

Let $a$, $b$ and $c$ denote some given, pairwise relatively prime positive integers. Prove that the equation $x^a + y^b=z^c$ has infinitely many solutions $(x,y,z)$ where $x$, $y$ and $z$ are positive integers.

Ali has $n$ coins in his purse, and Baba has $n-1$ purses, initially all empty. Baba is playing the following game: he divides the coins (all in the same purse at start) into two purses, with $a_1$ coins in one of them and $b_1$ coins in the other ($a_1,b_1>0$), and then he writes the product $a_1b_1$ on a blackboard. Then he continues in the same way: in the $k$th move ($k=2,3,\ldots$) he selects a purse containing at least two coins, divides them between two empty purses, with $a_k$ coins in one of them and $b_k$ in the other ($a_k,b_k>0$), and writes the product $a_kb_k$ on the board. The game terminates when there is 1 coin in each purse. Then Ali gives as many coins to Baba as the sum of all the products $a_k b_k$ on the board. What is the maximum number of coins that Baba may get?

Let $a,b,c>0$ and $x,y,z\ge 0$. Prove that if $x+aby \le a(y+z)$, $y+bcz \le b(z+x)$, and $z+cax \le c(x+y)$, then $x=y=z=0$ or $a=b=c=1$.

The points $A$, $B$, $C$, $D$ (in this order) lie on the same line. On the same side of the line, a regular triangle is drawn on each of the line segments $AB$, $BC$ and $CD$, with the third vertices being $E$, $F$, and $G$, respectively. Let the distances between the adjacent points on the line be $AB=a$, $BC=b$, $CD=c$. Show that the measure of $\angle EFG$ equals $120^{\circ}$ if and only if $a+c=b$ or $\frac1a+\frac1c=\frac1b$.

Is it possible that $x$, $\frac{14x+5}{9}$ and $\frac{17x-5}{12}$ are all integers?

In an acute-angled triangle $ABC$, a tangent is drawn to the inscribed circle, parallel to side $BC$. The tangent intersects side $AC$ at point $D$. $F$ is the orthogonal projection of point $D$ onto the side $BC$. Show that $AB=AD+BF$.

The positive integers are coloured in the following manner: the colour of $a+b$ is always uniquely determined by the colours of $a$ and $b$; that is, if the colour of $a$ and $a'$ is the same, and the colour of $b$ and $b'$ is the same, then $a+b$ and $a'+b'$ also have the same colour. Prove that if there is a colour that is used more than once then the colouring becomes periodic from some number onwards.

Solve the following simultaneous equations, where $x_1, x_2,\ldots,x_n$ are positive real numbers, and $n$ is a positive integer: $$\begin{align*} x_1+x_2+\ldots +x_n & =9,\\ \frac1{x_1}+\frac1{x_2}+\ldots +\frac1{x_n} & =1. \end{align*}$$

ErWin Layup is the best penalty taker of all times in the basketball league of Nowhereland. Although he missed the very first penalty throw of his career, altogether he has only missed 2020 out of his total of $222 222$ throws. Statisticians in Nowhereland consider a basketball penalty throw interesting if the ratio of successful penalty throws to all penalty throws, calculated immediately after the throw and expressed as a percentage, is a positive integer. (For example, if a player scores 12 out of a total of 40 throws then his last throw is interesting, since $\frac{12}{40} \cdot 100 = 30 \in \mathbb{N}^+$, while the following throw, which is the 41st, cannot be interesting, whether successful or not.) What is the minimum number of interesting penalty throws that ErWin Layup may have had?

Ann and Barbara divided between themselves the 81 cards of the game of SET . Ann received 40 cards and Barbara received 41. Each girl counted the number of ways they can form a SET of three cards out of the cards held by her. What may be the sum of the numbers they obtained?

The base of a right pyramid is a square $ABCD$, and the apex of the pyramid is $E$. The skew edges $AB$ and $CE$ are connected by a transversal that is normal to both of them. The feet of the normal transversal are point $P$ on the line segment $AB$, and point $Q$ on the line segment $CE$. Given that $Q$ bisects the edge $CE$, determine the ratio $AP: PB$, and calculate the angle enclosed between the lateral faces and the base of the pyramid.

The centre of the circumscribed circle of a cyclic quadrilateral $ABCD$ is $O$. The rays $AB$ and $DC$ intersect at point $E$. In the circle $BCE$, the point diametrically opposite to $E$ is $F$. Show that the lines $AC$, $BD$ and $OF$ are concurrent.

Prove that if $n\ge 3$, then there exist $n$ distinct positive integers such that the sum of their reciprocals is 1.

Given a convex angle and a line segment of length $k$, determine the locus of those points inside the angle through which there exists a line cutting off a triangle of perimeter $k$ from the angle.

Find all pairs of relatively prime integers $(x,y)$ such that $x^2 + x = y^3 + y^2$.

Two players are taking turns in selecting one of the coefficients $a$, $b$ and $c$ of the polynomial $x^3+ax^2+bx+c$, and giving it some integer value of their choice. Prove that the starting player can achieve that (after the three steps) all three roots of the polynomial should be integers (i.e. that the polynomial can be expressed as a product of three polynomials of integer coefficients).

There are $n$ points in the plane. We know that for any $k$ points ($k\ge 2$), it is possible to select two of them with distance at most 1. Show that the points can be covered with $k-1$ disks of unit radius.

Let $H$ be an equilateral triangle of unit area, let $O$ be a fixed point, and for any point $P$ let $H_P$ denote the triangle obtained from triangle $H$ by a parallel shift with vector $\overrightarrow{OP}$. Consider the set $N$ of points $P$ for which the area of the intersection $H\cap H_P$ is at least $4/9$. What is the area of $N$?

How many different strings of 2021 letters can be made of letters A, B and C such that the number of A's is even and the number of B's is of the form $3k+2$?

Given six points in the space, no four of which are coplanar, prove that they can be divided into two sets of three such that the two triangular plates spanned by the two sets of three points should intersect each other.

The feet of the altitudes drawn from vertices $A$, $B$, $C$ of an acute-angled triangle $ABC$ are $A_1$, $B_1$ and $C_1$, respectively; and the midpoints of the altitudes $AA_1$ and $BB_1$ are $G$ and $H$, respectively. Prove that the circumscribed circle of triangle $C_1GH$ passes through the midpoint $F$ of side $AB$.

The population of an island consists of underdogs and overlords. When a stranger visited the island, he was invited for dinner with a company of inhabitants. At the end, he asked each member of the company how many overlords were present. The underdogs all gave figures less than the true value and the overlords all gave figures larger than the true value. Is it true that the number of overlords can always be determined from the answers?

Solve the following simultaneous equations over the set of real numbers: $$\begin{align*} x+y^2 & =z^3,\\ x^2+y^3 & =z^4,\\ x^3+y^4 & =z^5. \end{align*}$$

Triangle $ABC$ is not isosceles. The interior angle bisectors drawn from vertices $A$ and $B$ intersect the opposite sides at points $A'$ and $B'$, respectively. Prove that the perpendicular bisector of $A'B'$ will pass through the centre of the inscribed circle if and only if $AB' + BA' = AB$.

The diagonals of a convex quadrilateral $ABCD$ intersect at $M$. The area of triangle $ADM$ is greater than that of triangle $BCM$. The midpoint of side $BC$ of the quadrilateral is $P$, and the midpoint of side $AD$ is $Q$, $AP+AQ=\sqrt2\,$. Prove that the area of quadrilateral $ABCD$ is smaller than 1.

There are 10 countries on an island, some of which share a border, and some do not. Each country uses a currency of its own. Every country operates a single exchange office, by the following rules: if you pay 10 units of the currency of that country, you will get 1 unit of each of the currencies of the bordering countries. Initially Arthur and Theodore each own 100 units of the currency of every country. Then each of them shops around the exchange offices of various countries in any order they like, and keeps exchanging money while they can (that is, while they have at least 10 units of a kind). Prove that Arthur and Theodore will have the same number of Bergengocian dollars at the end (the Bergengocian dollar is the currency of one of the countries on the island).

Prove that $\sum_{i=0}^n \sum_{j=i}^n \binom{n}{i} \binom{n+1}{j+1} = 2^{2n}.$

Find all positive integers $n$ for which $n!$ ends in exactly $19 531$ zeros.

Let $ABP$, $BCQ$, $CDR$, $DAS$ denote the regular triangles drawn on the sides of paralellogram $ABCD$ on the outside. What requirements does the parallelogram need to meet in order for $PQRS$ to be a square?

In the Cartesian coordinate plane, represent the points for which the following inequality holds: $|x|+|y|+|x+y|\le 2.$ What is the area of the resulting figure?

What is the maximum area of a square that can be obtained from a triangular sheet of paper with sides 3 cm, 4 cm and 5 cm by cutting along at most three straight lines?

What distinct positive primes $p$ and $q$ satisfy the equality $p-4p^2+p^3=q-4q^2+q^3$?

The points $D$ and $E$ divide side $AB$ of triangle $ABC$ into three equal parts. Let $P$ be an arbitrary interior point of line segment $DE$. Draw parallels to line $PC$ through the points $D$ and $E$. These lines intersect sides $AC$ and $BC$ at points $Q$ and $R$, respectively. Show that the area of quadrilateral $PRCQ$ equals the area of triangle $APQ$.

Solve the equation $\frac{x-1}{\sqrt x}-\frac{\sqrt x}{x-2}=\frac{\sqrt x}{2x-1}$ over the set of real numbers.

Let $E$ and $F$ be the points lying closer to vertex $A$ which divide the sides $AB$ and $AD$ of a quadrilateral $ABCD$, respectively, in a $1:2$ ratio. Let $G$ be the point lying closer to $B$ which divides side $BC$ in a $1:2$ ratio. Reflect point $G$ in the point $E$, and reflect its image in the point $F$. Prove that the final image lies on a side of the quadrilateral. Which side is it, and in what ratio is it divided by the final image?

Solve the following equation over the set of real numbers: ${(y^2+y-x-1)}^2+\left(x+\frac1x \right)^{2}=4.$

Find the positive integer solutions of the equation ${(a+1)}^4\cdot {(b+1)}^4\cdot {(c+1)}^4=(40a+1)\cdot(40b+1)\cdot(40c+1).$

The coordinates of a ship are $x=2$, $y=0$. The shoreline is given by the curve of equation $y=\sqrt{2x+1}$. At what angle should the ship deviate from the direction due north in order to reach the closest point of the shore in a straight line? (Assume that the $x$-axis points towards the east.)

In England, a man lost his wedding ring, and set out to search for it with the help of a friend and a metal detector. They did not find the ring, but they did find some gold coins from the time of King Henry VIII, delivering $100 000$ pounds to the two friends. The one-pound coins were preserved in very good condition, and during the 500 years elapsed, their average annual increase in value was between 1.42% and 1.43%. How many coins may they have found?

Two sides of a triangle are 3 cm and 4 cm long. What is the angle enclosed by the sides if the medians drawn from the opposite vertices are perpendicular to each other?

The first row of seats in an auditorium consists of 24 seats. 20 seats are already taken. What is the probability that there are 2 adjacent vacant seats?

Find all pairs $(x,y)$ of positive integers such that $\frac1x+\frac1y=\frac{2}{1893}.$

The sides of a triangle are 5 cm, 5 cm and 6 cm long. The sides, and the tangents drawn to the incircle parallel to the sides form a hexagon. What is the area of this hexagon?

How many different right-angled triangles are there in which the measures of the sides are integers, and one side is $2^n$ units long? (Where $n$ is a positive integer: express your answer in terms of $n$.)

The length of the line segment $MN$ joining the midpoints of sides $AB$ and $CD$ in a convex quadrilateral $ABCD$ is the arithmetic mean of the lengths of sides $AD$ and $BC$. Show that the quadrilateral $ABCD$ is a trapezium.

Solve the following equation over the set of pairs of natural numbers: $2y^2-2x^2-3xy+3x+y=13.$

Solve the following equation over the set of real numbers: $4^x+9^x+36^x+\sqrt{\frac12-2x^2}=1.$

The height of a lateral face of a right pyramid with a square base is twice as long as the base edge. At what percentage of this height (counting from the base) do we need to cut the pyramid with a plane parallel to the base so that the total area of the lateral surface plus top square of the resulting frustum is equal to half the lateral surface area of the original pyramid?

Two tenth-grade students and two eleventh-grade students sat down to solve the exercises of type C in the April issue of KöMaL. (There are seven exercises each month. Exercises 1-5 are for students in grade 10 at most, while exercises 3-7 may be solved by 11th and 12th grade students.) After an hour, they observed that each exercise was solved by exactly one of them, and that everyone solved at least one exercise. In how many different arrangements may they have solved the exercises? (Two arrangements are considered different if there is at least one exercise that is solved by a different student.)

The altitude drawn from vertex $A$ of an isosceles triangle $ABC$ intersects the leg $BC$ at $T$. Let $M$ denote the orthocentre, and let $O$ be the centre of the inscribed circle. Prove that if line $OT$ is parallel to the base $AB$, then $MC=2AM$.

A farmer brought 1225 packets of seeds to an agricultural fair: 1 packet of 1 gram of seeds, 2 packets of 2 grams, 3 packets of 3 grams, ..., $k$ packets of $k$ grams of seeds in each - every positive integer $1$ to $k$ occurred. What was the average mass of seeds in a packet?

The diagonals of a convex quadrilateral $ABCD$ intersect at $M$. The area of triangle $ABM$ is greater than the area of triangle $CDM$. The midpoint of side $BC$ of the quadrilateral is $P$, and the midpoint of side $CD$ is $Q$, $AP+AQ=\sqrt2\,$. Prove that the area of quadrilateral $ABCD$ is less than 1.

The areas of two faces of a cuboid are 40 and 56 units. The length of the diagonal of the cuboid is $\sqrt{138}$ units of length. Calculate the possible surface area and the volume of the cuboid.

Between 4 and 9, some digits of 4 are inserted, followed by the same number of digits of 8 (e.g. 4489). Prove that the resulting number is a perfect square.

Solve the following simultaneous equations over the set of real numbers: $$\begin{align*} x+y+\frac xy & =19,\\ \frac{x(x+y)}{y} & =60. \end{align*}$$

In a unit circle, the diameter $AB$ and the chord $AC$ enclose a $30^\circ$ angle. Let $B'$ denote the reflection of $B$ about the point $C$. Determine the distances between $B$ and the points where the tangents drawn from $B'$ to the circle intersect the line $AB$.

Some numbers are selected from the set of the first 21 positive integers such that the absolute values of the differences of all pairs of selected numbers should be different. What is the largest possible number of different absolute values obtained? Give an example of a case when this occurs.

The convex heptagon $A_1A_2A_3A_4A_5A_6A_7$ has a circumscribed circle centred at an interior point of the heptagon. Prove that the sum of the interior angles at the vertices $A_1$, $A_3$ and $A_5$ is less than $450^{\circ}$.

There were $n$ teams participating in a basketball championship. Every team played every other team exactly once, and there was no draw. At the end of the championship, the $i$th team had $x_i$ games won and $y_i$ games lost ($i=1,2,\ldots,n$). Prove that $x_1^2+x_2^2+\cdots+x_n^2=y_1^2+y_2^2+\cdots+y_n^2.$

12 round muffins of diameter 9 cm are arranged along the edge of a round tray of radius 30 cm such that they all touch the edge of the tray, and the neighboring muffins are separated by the same distance from each other. What is this equal distance?

The text was changed. The correct text: Julie's grandmother bakes cookies every Monday. She always selects out of her infinite number of recipes at random. $60\%$ of her recipes contain chocolate chips. Julie is quite picky about cookies: she only likes $90\%$ of grandma's chocolate chip cookies, and only $30\%$ of the other kinds of cookies. On a special Monday, grandmother is making two different kinds of cookies. Find the probability that Julie will like exactly one of them.

Solve the equation $x+\frac{1}{y+\frac{1}{\frac{5z}{4}+\frac{13}{6v}}}=\frac{135}{113}$ where $x$, $y$, $z$, $v$ are positive integers.

Prove that $\frac23\le a_{n+1}-a_n<1$ holds for all the terms of the sequence $a_n=\frac{(n-1)n}{n+1}$, with $n\ge 1$.

From the midpoint of each side of an acute-angled triangle, drop perpendiculars to the other two sides. Prove that the area of the hexagon formed by the perpendiculars is half the area of the triangle.

Jack Russell fashioned a springboard in his back yard. He carried out measurements to determine that by bending the springboard $x$ decimetres below horizontal, he could jump to a height of $0.5x^2+ax+b$ decimetres. Unfortunately, he forgot the values of $a$ and $b$. He only remembers that bending by $10$ cm let him jump $35$ cm, and a deformation four times as large resulted in a jump four times as high. What may have been Jack's measured values of $a$ and $b$?

The lengths of the sides of a trapezoid circumscribed about a circle are integers that are consecutive terms of an arithmetic progression in some order. Given that the radius of the incircle and the length of the shorter base are both 6, how long are the other three sides?

Prove that the area bounded by the graphs of the functions $y=1-|x-1|$ and $y=|2x-a|$ is less than $\frac13$ for $1<a<2$.

Let $m$ be a positive integer. Show that $a)$ there exist three $m$-digit powers of 2; $b)$ there exist at most four $m$-digit powers of 2.

Point $P$ of side $AB$ in a square $ABCD$ is connected to $D$, and point $Q$ of side $BC$ is connected to $A$. The intersection of the resulting line segments is denoted by $R$. The area of triangle $ARD$ is 1200, the area of triangle $APR$ is 600, and the area of quadrilateral $PBQR$ is $3380-240\sqrt{95}$ units of area. What is the area of quadrilateral $RQCD$?