KöMaL 2019

A $k\times k$ array contains each of the numbers $1,2,\ldots,m$ exactly once, with the remaining entries all zero. Suppose that all the row sums and column sums are equal. What is the smallest possible value of $m$ if $k=3^n$ ($n\in\mathbb{N^+}$)?

Let $f$ be a function defined on the positive integers with $f(n)\ge 0$ and $f(n)\le f(n+1)$ for all $n$. Prove that if $\sum\limits_{n=1}^\infty \frac{f(n)}{n^2}$ diverges, there exists a sequence $a_1,a_2,\dots$ such that the sequence $\frac{a_n}{n}$ hits every rational number, while $a_{n+m}\le a_n+a_m+f(n+m)$ holds for every pair $n$, $m$.

Convex quadrilateral $ABCD$ is inscribed in circle $\Omega$. Its sides $AD$ and $BC$ intersect at point $E$. Let $M$ and $N$ be the midpoints of the circle arcs $AB$ and $CD$ not containing the other vertices, and let $I$, $J$, $K$, $L$ denote the incentres of triangles $ABD$, $ABC$, $BCD$, $CDA$, respectively. Suppose $\Omega$ intersects circles $IJM$ and $KLN$ for the second time at points $U\ne M$ and $V\ne N$. Show that the points $E$, $U$, and $V$ are collinear.

The incircle of tangential quadrilateral $ABCD$ intersects diagonal $BD$ at $P$ and $Q$ ($BP<BQ$). Let $UV$ be the diameter of the incircle perpendicular to $AC$ ($BU<BV$). Show that the lines $AC$, $PV$ and $QU$ pass through one point.

Show that for every odd integer $N>5$ there exist vectors $\mathbf{u},\mathbf{v},\mathbf{w}$ in (three-dimensional) space which are pairwise perpendicular, not parallel with any of the coordinate axes, have integer coordinates, and satisfy $|\mathbf{u}|=|\mathbf{v}|=|\mathbf{w}|=N$.

We have attached a clock hand to every face of a convex polyhedron. Each hand always points towards a neighboring face (two faces of the polyhedron are neighbors if they share an edge). At the end of every minute, exactly one of the hands turns clockwise to point at the next face. Suppose that the hands on neighboring faces never point towards one another. Show that one of the hands makes only finitely many turns.

Let $p$ be a prime number. How many solutions does the congruence $x^2+y^2+z^2+1\equiv0\pmod{p}$ have among the modulo $p$ remainder classes?

In a simple graph on $n$ vertices, every set of $k$ vertices has an odd number of common neighbours. Prove that $n+k$ must be odd.

The circles $\Omega$ and $\omega$ in its interior are fixed. The distinct points $A$, $B$, $C$, $D$, $E$ move on $\Omega$ in such a way that the line segments $AB$, $BC$, $CD$ and $DE$ are tangents to $\omega$. The lines $AB$ and $CD$ meet at point $P$, the lines $BC$ and $DE$ meet at $Q$. Let $R$ be the second intersection of the circles $BCP$ and $CDQ$, other than $C$. Show that $R$ moves either on a circle or on a line.

Given are two polyominos, the first one is an L-shape consisting of three squares, the other one contains at least two squares. Prove that if $n$ and $m$ are co-prime then at most one of the $n\times n$ and $m\times m$ boards can be tiled by translated copies of the two polyominos.

Let $c>0$ be a real number, and suppose that for every positive integer $n$, at least one percent of the numbers $1^c,2^c,3^c,\ldots,n^c$ are integers. Prove that $c$ is an integer.

Let $k$, $s$ and $n$ be positive integers such that $s<(2k+1)^2$, and let $R$ be the set of lattice points $(x,y)$ in the plane, satisfying $1\le x,y\le n$. On the grid $R$ we perform the following procedure. Initially, we colour one point of $R$ green; all other points in $R$ are coloured white. On every move, we choose a square $S$, consisting of ${\color{red}(2k+1)\times(2k+1)}$ lattice points in such a way that the center of $S$ is green and it contains at least $s$ white points; then we re-colour $s$ white points of $S$ to green. We repeat this step as long as there is a suitable square $S$. We say that the number $s$ is $k$-sparse , if there exists a positive real number $C$ such that, for every $n$, for every choice of the initial green point, and for every valid sequence of moves, the total number of green points in the grid cannot exceed $Cn$. Determine the least $k$-sparse positive integer $s$ in terms of $k$.

Let $a$ be an integer, and let $p$ be a prime divisor of $a^3+a^2-4a+1$. Show that there is an integer $b$ such that $p\equiv b^3\pmod{13}$.

Let $P$ be a point inside the acute triangle $ABC$, and let $Q$ be the isogonal conjugate of $P$. Let $L$, $M$ and $N$ be the midpoints of the shorter arcs $BC$, $CA$ and $AB$ of the circumcircle of $ABC$, respectively. Let $X_A$ be the intersection of ray $LQ$ and circle $PBC$, let $X_B$ be the intersection of ray $MQ$ and circle $PCA$, and let $X_C$ be the intersection of ray $NQ$ and circle $PAB$. Prove that $P$, $X_A$, $X_B$ and $X_C$ are concyclic or coincide.

Prove that every polygon that has a center of symmetry can be dissected into a square such that it is divided into finitely many polygonal pieces, and all the pieces can only be translated. (In other words, the original polygon can be divided into polygons $A_1, A_2,\dots, A_n$, a square can be divided into polygons a $B_1, B_2,\dots, B_n$ such that for $1\le i\le n$ polygon $B_i$ is a translated copy of polygon $A_i$.)

Find all functions $f\colon \mathbb{R}\to \mathbb{R}$ (functions with domain $\mathbb{R}$ and values from $\mathbb{R}$) which satisfy the following two conditions: $(i)$ $f(x+1)=f(x)+1$; $(ii)$ $f(x^2)=\big(f(x)\big)^2$.

For every $n$ non-negative integer let $S(n)$ denote a subset of the positive integers, for which $i$ is an element of $S(n)$ if and only if the $i$-th digit (from the right) in the base two representation of $n$ is a digit 1. Two players, $A$ and $B$ play the following game: first, $A$ chooses a positive integer $k$, then $B$ chooses a positive integer $n$ for which $2^n\ge k$. Let $X$ denote the set of integers $\{0,1,\dots,2^n-1\}$, let $Y$ denote the set of integers $\{0,1,\dots,2^{n+1}-1\}$. The game consists of $k$ rounds, and in each round player $A$ chooses an element of set $X$ or $Y$, then player $B$ chooses an element from the other set. For $1\le i\le k$ let $x_i$ denote the element chosen from set $X$, let $y_i$ denote the element chosen from set $Y$. Player $B$ wins the game, if for every $1\le i\le k$ and $1\le j\le k$ $x_i<x_j$ if and only if $y_i<y_j$ and $S(x_i)\subset S(x_j)$ if and only if $S(y_i)\subset S(y_j)$. Which player has a winning strategy?

In quadrilateral $ABCD$, $AB=BC=\frac1{\sqrt2}DA$, and $\angle ABC$ is a right angle. The midpoint of side $BC$ is $E$, the orthogonal projection of $C$ on $AD$ is $F$, and the orthogonal projection of $B$ on $CD$ is $G$. The second intersection point of circle $BCF$ (with center $H$) and line $BG$ is $K$, and the second intersection point of circle $BHC$ and line $HK$ is $L$. The intersection of lines $BL$ and $CF$ is $M$. The center of the Feurbach circle of triangle $BFM$ is $N$. Prove that $\angle BNE$ is a right angle.

We choose a random permutation of numbers $1, 2,\dots, n$ with uniform distribution. Prove that the expected value of the length of the longest increasing subsequence in the permutation is at least $\sqrt{n}\,$.

An illusionist and his assistant are about to perform the following magic trick. Let $k$ be a positive integer. A spectator is given $n = k! + k - 1$ balls numbered $1, 2,\dots, n$. Unseen by the illusionist, the spectator arranges the balls into a sequence as he sees fit. The assistant studies the sequence, chooses some block of $k$ consecutive balls, and covers them under her scarf. Then the illusionist looks at the newly obscured sequence and guesses the precise order of the $k$ balls he does not see. Devise a strategy for the illusionist and the assistant to follow so that the trick always works. (The strategy needs to be constructed explicitly. For instance, it should be possible to implement the strategy, as described by the solver, in the form of a computer program that takes $k$ and the obscured sequence as input and then runs in time polynomial in $n$. A mere proof that an appropriate strategy exists does not qualify as a complete solution.)

Let $n \ge 3$ be a positive integer. We say that a set $S$ of positive integers is good if $|S| = n$, no element of $S$ is a multiple of $n$, and the sum of all elements of $S$ is not a multiple of $n$ either. Find, in terms of $n$, the least positive integer $d$ for which there exists a good set $S$ such that there are exactly $d$ nonempty subsets of $S$ the sum of whose elements is a multiple of $n$.

In a forest there are $n$ different trees (considered as points), no three of which lie on the same line. John takes photographs of the forest such that all trees are visible (and no two trees are behind each other). What is the largest number of orders of in which the trees that can appear on the photos?

Let $k\ge 2$ be an integer. We want to determine the weight of $n$ balls. One try consists of choosing two balls, and we are given the sum of the weights of the two chosen balls. We know that at most $k$ of the answers can be wrong. Let $f_k(n)$ denote the smallest number for which it is true that we can always find the weights of the balls with $f_k(n)$ tries (the tries don't have to be decided in advance). Prove that there exist numbers $a_k$ and $b_k$ for which $\big|f_k(n)-a_kn\big|\le b_k$ holds.

We call a diagonal of a polygon nice, if it is entirely inside the polygon or entirely outside the polygon. Let $P$ be an $n$-gon with no three of its vertices being on the same line. Prove that $P$ has at least $\frac32(n-3)$ nice diagonals.

Find all functions $f\colon \mathbb{R}\to\mathbb{R}$ which satisfy the following equality for all $x,y\in \mathbb{R}$: $f(x)f(y)-f(x-1)-f(y+1)=f(xy)+2x-2y-4.$

Let $T$ be any triangle such that its side-lengths $a$, $b$, and $c$ and its circumradius $R$ are positive integers. Show that $a)$ the inradius $r$ of $T$ is a positive integer; $b)$ the perimeter $P$ of $T$ is a multiple of four; and $c)$ all three of $a$, $b$, and $c$ are even.

A set of small plastic figures used as a visual aid in primary-school mathematics consists of $48$ pieces. The figures can be classified according to four different attributes: their size may be large or small, they may have or not have a hole in the middle, their colour may be red, yellow, blue or green, and their shape may be circular, square or triangular. Every possible combination of the four properties (e.g. small, blue circular disc with a hole) occurs exactly once. How many pieces $x$ are there in the set, such that there exists a piece $y$ for which the two conditions below both hold? 1. If $x$ is red or has no hole, then $y$ is a small yellow square. 2. If $y$ is small or blue, then $x$ is a green triangle or some figure without a hole.

One of $4999$ distinct given integers is $42$. Prove that it is possible to select some numbers such that their sum is divisible by $5000$.

The base of an isosceles triangle is $a$, its apex angle is smaller than $120^{\circ}$, and the altitude drawn to the base is $m$. Each vertex of the triangle is reflected in the line of the opposite side. The three reflections form a new isosceles triangle, with a base $a'$ and an altitude $m'$ drawn to the base. Show that $\frac{a'}{a}+\frac{m'}{m}=4.$

The graph of the cubic polynomial $x^3 +a x^2 + bx + c$ and the circle of radius $10$ centred at the origin intersect at six distinct points $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, $P_6$. Express the centre of mass of the system $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, $P_6$ in terms of the coefficients $a$, $b$, $c$.

Is it true that if five out of the midpoints of the six edges of a tetrahedron lie on the same sphere, then the sixth midpoint also lies on that sphere?

In a set of $2n$ consecutive integers, what is the maximum possible number of elements that are divisible by at least one of the numbers $n+1$, $n+2$, $\ldots$, $2n$?

The feet of the altitudes of an acute-angled triangle $ABC$ drawn to the sides $BC$, $CA$, $AB$ are $D$, $E$, $F$, respectively. The orthocentre of triangle $ABC$ is $M$. Let $k_1$ denote the circle of diameter $AB$, and let $k_2$ be the circumscribed circle of triangle $DEM$. Let $P$ be an interior point of the arc $EM$ of $k_2$ that does not contain point $D$. Let line $DP$ intersect circle $k_1$ again at point $Q$, and let the midpoint of line segment $PQ$ be $R$. Show that lines $AQ$, $MP$, $FR$ are concurrent.

The bases of a trapezium $ABCD$ are $AB$ and $CD$, the intersection of the diagonals is $M$. Diagonal $AC$ bisects the angle $BAD$, $AM=BC$ and $BM=CD$. Find the angles of the trapezium.

We have $3n+1$ coins, $n$ of which have 0 on one side and $11$ on the other. Another $n$ coins have 0 on one side and $44$ on the other, and the remaining $n+1$ coins have 0 on one side and $99$ on the other. All the coins are tossed at the same time. What is the probability that the sum of the resulting numbers is divisible by 7? (The coins are fair coins.)

A circle $k_A$ is centred at $A$, and a circle $k_B$ is centred at $B$. Line $l_1$ touches $k_A$ at $A_1$ and $k_B$ at $B_1$. Line $l_2$ touches $k_A$ at $A_2$ and $k_B$ at $B_2$. Prove that the orthogonal projections of the line segments $A_1A_2$ and $B_1B_2$ onto the line $AB$ are of equal length.

Given that $x^2+y^2+z^2=3$, where $x$, $y$, $z$ are positive numbers, prove that $2^{\tfrac1x} + 2^{\tfrac1y} + 2^{\tfrac1z} \ge 6$.

We are given 6 points in the plane such that all pairwise distances are different. Show that there exist two triangles with the following property: each vertex is among the 6 given points, and the two triangles share a side in common such that this side is the shortest one in one triangle and the longest one in the other.

Let $f(x)$ be a polynomial with integer coefficients. Let $f^{(n)}$ denote the $n$-fold application of $f$: $f^{(n)}(x)=\underbrace{f\Big(f\big(\ldots f}_{n}(x)\ldots\big)\!\Big).$ Let $k(f)$ denote the smallest positive integer $k$ for which $f^{(k)}(x)\equiv x\pmod{13}$ holds for all integers $x$, provided that there exists such a $k$, and let $k(f)=0$ otherwise. Prove that there is a maximum value of $k(f)$, and determine this maximum value.

The excircle of triangle $ABC$ opposite to vertex $A$ touches line $AC$ at point $B_1$. The line segment $BB_1$ intersects the excircle at $B_2$, and the tangent drawn to the excircle at $B_2$ intersects side $BC$ at $B_3$. Similarly, the inscribed circle of the triangle touches side $AB$ at point $C_1$, line segment $CC_1$ intersects the incircle at $C_2$, and the tangent drawn to the incircle at $C_2$ intersects side $BC$ at $C_3$. Prove that $B_2B_3=C_2C_3$.

After the elections in Nowhereland, there are $50 < n < 100$ representatives in the parliament, all from a single party called the Blue Party. (The Blue Party has a single president.) According to the law, a party in the parliament may be divided into two parties as long as the following conditions are met:

• The president of the old party is not allowed to become a member of the newly formed parties. His or her parliament mandate will terminate, thereby reducing the total number of representatives.
• Every other member may decide which new party to join.
• Each of the new parties must have at least one member among the representatives.
• Each of the new parties must elect a president from their representatives. If at least one such splitting of a party results in all parties in the parliament having the same number of members, the parliament will be dissolved. What should be the value of $n$ so that this could never happen?

The second intersections of three concurrent unit circles are $A$, $B$ and $C$. What is the radius of the circle $ABC$?

In a convex quadrilateral $ABCD$, point $E_1$ lies on side $AD$, point $F_1$ lies on side $BC$, $E_2$ lies on diagonal $AC$, and $F_2$ lies on diagonal $BD$. Given that $AE_1:E_1D=BF_1:F_1C=AE_2:E_2C=BF_2:F_2D=AB:CD.$ and no pair of points coincide, prove that the lines $E_1F_1$ and $E_2F_2$ are perpendicular.

Is there a function $f\colon \mathbb{R}\to \mathbb{R}$ with the following properties: (1) if $x_1\ne x_2$ then $f(x_1)\ne f(x_2)$, (2) there exist appropriate constants $a,b>0$ such that $f(x^2)- \big(f(ax+b)\big)^2\ge \frac14.$ for all $x\in\mathbb{R}$?

The sultan imprisoned all the 1024 mathematicians of his empire. They were not allowed to keep any of their possessions except for a single copper coin each. The mathematicians know that there are 1024 of them, but they are not able to communicate with one another in any way. On his birthday the sultan offered them the following game: they are taken to the prison yard one by one. Each of them may say either 0 or 1 when taken there. If the sum of the numbers they say is 1, then he will let them all go free. (The mathematicians cannot signal to each other, they do not know how many others have been to the yard before them, or what those before them have done in the yard.) What are the chances that they can get out of the prison?

The quadrilateral $ABCD$ is cyclic. Given that $AB+BC=AD+DC$ and $BA+AC=BD+DC$, show that $ABCD$ is a rectangle.

A parabola is reflected in a line that passes through its focus and encloses an angle $\alpha$ with its axis. Show that the parabola and its reflection intersect at an angle of $\alpha$.

A positive integer $n$ is not divisible by 3. The sum of its positive divisors that leave a remainder of 1 when divided by 3 is $A(n)$, and the sum of its positive divisors that leave a remainder of 2 when divided by 3 is $B(n)$. Find those numbers $n$ for which $\big|A(n)-B(n)\big|<\sqrt{n}\,$.

Given some unit circles on the plane, we coloured each centre blue. On the circumferences of the circles, we marked some points red such that there should be exactly 2 red points on the circumference of each circle. What is the maximum possible number of blue points if there are 25 coloured points altogether?

In a triangle $ABC$, $\angle ACB=90^{\circ}$ and $AC>BC$. Let $X$ be the midpoint of the arc $AB$ of the circumscribed circle that does not contain $C$. The perpendicular drawn to $CX$ at $X$ intersects line $CA$ at $P$. Show that $AP=BC$.

Let $p$ denote an odd prime. If each of the numbers $\binom{p-2}{0}, \binom{p-2}{1}, \ldots, \binom{p-2}{p-2}$ are divided by $p$, how many different remainders are obtained?

The inscribed circle of triangle $ABC$ is centred at $I$, and touches sides $BC$, $CA$ and $AB$ at points $D$, $E$ and $F$, respectively. Let $M$ be an arbitrary point in the interior of side $BC$, different from $D$. Let the lines $DI$ and $EF$ intersect at $T$, and let $K$ denote the midpoint of line segment $MT$. Prove that the circles $DEF$, $TDM$ and $KIT$ are concurrent.

Let $P$ be an arbitrary point of a given ellipse, different from the endpoints of the major axis. $P$ is connected to the foci $F_1$ and $F_2$. The angle bisector of angle $\angle F_1PF_2$ intersects $F_1F_2$ at $E$. The circle which passes through $P$ and touches $F_1F_2$ at $E$ intersects $PF_1$ at $G$ and $PF_2$ at $H$. Show that the length of $GH$ does not depend on the choice of $P$.

Arthur Dumpling (Hungarian cartoon character: a fat bird who loves chocolate of all kinds) lives at 1 Sweet Street. The chocolate shop is operating at number $n$, the far end of the street. Arthur's daily fitness programme is as follows: he starts in front of number 2. When he stands in front of number $k$ (where $1<k<n$), he tosses a fair chocolate coin. If it shows heads, he moves to number $(k-1)$. If it shows tails, he moves to number $(k+1)$. If he reaches the chocolate shop, he enters and throws a chocolate ball down his throat, and then moves to number $(n-1)$. If he arrives back home, the fitness programme terminates. On average, how many chocolate balls does Arthur throw down his throat per day?

Let us define a function $f$ as follows. For any acute-angled triangle $XYZ$, if $P$ is a point on $YZ$, then $f(P;XYZ)$ is defined as the line joining the feet of perpendiculars from $P$ to lines $XY$; $XZ$. Let $ABC$ be a triangle with orthocenter $H$. Let $A'B'C'$ be the orthic triangle of $ABC$. Let $A''\equiv f(B';HCA) \cap f(C';HAB)$. Similarly, points $B''$; $C''$ are defined. Show that the lines $AA''$; $BB''$; $CC''$ are concurrent.

Assume that a certain football team have played 1000 games altogether, and scored 1000 points altogether since the team was founded. (A team score 3 points for every game they win, 1 point for a draw and no points for games they lose.) Prove that there are at most ${(2.9)}^{1000}$ possible sequences of the 1000 scores.

Prove that every integer greater than $1$ can be represented as a sum of numbers of the form $2^p\cdot 3^q$ greater than $1$ such that no term of the sum is a divisor of another term. (For example, $23=9+8+6$, $11=9+2$ or $12=12$.)

Let $F$ be a point on the extension of side $AD$ of parallelogram $ABCD$ beyond vertex $D$. Line segment $BF$ intersects side $CD$ at $G$ and diagonal $AC$ at $E$. Show that $\frac{1}{BE}=\frac{1}{BG}+\frac{1}{BF}$.

In the interior of an isosceles triangle, what is the locus of points for which the distance from the base is the geometric mean of the distances from the legs?

Prove that if none of the angles $\alpha$, $\beta$, $\gamma$, $\delta$ of a convex quadrilateral are right angles then $\tan \alpha+\tan \beta+\tan \gamma+\tan \delta=\tan \alpha\cdot\tan \beta\cdot\tan \gamma\cdot \tan \delta(\cot \alpha+\cot\beta+\cot\gamma+\cot\delta)$.

The edges of a complete graph on $n\ge 8$ vertices are coloured in two colours. Prove that the number of cycles formed by four edges of the same colour is more than $\frac{{(n-5)}^4}{64}$.

From a point $M$, two tangents are drawn to a right-angled hyperbola of centre $O$. One tangent intersects an asymptote at point $P$, and the other tangent intersects the other asymptote at point $Q$. Prove that line $OM$ bisects the line segment $PQ$.

A polyhedron $P$ is divided into the smaller polyhedra $P_1, \ldots, P_k$ and also divided into the smaller polyhedra $Q_1, \ldots, Q_k$ such that the polyhedra $P_i$ and $Q_i$ are congruent for all $i=1, \ldots, k$. Show that it is possible to select some points in the interior of $P$ such that there are the same number of points (at least one) in the interior of polyhedra $P_i$ and $Q_i$ for all $i=1, \ldots, k$. (The position of each polyhedron is fixed in the space.)

Let $P$ be a point in the interior of a regular octagon $ABCDEFGH$. Show that the sum of the areas of triangles $ABP$, $CDP$, $EFP$ and $GHP$ equals the sum of the areas of triangles $BCP$, $DEP$, $FGP$ and $HAP$.

Every entry in a $2019\times 2019$ table is either $(+1)$ or $(-1)$. If the sum of each row and the sum of each column are calculated, how many different numbers may be obtained at most?

In a square $ABCD$, let $F$ be an interior point of side $AB$, and let $E$ be an interior point of side $AD$. Draw a perpendicular to line $CE$ at point $E$, and a perpendicular to line $CF$ at point $F$. Denote the intersection of the two perpendiculars by $M$. Given that the area of triangle $CEF$ is half the area of pentagon $BCDEF$, prove that point $M$ lies on diagonal $AC$ of the square.

An $n \times n$ table of real numbers in each field is called a zero square if the sum of the numbers in every $2 \times 2$ square part of it (therefore in the whole table, too) is zero. (The diagram shows a $3\times3$ example.) 2 -3 4 -4 5 -6 1 -2 3 What is the largest possible $n$ for which there exists an $n \times n$ zero square such that the entries are not all zeros?

The convex quadrilateral $ABCD$ is not a trapezium, and diagonals $AC$ and $BD$ are equal in length. Let $M$ denote the intersection of the diagonals. Show that the other intersection (different from $M$) of the circles $ABM$ and $CDM$ lies on the angle bisector of angle $BMC$.

Prove that the set $\{1,2,3,4,5,6,7,8,9,10,11,12,13\}$ has an odd number of nonempty subsets in which the arithmetic mean of the elements is an integer.

Let $D$ be a point in the interior of side $AB$ in triangle $ABC$, and $E$ be a point in the interior of side $AC$. The intersection of line segments $BE$ and $CD$ is $M$. Let $x$ denote the area of triangle $BCM$, and let $y$ denote the area of triangle $EDM$. Prove that $T_{ABC}\ge x \frac{\sqrt x+\sqrt y}{\sqrt x-\sqrt y}$.

For which positive integers $n$ is there an appropriate order $a_1,a_2,\dots,a_n$ of the first $n$ positive integers such that the numbers $a_1+1,a_2+2,\dots,a_n+n$ are all perfect powers? (A number is called a perfect power if it can be represented in the form $a^b$, where $a, b\ge 2$ are integers.)

Let $n\ge3$, and consider the graph in which the vertices are the grid points $(i,j)$, where $1\le i,j\le n$, and the distinct points $(i,j)$ and $(k,l)$ are connected by an edge if and only if $i^2+j^2+k^2+l^2$ is divisible by $3$. For what values of $n$ is it possible to walk the edges of the graph by traversing each edge exactly once?

In a right-angled triangle $ABC$, point $D$ lies in the interior of leg $AC$, and point $E$ lies on the extension of hypotenuse $AB$ beyond $B$. The second intersection of circles $ADE$ and $BCE$ (different from $E$) is $F$. Show that $\angle CFD=90^\circ$.

The base of a pyramid is a convex polygon, and the areas of the lateral faces are equal. Select an arbitrary point on the base, and consider the sum of the distances of this point from the lateral faces. Prove that the sum is independent of the choice of the point on the base.

Prove that there are infinitely many pairs of positive integers $(a,b)$ for which $2019 < \frac{2^a}{3^b} < 2020.$

Solve the equation $\cos(3x)+\cos^2x=0.$

The sides of quadrilateral $ABCD$ are $AB=8$, $BC=5$, $CD=17$ and $DA=10$. The intersection of diagonals $AC$ and $BD$ is $E$, and $BE:ED=1:2$. What is the area of the quadrilateral?

Two players, First and Second take turns in writing a number 0 or 1 in the fields of a $19\times 19$ table, initially all blank. When all fields are filled in, they calculate the sum of each row, and the sum of each column. Let the largest row sum be $A$, and let the largest column sum be $B$. If $A>B$ then First wins the game. Second wins if $A<B$, and it is a draw if $A=B$. Does either of the players have a winning strategy?

Let $G$ denote the inscribed sphere of a tetrahedron $ABCD$, and let $G_A$ be the escribed sphere touching the face $BCD$. Let $T$ be the tetrahedron formed by the points of tangency of $G$ on the planes of the faces, and let $T_A$ be the tetrahedron formed by the points of tangency of $G_A$ on the planes of the faces. Show that $\frac{V^3(T)}{V^3(T_A)} =\frac{V^2(G)}{V^2(G_A)}$ holds for the volumes of the tetrahedra and the spheres.

Are there positive integers $n$ and $k$ such that $20^k+19^k=2019^n-10^n?$

Given a circle $k$ in the plane, determine the locus of the orthocentres of all triangles inscribed in $k$.

Consider the quadratic function $f(x)=x^2+bx+c$ defined on the set of real numbers. Given that the zeros of $f$ are some distinct prime numbers $p$ and $q$, and $f(p-q)=6pq$, determine the primes $p$ and $q$, and determine the function $f$.

Let $D$ and $E$ be points on leg $BC$ of a right-angled triangle with hypotenuse $AB$ such that $\angle DAC = \angle EAD= \angle BAE$. The feet of the perpendiculars dropped from vertex $C$ onto the line segment $AD$ and from point $D$ onto the hypotenuse $AB$ are $F$ and $K$, respectively. Line $CK$ intersects line segment $AE$ at point $H$, and the parallel line drawn from point $H$ to the line $AD$ intersects line segment $BC$ at point $M$. Show that point $F$ is the circumcentre of triangle $CHM$.

Let $P$ be an arbitrary point in the interior of triangle $ABC$. Lines $AP$, $BP$ and $CP$ intersect sides $BC$, $AC$, and $AB$ at points $A_1$, $B_1$ and $C_1$, respectively. Prove that $\frac{AP}{A_1P}\cdot\frac{BP}{B_1P}\cdot\frac{CP}{C_1P}\ge 8.$

For a positive integer $c$, define the sequence $\{ a_n\}$ by the following recurrence relation: $a_0=c$ and $a_{n+1} = \big[ a_{n} + \sqrt{a_{n}}\,\big]$ for $n \ge 0$. Prove that if $2019$ occurs as a term of the sequence, then no preceding term is a perfect square, but there are infinitely many perfect squares among the following terms.

In the plane $\Sigma$, given a circle $k$ and a point $P$ in its interior, not coinciding with the center of $k$. Call a point $O$ of space, not lying on $\Sigma$, a proper projection center if there exists a plane $\Sigma'$, not passing through $O$, such that, by projecting the points of $\Sigma$ from $O$ to $\Sigma'$, the projection of $k$ is also a circle, and its center is the projection of $P$. Show that the proper projection centers lie on a circle.

A function $f\colon \mathbb R \to \mathbb R$ is called area preserving if the area of the triangle formed by the points $\big(a,f(a)\big)$, $\big(b,f(b)\big)$ and $\big(c,f(c)\big)$ is equal to the area of the triangle formed by points $\big(a+x,f(a+x)\big)$, $\big(b+x,f(b+x)\big)$ and $\big(c+x,f(c+x)\big)$ for all $a<b<c$ and $x$. Which continuous functions $f$ are area preserving?

Solve the following simultaneous equations: $$\begin{align*} x[x]+y[y] & =1,\\ [x]+[y] & =1. \end{align*}$$

In a triangle $ABC$, $BC<AC$ and $\angle ACB$ is a right angle. The tangents drawn from point $A$ to the circle of diameter $BC$ touch it at $C$ and $D$. The line of tangent $AD$ intersects line $BC$ at point $E$. The midpoint of line segment $BC$ is $O$. Prove that the area of triangle $DEO$ equals the area of triangle $AEB$.

The centre of the circumscribed circle of an acute angled triangle $ABC$ is $O$, and the reflections of $O$ in sides $BC$, $CA$ and $AB$ are $O_A$, $O_B$, and $O_C$, respectively. Show that the lines $AO_A$, $BO_B$ and $CO_C$ are concurrent.

Thirty students were taking an exam in a subject called Tautologics''. The students were sitting down in a classroom, and the teacher asked them a single question, How many of the students sitting in this room are going to fail this exam altogether?'' The students had to name a number, one by one, in a row. After each answer, the teacher immediately announced the result which was either pass'' or fail''. When the exam was over, the Student Union requested an inspection by the School District. If it turns out that any student gave the correct answer but still failed the exam, all the results will be cancelled, and everyone will receive a ``pass'' grade. Is there a strategy for the students to achieve that everyone should pass the exam?

The midpoint of side $AB$ of an acute angled triangle $ABC$ is $F$, and the line $e$ through the point $F$ halves the perimeter of triangle $ABC$. Line $e$ intersects the lines of sides $BC$ and $CA$ at points $D$ and $E$, respectively. Show that the perpendiculars drawn to $AB$ at $F$, to $BC$ at $D$, and to $CA$ at $E$ are concurrent.

Let $p$ be an at most 1998th-degree polynomial such that the values $p(1), p(2),\dots,p(2000)$ form a permutation of the numbers $1,2,\dots,2000$. Does it follow that $p(1)$ and $p(2000)$ are $1$ and $2000$ in some order?

In the quadrilateral $ABCD$, $AB=AD$ and $BC=DC$. Let $M$ be the intersection of the perpendiculars drawn to side $AB$ at $B$, and side $CD$ at $D$. Show that $\angle AMD=\angle BMC$.