KöMaL 2017

In the triangle $ABC$, the incircle is denoted by $\omega$. The excircle opposite to $A$ is tangent to the line $BC$ at point $A_1$. Let $X$ be an arbitrarily chosen point on the extension of the segment $AA_1$ beyond $A_1$, and let the line $BC$ meet the tangents drawn from $X$ to $\omega$ at $Y$ and $Z$, such that $BY<BZ$. Show that $XY-XZ$ does not depend on the position of the point $X$.

Let $f(x)$ and $g(x)$ be nonzero polynomials such that the degree of $f(x)$ is higher than that of $g(x)$. Suppose that for infinitely many prime numbers $p$, the polynomial $pf(x)+g(x)$ also has a rational root. Show that $f(x)$ has a rational root.

Prove that among any ${\color{red}4097}$ distinct $0$-$1$ sequences of length $24$, there are two which differ from each other at no more than $7$ positions.

Let $f_1,f_2,\ldots$ be an infinite sequence of continuous $\mathbb{R}\to\mathbb{R}$ functions such that for arbitrary positive integer $k$ and arbitrary real numbers $r>0$ and $c$ there exists a number $x\in(-r,r)$ with $f_k(x)\ne cx$. Show that there exists a sequence $a_1,a_2,\ldots$ of real numbers such that $\sum_{n=1}^\infty a_n$ is convergent, but $\sum_{n=1}^\infty f_k(a_n)$ is divergent for every positive integer $k$.

In a convex quadrilateral $ABCD$, the perpendicular drawn from $A$ to line $BC$ meets the lines $BC$ and $BD$ at $P$ and $U$, respectively. The perpendicular drawn from $A$ to line $CD$ meets the lines $CD$ and $BD$ at $Q$ and $V$, respectively. The midpoints of the segments $BU$ and $DV$ are $S$ and $R$, respectively. The lines $PS$ and $QR$ meet at $E$. The second intersection point of the circles $PQE$ and $RSE$, other than $E$, is $M$. The points $A$, $B$, $C$, $D$, $E$, $M$, $P$, $Q$, $R$, $S$, $U$, $V$ are distinct. Show that the center of the circle $BCD$, the center of the circle $AUV$ and the point $M$ are collinear.

Let $c\ge3$ be an integer, and define the sequence $a_1,a_2,\dots$ by the recurrence $a_1=c^2-1$, $a_{n+1}=a_n^3-3a_n^2+3$ $(n=1,2,\ldots)$. Show that for every integer $n\ge2$, the number $a_n$ has a prime divisor that does not divide any of $a_1,\ldots,a_{n-1}$.

Do there exist bijective functions $f,g\colon \mathbb{Q}\to\mathbb{Q}$ such that the function $f\big(g(x)\big)$ is strictly increasing, but the function $g\big(f(x)\big)$ is strictly decreasing?

Let $A$ and $B$ be two vertices of a convex polygon $\mathcal{P}$ with maximum distance from each other. Let the perpendicular bisector of the segment $AB$ meet the boundary of $\mathcal{P}$ at points $C$ and $D$. Show that the perimeter of $\mathcal{P}$ is less than $2(AB+CD)$.

Prove that the inequality $\frac1{\sqrt{2x}} + \frac1{\sqrt{2y}} + \frac2{\sqrt{x+y}} + 2 \ge \frac4{\sqrt{x+2}} + \frac4{\sqrt{y+2}}$ holds for all pairs $(x,y)$ of positive real numbers.

We are given $2k$ lines, $e_1, \ldots, e_{2k}$ in a plane $S$, further given a line $g$ which has an angle $\alpha$ with $S$. Show that the sum of the pairwise angles between the lines $e_1, \ldots,e_{2k}, g$ is at most $(k^2+k)\cdot \dfrac \pi 2 + k\alpha.$

Let $k\ge2$ be an integer. Determine all those polynomials $p(x)$ with real coefficients for which $p(x) \cdot p(2x^k-1) = p(x^k) \cdot p(2x-1).$

For all primes $p\ge3$, let $S(p) = \sum_{k=1}^{\frac{p-1}2} \tan \frac{k^2\pi}{p}.$ $(a)$ Show that $p\equiv1\pmod4$ implies $S(p)=0$. $(b)$ Show that if $p\equiv3\pmod4$, then $\dfrac{S(p)}{\sqrt{p}}$ is an odd integer.

Let $m$ and $n$ be positive integers, and let $H$ denote a subset of the set $\{1,2,\ldots,m\}\times\{1,2,\ldots,n\}$. Show that if $|H|>m+(m+n)\log_2n$, then there exist integers $1\le u<v\le m$ and $1\le x<y<z\le n$ such that the pairs $(u,x)$, $(u,y)$, $(v,x)$ and $(v,z)$ are elements of $H$.

A circle $\omega$ lies in a circle $\Omega$ such that their common center is the point $O$. Fix a point $A\ne O$ inside $\omega$. Let $X$ denote an arbitrary point on the circumference of $\Omega$, and let $Y$ denote the second intersection point of $\Omega$ and the line $AX$. Let $Z$ denote the intersection of $\omega$ and the line segment $AX$. Let $M$ denote the point on the line segment $AZ$ for which $MX\cdot MZ\cdot AY = MA\cdot MY\cdot XZ$. Let $x$ and $y$ denote the tangents of the circle $\Omega$ at the points $X$ and $Y$, respectively. Let $t$ denote the line that passes through $M$ and either also passes through the intersection of $x$ and $y$, or is parallel to both $x$ and $y$. Finally, let $T$ denote the intersection of $t$ and the line $OZ$. Show that the locus of the points $T$, as $X$ is varied, is an ellipse, and the line $t$ is a tangent of this ellipse.

A positive integer $n$ satisfies the following: it is possible to select some integers such that if we randomly choose two different integers from this list, say, $i$ and $j$, then $i+j$ $\mathrm{mod\ } n$ is equal to one of the numbers $0,1,\dots,n-1$ with equal probability. Find all numbers $n$ with this property.

An airline operates flights between any two capital cities in the European Union. Each flight has a fixed price which is the same in both directions. Furthermore, the flight prices from any given city are pairwise distinct. Anna and Bella wish to visit each city exactly once, not necessarily starting from the same city. While Anna always takes the cheapest flight from her current city to some city she hasn't visited yet, Bella always continues her tour with the most expensive flight available. Is it true that Bella's tour will surely cost at least as much as Anna's tour?

Fix a triangle $ABC$. We say that triangle $XYZ$ is elegant if $X$ lies on segment $BC$, $Y$ lies on segment $CA$, $Z$ lies on segment $AB$, and $XYZ$ is similar to $ABC$ (i.e., $\angle A=\angle X$, $\angle B=\angle Y$, $\angle C=\angle Z$). Of all the elegant triangles, which one has the smallest perimeter?

Let $n\ge 2$ be an integer. We call an ordered $n$-tuple of integers primitive if the greatest common divisor of its components is $1$. Prove that for every finite set $H$ of primitive $n$-tuples, there exists a non-constant homogenous polynomial $f(x_1,x_2,\dots,x_n)$ with integer coefficients whose value is $1$ at every $n$-tuple in $H$.

A regular triangle has side length $n$. We divided its sides into $n$ equal parts and drew a line segment parallel with each side through the dividing points. A lattice of $1+2+\cdots+(n+1)$ intersection points is thus formed. For which positive integers $n$ can this lattice be partitioned into triplets of points which are the vertices of a regular triangle of side length $1$?

Triangle $ABC$ has orthocenter $H$. Let $D$ be a point distinct from the vertices on the circumcircle of $ABC$. Suppose that circle $BHD$ meets $AB$ at $P\ne B$, and circle $CHD$ meets $AC$ at $Q\ne C$. Prove that as $D$ moves on the circumcircle, the reflection of $D$ across line $PQ$ also moves on a fixed circle.

Let $\mathbb{Z}^+$ denote the set of positive integers. Find all functions $f\colon \mathbb{Z}^+\to\mathbb{Z}^+$ which satisfy the following: $\bullet$ $f(mn)=f(m)f(n)$ for all $m,n\in\mathbb{Z}^+$, and $\bullet$ $f^{(n)}(n)=n$ for all $n\in\mathbb{Z}^+$ (in other words, $f\Big(f\big(\dots \big(f(n)\big)\dots\big)\!\Big)=n$, where there are $n$ pairs of brackets on the left-hand side).

$100$ betyárs stand on the Hortobágy plains. Every betyár's field of vision is a $100$ degree angle. After each of them announces the number of other betyárs they see, we compute the sum of these $100$ numbers. What is the largest value this sum can attain?

Let $S$ be a finite set of rational numbers. For each positive integer $k$, let $b_k=0$ if we can select $k$ (not necessarily distinct) numbers in $S$ whose sum is $0$, and $b_k=1$ otherwise. Prove that the binary number $0.b_1b_2b_3\dots$ is a rational number. Would this statement remain true if we allowed $S$ to be infinite?

Let $a>0$ be a real number. Find the minimal constant $C_a$ for which the inequality $C_a\sum_{k=1}^n \frac1{x_k-x_{k-1}} >\sum_{k=1}^n \frac{k+a}{x_k}$ holds for any positive integer $n$ and any sequence $0=x_0<x_1<\cdots<x_n$ of real numbers.

For which $n$ can we partition a regular $n$-gon into finitely many triangles such that no two triangles share a side?

For which pairs $(m,n)$ does there exist an injective function $f\colon \mathbb{R}^2\to\mathbb{R}^2$ under which the image of every regular $m$-gon is a regular $n$-gon. (Note that $m,n\ge 3$, and that by a regular $N$-gon we mean the union of the boundary segments, not the closed polygonal region.)

We say that a strictly increasing positive real sequence $a_1,a_2,\dots$ is an elf sequence if for any $c>0$ we can find an $N$ such that $a_n<cn$ for $n=N,N+1,\dots\,.$ Furthermore, we say that $a_n$ is a hat if $a_{n-i}+a_{n+i}<2a_n$ for $1\le i\le n-1$. Is it true that every elf sequence has infinitely many hats?

Show that every integer can be represented in the form $x^{2}+y^{2}-z^{2}$ with appropriate positive integers $x$, $y$, $z$.

The circle $k$ centred at $O$ intersects line $e$ at points $A$ and $B$, and it intersects the perpendicular bisector of line segment $OB$ at points $C$ and $D$. Prove that the angle bisector of angle $COA \sphericalangle$ and line $e$ enclose an angle of 60 degrees.

Let $p$ denote an arbitrary (positive) prime number. Find all positive integers $b$ such that the roots of the quadratic $x^2-bx+bp = 0$ are integers.

The escribed circles drawn to sides $AC$ and $BC$ of a triangle $ABC$ touch the sides at points $K$ and $L$, respectively. Prove that the line passing through the midpoints of line segments $KL$ and $AB$ is parallel to the angle bisector of $ACB\sphericalangle$, and it halves the perimeter of the triangle.

Find all ordered triples $(p,q,n)$ (where $n\ge 2$ is a positive integer) such that there exist exactly $n$ real numbers, not necessarily different, whose product is $p$ and whose sum is $q$.

Prove that each term of the sequence $a_1 =1,$ $a_n =\frac{4a_{n-1}+\sqrt{7a^2_{n-1}-3}}{3}; n\ge2$ is rational.

The intersection of five closed half planes is bounded. Show that it is possible to select four out of them such that their intersection is also bounded.

Let $f$ be a positive bounded function defined on the interval $[0;1]$. Prove that there are numbers $x_1$ and $x_2$ for which $\frac{(x_2-x_1)f^2(x_1)}{f(x_2)} >\frac{f(0)}4.$

Find all convex polyhedra $P$ whose interior contains a point $O$ such that every plane passing through $O$ intersects $P$ in a parallelogram centred at $O$.

The centre of the inscribed circle of $ABC\triangle$ is $K$, and the centre of the excircle drawn to side $AB$ is $L$. Prove that the line segment $KL$ and the arc $AB$ of the circumscribed circle not containing $C$ bisect each other.

Solve the following simultaneous equations in the set of real numbers: $\sqrt{x_1}+\sqrt{x_2}+\ldots+\sqrt{x_{2016}} =\sqrt{2017}\,,$ $x_1+x_2+\ldots+x_{2016} =2017.$

Prove that if all three roots of the equation $x^3-px^2+qx-r=0$ are positive then the sum of the reciprocals of the roots is at most $\frac{p^2}{3r}$.

Let $K$ denote the centre of the incircle of a circumscribed quadrilateral $ABCD$. $M$ is a point on the line segment $AK$, and $N$ is a point on the line segment $CK$, such that $2 MBN\sphericalangle= ABC\sphericalangle$. Prove that $2 MDN\sphericalangle=ADC\sphericalangle$.

Let $a_1,a_2,\dots,a_n$ be real numbers. Consider the $2^n-1$ (nonempty) sums composed out of these numbers. How many of these may be positive?

A table is filled in with digits of 0 and 1 such that there are no two identical rows, but there are two identical rows in any $4\times 2$ sub-table formed by two columns and four rows. Show that there exists a column in which one kind of digit occurs exactly once.

Point $P$ is said to be a centre of convexity of a point set $\mathcal{H}$ if $\mathcal{H}\cup H_P'$ is convex, where $H_P'$ is the reflection of the point set $\mathcal{H}$ in the point $P$. Show that $a)$ every convex quadrilateral of the plane has three distinct non-collinear centres of convexity; $b)$ a tetrahedron has no centre of convexity.

Determine all pairs $(n,k)$ of positive integers for which $\big(2^{2^n}+1\big)\big(2^{2^k}+1\big)$ is divisible by $nk$.

Points $A$ and $B$ are separated by a unit distance. The unit circles $k_A$ and $k_B$ centred at $A$ and $B$ intersect at points $C$ and $D$, respectively. Let $k_C$ denote the circle centred at $C$ and passing through $D$. $F$ is that intersection of line $AC$ and circle $k_C$ which is farther away from $A$. $G$ is the other intersection of circle $k_A$ and line $DF$. Show that $\angle GAD=90^\circ$.

Rudi chose a positive integer $k$, and observed that $4^k$ and $5^k$ began with the same digit in decimal notation. Prove that this digit may only be a $2$ or a $4$.

Assume that $a<b<c<d$ and $a+d\ne b+c$. Show that the equation $\frac1{a-x}-\frac1{b-x}-\frac{1}{c-x}+\frac1{d-x}=0$ has exactly two distinct roots, such that one of them lies in the interval $(b,c)$, and the other one lies outside the interval $(a,d)$.

Let $r$ and $R$, respectively, denote the radii of the inscribed and circumscribed circles of a right-angled triangle $ABC$. Let $CD$ be the altitude drawn to the hypotenuse $AB$. Draw the square $CEFG$ of side length $CD$ that has vertex $E$ on side $AC$ and vertex $G$ on side $BC$. Let $T$ and $t$, respectively, denote the parts of the area of the square $CEFG$ which lie inside and outside the triangle $ABC$. Prove that $\frac{t}{T}=\frac{r}{2R}.$

$M$, $N$, $P$, $Q$, and $R$, respectively, are the midpoints of the sides $AB$, $BC$, $CD$, $DE$ and $EA$ of a convex pentagon $ABCDE$. Show that if the line segments $AP$, $BQ$, $CR$ and $DM$ are concurrent then the common point also lies on line segment $EN$.

In a triangle $ABC$, $AB=BC$. $D$ is an interior point of the triangle such that $\angle ADC= 2 \angle ABC$. Prove that the distance of point $B$ from the exterior angle bisector of $\angle ADC$ is the arithmetic mean of the line segments $AD$ and $DC$.

A sack contains 100 red balls and 100 blue balls. Balls are drawn at random one by one, without replacement, until all 100 red balls are drawn out. Determine the expected value of the number of balls remaining in the sack.

In an acute-angled triangle, $f$ and $g$ are the line segments that divide the angle enclosed by sides $a$ and $b$ into three equal parts. Prove that $\frac{f+g}{2}< \frac{2}{\frac{1}{a} +\frac{1}{b}}.$

The sum of the real numbers $a$, $b$, $c$ and $d$ is $0$. Let $M=ab+bc+cd$ and $N=ac+ad+bd$. Prove that at least one of the sums $20M+17N$ and $20N+17M$ is non-positive.

In a triangle $ABC$, $AC<AB$ and the median $AF$ divides the angle at $A$ in a $1:2$ ratio. The perpendicular drawn to $AB$ at $B$ intersects line $AF$ at $D$. Show that $AD=2AC$.

Let $A$ be a finite set of real numbers. A partition of the elements of $A$ into at least two subsets is said to be a tiling if the (pairwise disjoint) subsets have at least two elements each, and are obtained from each other by a translation. Prove that $A$ has an even number of tilings.

The diagonals of a convex quadrilateral $ABCD$ intersect at point $E$. Prove that the centres of the Feuerbach circles of the triangles $ABE$, $BCE$, $CDE$ and $DAE$ either form a parallelogram or are collinear.

Prove that the number $a_n=1001001\ldots 1001$ (where $n$ denotes the number of ones) cannot be a prime.

Let $p$, $q$ and $r$ denote three distinct primes, and let $n=pq^2r^3$. Show that $(n,1)+(n,2)+\dots+(n,n) =qr^2(2p-1)(3q-2)(4r-3),$ where $(a,b)$ denotes the greatest common divisor of $a$ and $b$.

In a triangle $ABC$, $AB=1$, $\angle BAC =135^{\circ}$, $\angle ABC =30^{\circ}$. Find the parameter of the parabola with symmetry axis $AB$ that is tangent to the lines $AC$ and $BC$.

Let $\|x\|$ denote the distance of the real number $x$ from the closest integer. Let $a_1,a_2,\ldots,a_n$ be positive integers. Prove that there exists a real number $x$ such that $\|a_ix\|\ge \frac{1}{2n}$ for all $1\le i\le n$.

Line $g$ encloses an angle $\alpha$ with the plane determined by lines $e$ and $f$. Show that $\angle(e,f)+\angle(e,g)+\angle(g,f)\le \pi +\alpha.$

Find the largest positive integer $n$ for which there exists exactly one integer $k$ such that $\frac{10}{11}<\frac{n}{k+n}<\frac{11}{12}?$

Points $A$, $B$, $C$ and $D$, in this order, lie on a straight line. Point $E$ does not lie on the line, and $AEB\sphericalangle=BEC\sphericalangle =CED\sphericalangle = 45^\circ.$ Let $F$ and $G$ be the midpoints of $AC$ and $BD$, respectively. What is the measure of angle $FEG$?

What is the maximum possible value of the sum $PA+PB+PC+PD$ if $P$ is a point of the unit square $ABCD$?

$a)$ Is it true that for any irrational number $a$ there exists an irrational number $x$ such that $a+x$ is rational and $ax$ is irrational? $b)$ Is it true that for any irrational number $a$ there exists an irrational number $y$ such that $a+y$ is irrational and $ay$ is rational?

In the sequence $a_1$, $a_2$, $a_3$, ... of positive integers, $a_n\cdot a_{n+1} = a_{n+2}\cdot a_{n+3}$ for all positive integers $n$. Show that the sequence is eventually periodic.

Let $d(n)$ denote the number of (positive) factors of the number $n$. Prove that $n+d(1)+d(2)+\cdots +d(n)\le d(n+1)+d(n+2)+\dots +d(2n).$

What is the maximum number of faces of a convex polyhedron if it is possible to select three faces such that every edge of the polyhedron lies on a selected face?

Define the sequence $a_1, a_2, \dots$ with the following recurrence relation: $a_1=4, a_2=2 \text{and} a_{n+1}=\frac{na_n^2}{na_n^2-(n+1)a_n+n+1}, \text{ if } n\ge 2.$ Prove that $a_1+2\cdot a_2+3\cdot a_3+\cdots +n\cdot a_n=a_1\cdot a_2\cdot a_3\cdot\,\cdots \cdot a_n$ for all $n\ge 1$.

What is the minimum possible area of a triangle that contains a unit square?

Let $k$ and $m$ be two distinct 14-digit positive integers, each containing two of each digit 1, 2, 3, 4, 5, 6 and 7 (like 22133456456717, for example). Prove that $\frac km$ cannot be an integer.

How many different convex polyhedra are determined by the vertices of a cube? (Two polyhedra are considered different if they are not congruent.)

Prove that there are infinitely many number pairs $(a,b)$, such that $a+\frac{1}{b}=b+\frac{1}{a}$, where $a\ne b$. Find the possible values of $ab$.

From his third birthday onwards, Sebastian always gets a birthday cake shaped like a triangular prism, with one candle in each of the three upper vertices, and as many further candles on the top as needed to make the total equal to his age. No three candles are collinear. Sebastian wants to cut the cake into triangular pieces with vertices at the positions of the candles, without other candles in the interior of the triangles. How many pieces can he form on his $k$th birthday?

The trapezium $ABCD$ has an inscribed circle. The circle touches base $AB$ at point $T$, and the parallel base $CD$ at point $U$. Let $M$ denote the intersection of the lines of legs $AD$ and $BC$, and let $V$ be the intersection of side $AB$ with line $MU$. Show that $AT=VB$.

Solve the following equation on the set of positive integers: $x-y-\frac xy-\frac{x^3}{y^3}+\frac{x^4}{y^4} = 2017.$

The circles $S_1$, $S_2$, $S_3$ pairwise touch each other on the outside. Let $A$, $B$ and $C$ denote the common points of the circles $S_1$ and $S_2$, $S_1$ and $S_3$, $S_2$ and $S_3$, respectively. Line $AB$ intersects the circles $S_2$ and $S_3$ again at points $D$ and $E$, respectively. Line $DC$ intersects circle $S_3$ again at $F$. Prove that triangle $DEF$ is right-angled.

Two players, First and Second, play the following game: they place 2017 pebbles on the table. First starts by removing 1 pebble. Then Second may choose to remove either 1 or 2. Then First may remove 1, 2, 3 or 4. Then Second may remove any number from 1 to 8. And so on, the player in the $i$th step needs to remove at least 1 and at most $2^{i-1}$ pebbles. The player removing the last pebble from the table wins the game. Who has a winning strategy?

In a triangle $ABC$, $AB\ne BC$. The angle bisector drawn from point $B$ intersects side $AC$ at point $D$, and intersects the circumscribed circle again at point $E$. The circle of diameter $DE$ intersects the circumscribed circle again at a point $F$, different from $E$. Prove that the reflection of line $BF$ about the line $BD$ results in a median of triangle $ABC$.

Seven thieves have stolen some golden coins, and now each of them takes a share of the loot in the following manner. They proceed in alphabetical order of their names, and everyone takes as many coins as the sum of the digits of the number of coins in the heap not distributed yet. The last coin is removed when two full circles are completed. It turns out that everyone has received the same number of coins, only the chief got more. What was the position of the chief in the alphabetical order?

Prove that if $n-1$ and $n+1$ are both primes and $n> 6$ is an integer then $n^2(n^2+16)$ is divisible by 720.

Let $A_1$, $B_1$, $C_1$, $D_1$ denote the midpoints of the sides of a convex quadrilateral $ABCD$. Let $A_2$, $B_2$, $C_2$, $D_2$ denote the midpoints of the sides of the quadrilateral $A_1B_1C_1D_1$. The procedure is continued. Prove that if quadrilateral $A_1B_1C_1D_1$ is cyclic then quadrilateral $A_{2017}B_{2017}C_{2017}D_{2017}$ is also cyclic.

Given $n$ points in the plane, show that it is possible to select three, denoted by $A$, $B$ and $C$, such that $\angle ABC\le 180^\circ /n$.

Let $A$ be a four-element set of positive integers such that $ab+13$ is a perfect square for all $a,b\in A$, $a\ne b$. Prove that each element of $A$ leaves a remainder of 2 when divided by 4.

The degree of each vertex of a simple planar graph $G$ is 3, and $G$ can be drawn in the plane with its edges represented by non-intersecting unit line segments. Show that $G$ has at least $8$ vertices.

Let $K$ be a convex shape symmetric in the origin, let $e$ be a line through the origin, and let $e'$ be any line parallel to $e$. Let, furthermore $\# H$ denote the number of lattice points in a set $H$. Prove that $\# (K\cap e)+1 \ge \# (K\cap e')$.

An epidemic broke out in Smurf village after a few residents contracted a disease. Luckily, every smurf recovers from the illness in one day, and then they will be immune to the disease for another day. However, from the following day onwards they may catch the disease again. The transition between the sick and healthy states always occurs at night when the smurfs are sleeping. Unfortunately, the smurfs never give up their habit of visiting all their friends every day, not even when they are ill. So when a sick smurf meets a healthy but not immune one, the latter will inevitably catch the disease. Given that Smurf village has 100 inhabitants, show that the epidemic will necessarily terminate by the 101th day following the outbreak.

Let $A_1B_1$, $A_2B_2$, $A_3B_3$ and $A_4B_4$ be four parallel line segments of different lengths given in the plane. For any $1\le i<j\le 4$, let $M_{ij}$ denote the intersection of lines $A_iB_j$ and $A_jB_i$. Show that the lines $M_{12}M_{34}$, $M_{13}M_{24}$ and $M_{14}M_{23}$ are either concurrent or parallel.

Determine all positive integers $a$, $b$, $c$, $d$ such that $abcd-1\mid a+b+c+d$.

A plane figure $S$ has exactly two axes of symmetry. Show that $S$ has central symmetry, too.

Let $a_1 \ge a_2 \ge a_3 \ge \cdots \ge a_{2n-1} \ge a_{2n} \ge 0$, and $\sum_{i=1}^{2n} a_i = 1$. Prove that $a_1 a_2 + 3 a_3 a_4 + 5 a_5 a_6 + \ldots + (2n-1) a_{2n-1} a_{2n} \le \frac{1}{4}.$ When will the equality hold?

The midpoints of sides $BC$ and $CD$ of a convex quadrilateral $ABCD$ are $E$ and $F$, respectively. The line segments $AE$, $EF$ and $AF$ divide the quadrilateral into four triangles whose areas are four consecutive integers. What is the maximum possible area of triangle $ABD$?

$1\times 1$ squares are placed on a rectangle of dimensions $a\times b$, with sides parallel to the sides of the rectangle. Prove that the maximum number of such squares without overlaps is $[a] \cdot [b]$ (where $[x]$ stands for the greatest integer not greater than the number $x$).

Let $C$ denote an arbitrary point on the circumference of a circle of diameter $AB$. Let $T$ be the perpendicular projection of point $C$ onto the line segment $AB$. Draw the circle of centre $C$ that passes through $T$. Let the intersections of the two circles be $P$ and $Q$. Prove that line $PQ$ bisects the line segment $CT$.

Find all functions $f\colon \mathbb{R}\to\mathbb{R}$ such that the equation $x \cdot f(y) - y \cdot f(x) = f\left(\frac{y}{x}\right).$ holds for all $x\ne0$ and $y$.

We have placed chessmen on a $8\times 8$ chessboard. There is an odd number of chessmen standing in every row, and in every column. Prove that the total number of chessmen on the black fields of the chessboard is even.

Prove that the equation $5x^2-4y^2=2017$ has no integer solution.

The diagonal $AC$ bisects the angle lying at vertex $A$ of a cyclic quadrilateral $ABCD$. Let $E$ denote a point on the extension of side $AD$ beyond $D$. Show that $CE=CA$ holds if and only if $DE=AB$.

Let $p(x)$ be a polynomial of integer coefficients such that there exist $4$ distinct integers $a$, $b$, $c$, $d$ with $p(a)=p(b)=p(c)=p(d)=2000$. $a)$ Prove that there is no integer $x_0$ for which $p(x_0)=2017$. $b)$ Find a polynomial $p(x)$ that meets the above requirement, and find an integer $x_1$ for which $p(x_1)=2018$.

Given any points $A_1$, $A_2$, $A_3$, $A_4$, $A_5$ and $P$ in the plane, let $k_i$ denote the number of ways it is possible to select $i$ points out of $A_1$, $A_2$, $A_3$, $A_4$, $A_5$ such that the convex hull of the selected points should contain $P$. Show that $k_3=k_4$.

Let $P(a,b,c)$ be a fixed point in the 3D Cartesian coordinate system. Let $a,b,c>0$ and let $O$ denote the origin. Let $S$ be a plane passing through $P$ that intersects the positive coordinate axes at the points $X$, $Y$ and $Z$. Show that the volume of tetrahedron $OXYZ$ is a minimum if and only if $P$ is the centroid of triangle $XYZ\triangle$.

Determine all functions $f \colon \big(\mathbb{R} \setminus \{0,1\}\big)\to \mathbb{R}$ for which $f\left(\frac{x-1}{x}\right)+f\left(\frac{1}{1-x}\right)=4-\frac2x.$

Given $M$ unit vectors in the space ($M\ge 2$), prove that it is possible to select $M-1$ unit vectors such that the length of their sum is at least unity.

Let $I$ be the centre of the inscribed circle of triangle $ABC$. The inscribed circle touches the sides $BC$, $CA$ and $AB$ at the points $A_1$, $B_1$ and $C_1$, respectively. The lines $AB$ and $A_1B_1$ intersect at point $K$. Show that the circle centred at $K$ and passing through $C_1$, and the inscribed circles of the kites $AC_1IB_1$ and $BA_1IC_1$ have a tangent in common, and that tangent passes through the point $C$.

The number $2017-(2+0+1+7)$ is divisible by the number $(2+20+201)$, that is, 2017 has the following property: if the sum of the digits of the number is subtracted from it, the result will be a four-digit number that is divisible by the sum of the one-digit number formed by the first digit, the two-digit number formed by the first two digits, and the three-digit number formed by the first three digits. How many four-digit numbers have this property?

Prove that there is no triangle whose altitudes have lengths 20, 17 and 9 cm.

How many positive integers are there whose prime factorization contains only the two smallest primes, and whose third power has eight times as many positive divisors as the number has?

The length of one side of a parallelogram $ABCD$ is twice the length of the other side. Determine the ratio of the area of the parallelogram formed by the interior angle bisectors to the area of $ABCD$.

All through a football game, a record of the current score is kept and updated. How many different records are possible if the final result is 3-3? In what percentage of the possible records is there a part of the game when one team was leading by at least 2 goals?

Prove that the function $f\colon \mathbb{R}\to \mathbb{R}, x\mapsto \begin{cases} 2x+2-6k, & \text{if \ } x\in \left[6k; 6k+2\right[ ,\\ \frac{x-2+6k}{2}, & \text{if \ } x\in \left[6k+2; 6k+6\right[ \end{cases} (k\in \mathbb{Z})$ (where $k$ takes every integer value) is one-to-one and equal to its own inverse.

A cube is inscribed in a right circular cone: one face lies on the base of the cone, and the remaining four vertices are on the lateral surface of the cone. What is the surface area of the cube if the base radius of the cone is $\frac{\sqrt{3}}{2}\,$, and its slant height is three times as large?

In a 50-metre running race, if Martin gives a 4-metre advantage to Bill, he will just catch up with him at the finish line. If Bill gives 15 metres of advantage to Henry in a 200-metre race, they will finish side by side. How many metres of advantage may Martin give to Henry in a 1000-metre race so that the two of them finish together? (Assume that each of the three runners maintains the same constant running speed throughout the races.)

Prove that if the interior angles of a convex hexagon are equal then the difference of each pair of opposite sides is the same.

The positive numbers $x$, $y$ satisfy the equation $x^3+y^3=x-y$. Prove that $x^2+y^2<1$.

On each side of a square, either a regular triangle or a square of the same side length is drawn, on the outside. Altogether, there are two triangles and two new squares drawn. What is the radius of the smallest possible circle that completely covers the complex figure obtained in this way?

An $n$-element set has half as many ${(k-1)}$ element subsets as $k$-element subsets, and $\frac 74$ times as many ${(k + 1)}$ element subsets as $k$-element subsets. Determine the number of $k$-element subsets of the set.