KöMaL 2014

Prove that for every simple graph $G$ with $n$ vertices there exist some simple graphs $S_1$, ..., $S_k$ with the following properties: ($a$) every $S_i$ is a complete bipartite graph; ($b$) every edge of $G$ is contained by an odd number of graphs $S_i$; ($c$) every edge of the complement of $G$ is contained by an even number of graphs $S_i$; ($d$) $k\le\frac34n$ .

The circles $k_1$, $k_2$ and $k_3$ are pairwise externally tangent to each other; the point of tangency between $k_1$ and $k_2$ is $T$. One of the external common tangents of $k_1$ and $k_2$ meets $k_3$ at points $P$ and $Q$. Prove that the internal common tangent of $k_1$ and $k_2$ bisects the arc $PQ$ of $k_3$ which is closer to $T$.

Suppose that the convex polygonal disks $P_1$, $P_2$ and $P_3$, lying in the plane, satisfy the property that for arbitrary points $A\in P_1$, $B\in P_2$ and $C\in P_3$, the area of the triangle $ABC$ is no more than a unit. Prove that there are two different ones among the disks $P_1$, $P_2$ and $P_3$ whose area is at most $8$ units in total.

Let $a_1,a_2,\dots,a_n$ and $b_1,b_2,\dots,b_n$ be complex numbers satisfying $\mathop{\rm Im} a_j\ge 1$ and $\mathop{\rm Im} b_j\le -1$ ($j=1,2,\dots,n$ ), and let $f(z) = \frac{(z-a_1)(z-a_2)\dots (z-a_n)}{(z-b_1)(z-b_2)\dots (z-b_n)}$. Prove that the function $f'(z)$ has no root in the set $|\mathop{\rm Im}z|<1$.

There is given a prime number $p$ and two positive integers, $k$ and $n$ . Determine the smallest nonnegative integer $d$ for which there exists a polynomial $f(x_1,\dots,x_n)$ on $n$ variables, with degree $d$ and having integer coefficients that satisfies the following property: for arbitrary $a_1,\dots,a_n\in\{0,1\}$, $p$ divides $f(a_1,\dots,a_n)$ if and only if $p^k$ divides $a_1+\dots+a_n$.

Let $n$ be a positive integer. Determine the smallest possible value of $\big|p(1)\big|^2+\big|p(2)\big|^2+\ldots+\big|p(n+3)\big|^2$ over all a monic polynomials $p$ with degree $n$.

Is there infinitely many positive integers that can be represented neither as the sum nor as the difference of two perfect powers?

In the triangle $A_1A_2A_3$, denote by $k_i$ the excircle opposite to $A_i$, and let $P_i$ be the point of $k_i$ for which the circle $A_{i+1}A_{i+2}P_i$ is tangent to $k_i$. ($i=1,2,3$; the indices are considered modulo $3$.) Show that the line segments $A_1P_1$, $A_2P_2$ and $A_3P_3$ are concurrent.

Basil and Peter play the following game. Basil writes 100 real numbers on the board. After that they move alternately; Peter is first. In every move, the next player chooses two numbers, erases them and replaces both of them by their mean. Peter wins if he can achieve that the sum of suitably chosen 50 numbers is equal to the sum of the other 50 numbers. Determine whether Basil can prevent this.

Prove that $\left(\frac{1+a}2\right)^{2x(x+y)} \left(\frac{1+b}2\right)^{2y(x+y)} \ge a^{x^2} b^{y^2} \left(\frac{a+b}2\right)^{2xy}$ holds for all real numbers $a,b>0$ and $x$ , $y$ .

Let $\mathcal{F}$ be a finite family of finite sets and let $A$ be an arbitrary finite set. We say that $\mathcal{F}$ shatters the set $A$ if for every $X\subseteq A$ there is a set $F\in \mathcal{F}$ such that $A\cap F=X$ . Show that $\mathcal{F}$ shatters at least $|\mathcal{F}|$ sets.

Prove that the equation $x^3 - x + 9 = 5 y^2$ has no solution among the integers.

There are given four rays, $a$ , $b$ , $c$ and $d$ in space, starting from the same point, laying in a plane $\varPi$. For an arbitrary acute angle $\varphi$, rotate $\varPi$ by angle $\varphi$ in positive direction around each of the four rays; denote the rotated planes by $A_\varphi$, $B_\varphi$, $\varGamma_\varphi$ and $\varDelta_\varphi$, respectively. Let $\varSigma_\varphi$ be the plane through the intersection line of $A_\varphi$ and $B_\varphi$, and the intersection line of $\varGamma_\varphi$ and $\varDelta_\varphi$. Show that the planes $\varSigma_\varphi$ share a common line.

Arthur and Ben have a chocolate table of size $k\times n$ , they play a game with this chocolate. They alternately eat parts of the chocolate; Arthur is first. In every step, the current player breaks the chocolate table into two rectangular parts along the lines and eats up the smaller part. (If eventually the two pieces are equal, he can choose which part to eat.) The player who first eats a single chocolate square in a step loses, the other player wins. Determine all pairs $(k,n)$ for which Arthur has a winning strategy.

In an acute triangle $ABC$ , the feet of the altitudes are $A_1$, $B_1$ and $C_1$, respectively. The midpoint of the side $BC$ is $F$ . The circle $BCB_1C_1$ intersects the line segment $AA_1$ at point $D$ . Let $T$ be that point on the line segment $DF$ for which the line $BT$ is tangent to the circle $AB_1C_1$. Let the line segment $C_1F$ meet the lines $BD$ and $BT$ at $P$ and $Q$ , respectively. Show that the quadrilateral $DPQT$ has an inscribed circle.

Prove that $\dfrac{7^{7^{k+1}}+1}{7^{7^{k}}+1}$ is a composite number for every nonnegative integer $k$ .

Let $a$ , $b$ and $c$ be three distinct positive reals. The logarithmic mean of $a$ , $b$ , $c$ is defined by $L(a,b,c) = 2\left( \frac{a}{(\ln a-\ln b)(\ln a-\ln c)} + \frac{b}{(\ln b-\ln c)(\ln b-\ln a)} + \frac{c}{(\ln c-\ln a)(\ln c-\ln b)} \right).$ Prove that $\sqrt[3]{abc} < L(a,b,c) < \frac{a+b+c}{3}$.

$a)$ Prove that for every infinite sequence $x_1,x_2,\ldots\in[0,1]$ there exists some $C>0$ such that for every positive integer $r$ there are positive integers $n$ , $m$ satisfying $|n-m|\ge r$ and $|x_n-x_m|<\frac{C}{|n-m|}$. $b)$ Show that for every $C>0$ there exists an infinite sequence $x_1,x_2,\ldots\in[0,1]$ and a positive integer $r$ such that $|x_n-x_m|>\frac{C}{|n-m|}$ holds true for every pair $n$ , $m$ of positive integers with $|n-m|\ge r$ .

Let $n\ge2$, and let $\mathcal{S}$ be a family of some subsets of $\{1,2,\ldots,n\}$ with the property that $|A\cup B\cup C\cup D|\le n-2$ for all $A,B,C,D\in\mathcal{S}$ . Show that $|\mathcal{S}|\le 2^{n-2}$.

We have $4n+5$ points on the plane, no three of them are collinear. The points are colored with two colors. Prove that from the points we can form $n$ empty triangles (they have no colored points in their interiors) with pairwise disjoint interiors such that all points occurring as vertices of the $n$ triangles have the same color.

Let $n\ge1$ be a fixed integer. Calculate the distance $\inf_{p,f} \max_{0\le x\le 1} \big|f(x)-p(x)\big|$, where $p$ runs over polynomials of degree less than $n$ with real coefficients and $f$ runs over functions $f(x) = \sum_{k=n}^\infty c_k x^k$ defined on the closed interval $[0,1]$, where $c_k\ge0$ and $\sum_{k=n}^\infty c_k=1$.

Is it true that for every infinite sequence $x_1,x_2,\ldots$ of integers satisfying $|x_{k+1}-x_k|=1$ for every positive integer $k$ , there exists a sequence $k_1<k_2<\ldots<k_{2014}$ of positive integers such that as well the indices $k_1,k_2,\ldots,k_{2014}$ as the numbers $x_{k_1},x_{k_2},\ldots,x_{k_{2014}}$ (in this order) form arithmethic progressions?

An infinite number of points have been marked in the square grid in such a way that no circle contains more than 2014 marked points. Show that there is a circular disk of diameter 100 (in the plane of the grid) that does not contain any marked point in its interior.

The konvex quadrilateral $ABCD$ has an inscribed circle with center $I$ . The rays $AB$ and $DC$ meet at point $F$ , the rays $AD$ and $BC$ meet at point $G$ . Let $\mathcal{E}$ be the ellipse with foci $F$ and $G$ that passes through points $B$ and $D$ , and let $\mathcal{H}$ be the hyperbola branch with foci $F$ and $G$ that passes through points $A$ and $C$ . Denote by $P$ and $Q$ the intersections of $\mathcal{E}$ and $\mathcal{H}$ . Show that the points $P$ , $Q$ and $I$ are collinear.

Let $k\ge1$ and let $I_1,\ldots,I_k$ be non-degenerate subintervals of the interval $[0, 1]$. Prove $\sum \frac1{|I_i\cup I_j|} \ge k^2$ where the summation is over all pairs $(i,j)$ of indices such that $I_i$ and $I_j$ are not disjoint.

What may be the number of people in a company in which every member has exactly three acquaintances, and any two people have an acquaintance in common exactly if the two of them do not know each other?

An ant starts crawling at constant speed from the left end of a 4-metre-long rubber rope towards the right end. It covers exactly one metre per minute. The rope is fixed at the left end, in a horizontal position. At the end of each minute, the rope is stretched uniformly by one metre. In which minute will the ant reach the right end of the rope? The ant is considered pointlike, the act of stretching takes negligible time, and the rope can be stretched to any length without breaking it.

Consider a sequence of reflections in the lines of all four sides of a square in some order. How many different transformations are obtained by the composition of four such reflections?

Let $d$($n$) denote the number of positive divisors of a positive integer $n$. Solve the equation $d$ ( $n^3$ ) =$n$.

Solve the equation $x^{4}-2\sqrt{3}x^{2}+x+3-\sqrt{3}=0$.

The radii of the escribed circles of a triangle are $r_a$, $r_b$ and $r_c$, the radius of the circumscribed circle is $R$. Determine the angles of the triangle if $r_a$+$r_b$=3$R$ and $r_b$+$r_c$=2$R$.

The diagonals of a cyclic quadrilateral ABCD intersect at $E$, the midpoints of sides $AB$ and $CD$ are $K$ and $M$, and the perpendicular projections of point $E$on the sides $BC$ and $AD$ are $L$ and $N$, respectively. Prove that the lines $KM$and $LN$ are perpendicular to each other.

Solve the equation $sin^5$x$+cos^5$x$+sin^4 x$=2.

Is it possible to divide the first $n$ prime numbers into two sets such that the sum of elements in each set is the same if $a$) n$=2013^{2014}$; $b$) n$=2014^{2013}$?

One face of a tetrahedron is a regular triangle of unit side. The length of the other three edges is $a$. What is the maximum possible area of the perpendicular projection of the tetrahedron on a plane?

The diagonals of a cyclic trapezium are perpendicular. Prove that the distance of the centre of the circumscribed circle from one base is equal to the distance of the intersection of the diagonals from the other base.

The surface area of a right circular cone is $A$, its volume is $V$. Prove that $A^3 \ge$72$\pi^. V^2$.

Solve the equation $x^{3}+\left(\frac{x}{2x-1}\right)^{3}=\frac{243}{64}$.

$\alpha$ and $\beta$ are different real numbers, at least one of which is not an integer. Is it necessarily true that there exists a positive integer $n$ such that $\alpha^n$-$\beta^n$ is not an integer?

Solve the equation $\frac{x\cdot 2014^{\frac{1}{x}}+\frac{1}{x}\cdot 2014^x}2 =2014$ on the set of positive numbers.

The sides of a triangle are $a$, $b$, $c$. A line passes through the centre of the inscribed circle, and intersects side $c$ at $P$, and side $b$ at $Q$. Let $AP$=$p$and $AQ$=$q$. Prove that $\frac{1}{p} + \frac{1}{q} = \frac{a+b+c}{bc}$ .

The perpendicular projections of the centroid $S$ of a triangle ABC on sides $BC$, $AC$ and $AB$ are $A_1$, $B_1$ and $C_1$, respectively. Prove that (with conventional notations) $a^2 \overrightarrow{SA_1} +b^2 \overrightarrow{SB_1}+c^2 \overrightarrow{SC_1} =\mathbf{0}$.

Find the smallest positive number $c$ with the following property: For any set $a_1,a_2,\ldots,a_n$ of real numbers, it is possible to select some elements of the set such that the distance of their sum from the nearest integer is at most $c$.

$K$ is the centre of the circumscribed circle of an acute triangle ABC . $O$ is the centre of the inscribed circle, and $M$ is the orthocentre. Is it possible for the points $O$, $K$ and $M$ to be the vertices of an isosceles triangle?

Consider the composition of the reflections in the planes of all six faces of a cube in some order. How many different transformations of the spaces are obtained?

Each member of a group of 9 people sends happy new year cards to exactly $k$ others. What may be the least possible value of $k$ such that we can be certain that there is a pair of members who mutually greet each other?

The rhombus $A_1 B_1 C_1 D_1$ lies in the interior of a parallelogram ABCD . The vectors $\overrightarrow{AB}$ and $\overrightarrow{A_1B_1}$ have the same direction, and the vectors $\overrightarrow{BC}$ and $\overrightarrow{B_1C_1}$ also have the same direction. Show that ABCD is a rhombus if and only if the sum of the areas of quadrilaterals $AA_1 D_1 D$ and BCC $_1 D_1$ is equal to the sum of the areas of quadrilaterals ABB $_1 A_1$ and CDD $_1 C_1$.

Let $x_1$,...,$x_n$ and $y_1$,...,$y_n$ be nondecreasing sequences of non-negative numbers, such that the sum of the $n$ terms is 1. $a$) What is the maximum possible value of $\min_{1\le i\le n} |x_i-y_i|$? $b$) What is the maximum possible value of $\sum_{i=1}^n{|x_i-y_i|}$?

Each angle of the triangle ABC is smaller than $120^o$. The isogonal point of the triangle is $P$. Draw lines through point $P$ parallel to the sides. The parallels intersect side $AB$ at $D$ and $E$, side $BC$ at $F$ and $G$, and side $CA$ at $H$ and $I$. Let $K$, $L$ and $M$ denote the isogonal points of triangles DEP , FGP , HIP , respectively. Show that triangle KLM is equilateral.

For what values of $n$ do the numbers $1!,2!, \ldots, n!$ leave pairwise different remainders when divided by $n$?

What angle may the hypotenuse of a right-angled triangle enclose with the median drawn to one of the legs?

The polygon $A_1 A_2 \ldots A_n$ has both an inscribed circle and a circumscribed circle. The centre of the inscribed circle is $O$, and the centre of the circle $OA_i A_{i+1}$ is $C_i$ ($i=1, 2,\ldots, n$ , and $A_{n+1}$=$A_1$). Prove that $C_1$,$C_2$,...,$C_n$ are concyclic.

Solve the equation $x^{2}-4x+3=\sqrt{1+\sqrt{x-1}}$.

Prove that for any numbers $0\le a_1\le\ldots\le a_{3n}$, $\bigg(\sum_{i=1}^{3n}a_i\bigg)^3 \ge 27n^2\bigg(\sum_{i=1}^{n}a_ia_{n+i}a_{2n+i}\bigg)$.

For a certain tetrahedron, there exists a sphere that touches all edges, and there exists another sphere that touches the three edges bounding one face and also the extensions of the other three edges. Show that there exists a sphere for each face that touches the three edges bounding that face and also the extensions of the other three edges.

The numbers $1,2,\ldots,9$ are written in the fields of a $3\times 3$ table so that the sum of the numbers is the same in each of the four $2\times 2$ squares. What may this sum be?

In a convex quadrilateral, the diagonals form four triangles of integer areas. Prove that the product of the four integers cannot end in 2014.

In a trapezium $ABCD$ , let $E$ and $F$ denote the midpoints of the bases $AB$ and $CD$ , respectively, and let $O$ denote the intersection of the diagonals. A line parallel to the bases intersects the line segments $OA$ , $OE$ and $OB$ at points $M$ , $N$ and $P$ , respectively. Show that the quadrilaterals $APCN$ and $BNDM$ have equal areas.

How many ordered pairs $(A,B)$ are there such that $A$ and $B$ are subsets of a fixed $n$ -element set, and $A\subseteq B$ ?

Prove that ${(1+a)}^4 {(1+b)}^4 \ge 64ab {(a+b)}^2$ for all $a,b\ge 0$.

The angle bisector drawn from right-angled vertex $C$ of triangle $ABC$ intersects the circumscribed circle at point $P$ , and the angle bisector from vertex $A$ intersects the circumscribed circle at point $Q$ . $K$ is the intersection of line segments $PQ$ and $AB$ . The centre of the inscribed circle is $O$ , and its point of tangency on side $AC$ is $E$ . Prove that points $E$ , $O$ and $K$ are collinear.

Show that if $\alpha$, $\beta$ and $\gamma$ are the angles of a triangle then $\sin\alpha\cdot \sin\beta\cdot \cos\gamma+\sin\alpha\cdot \cos\beta\cdot \sin\gamma+ \cos\alpha\cdot \sin\beta\cdot \sin\gamma\le \frac 98$.

Solve the equation $2\sin\frac{3x}{2}=3\sin \left(x+\frac{\pi}{3}\right)$.

Points $A$ , $B$ , $C$ and $D$ are not coplanar. Determine the locus of the points $P$ for which $PA^{2}+PC^{2}=PB^{2}+PD^{2}$.

The circles $k_0$, $k_1$, $k_2$, $k_3$ lie in the same plane and pairwise touch each other on the outside. The point of contact of $k_i$ and $k_j$ is $T_{ij}$. Let $k_0$ have centre $O$ and radius $r$ , and let the circle $T_{12}T_{23}T_{31}$ have centre $U$ and radius $R$ . Prove that $OU^2 = R^2-4Rr+r^2$.

The intersection of two given lines is off the page. Consider the line that passes through a given point on the page and through the intersection of the given lines. Of this line, construct the part on the page.

There are some points marked in the interior of a triangle, such that no three of them (including the vertices) are collinear. The points are connected to each other and to the vertices of the triangle so that the resulting line segments should not intersect, and the triangle should be divided into the largest possible number of smaller triangles. Prove that the number of small triangles formed is odd.

For what positive integers $n$ and $k$ is $\binom nk$ a power of a prime?

In an acute-angled triangle $ABC$ , $AB<AC$ . The centre of the circumscribed circle is $O$ , the orthocentre is $M$ . Construct the point $P$ on the side $BC$ for which $AOP\sphericalangle = PMA\sphericalangle$.

At which point in the interior of a triangle is the product of the distances from the sides a maximum?

Sir Bedevir will only enter in a tournament if he is certain that he will win with a probability of at least 1/2. In any combat of two knights, the probability of the victory of the parties are proportional to their fighting potentials. Bedevir's fighting potential is 1, and that of his $n$ th opponent is $\frac{1}{2^{n+1}-1}$. How many knights may have entered in the tournament if Bedevir, having carried out some careful calculations, also decided to enter?

Let $x_{1}, x_{2}, \ldots, x_{n}$ denote arbitrary real numbers. Prove that $\sqrt{\left(\sum_{k=1}^{n} \frac{x_{k}^{4}+k^{2}}{x_{k}^{2}}\right)^{2}-n^{2} (n+1)^{2}}\ge \sum_{k=1}^{n} \frac{x_{k}^{4}-k^{2}}{x_{k}^{2}}$.

Let $P$ be an exterior point of the ellipse $\mathcal E$ with foci $F_1$ and $F_2$. It does not lie on the line of the major axis. Let $M_1$ be the intersection of the line segment $PF_1$ with $\mathcal E$ , let $M_2$ be the intersection of the line segment $PF_2$ with $\mathcal E$ , and let $R$ be the intersection of the lines $M_1F_2$ and $M_2F_1$. Prove that the quadrilateral $PM_1RM_2$ has an inscribed circle.

Calculate the value of the sum $\sum_{j=0}^{n} \binom{2n}{2j} {(-3)}^{j}$.

The plane $S$ divides a regular octahedron into two parts. Determine the ratio of the volumes of the parts, given that the intersection of $S$ with the octahedron is a regular hexagon.

Given that no three of five points in the plane are collinear, prove that there are at most seven acute-angled triangles among the triangles determined by the five points.

Are there integers $n$ and $k$ such that $n^3-n-1=k^2-k+1$?

Let $H_{1}=\{1, 3, 5, \ldots, 2n-1\}$ and $H_{2}=\{1+k, 3+k, 5+k, \ldots, 2n-1+k\}$, where $n$ and $k$ are any positive integers. Is there an appropriate $k$ for every $n$ such that the product of all elements of the set $H_{1}\cup H_{2}$ is a perfect square?

For what values of the parameter $p$ is the inequality $\left(1+\frac{1}{\sin x}\right)^{3} \ge \frac{p}{\tan^2 x}$ true for all $0<x<\frac{\pi}{2}$?

Given that 100 closed unit discs, together, cover a different, 101st closed unit disc. Show that there is a disc among the 100 that covers the point on the top of another. (Top means greatest ordinate.)

The faces of a tetrahedron are congruent triangles with sides of lengths 13, $\sqrt{281}$ and $20$ units. Determine the dihedral angle enclosed by two faces.

Let $e_1,e_2,\dots,e_n$ denote different lines on the plane, and let $f$ be a line that is not parallel to any of them. Consider all lines $f_{\alpha}$ parallel to $f$ . Let $S_{\alpha}$ be the centre of mass of the points $f_{\alpha}\cap e_1, f_{\alpha}\cap e_2, \ldots, f_{\alpha}\cap e_n$. Show that the points $S_{\alpha}$ are collinear.

Is there a function of the form $f(x)=\frac{ax+b}{cx+d}$ for which $f(x_1)=x_2$, $f(x_2)=x_3$, $f(x_3)=x_4$, $f(x_4)=x_5$, $f(x_5)=x_1$ is true if $x_1,\ldots,x_5$ are appropriate pairwise different real numbers?

A positive integer $n$ is said to be exotic if it is divisible by the number of its positive factors. Prove the following statements: $a)$ If an exotic number is odd then it is a perfect square. $b)$ There are infinitely many exotic numbers.

The angles of a triangle are $\alpha$, $\beta$ and $\gamma$. Determine the angles of the triangle formed by the tangents drawn to the circumscribed circle at the vertices.

How many ordered triples of positive integers $a$ , $b$ , $c$ are there for which $[a,b,c]=10!$ and $(a,b,c)=1$? ($(a,b,c)$ denotes the greatest common divisor, and $[a,b,c]$ denotes the least common multiple.)

In a triangle $ABC$ , let $AD$ be an altitude, let $BE$ be an angle bisector, and let $CF$ be a median. Prove that the lines $AD$ , $BE$ and $CF$ are concurrent exactly if $ED$ is parallel to $AB$ .

Is there a solution of the equation $2012^{2015}=\binom n2 +\binom k2$ on the set of positive integers?

Prove that any region of space bounded by the planes of the lateral faces of a four-sided convex pyramid can be intersected with a plane such that the intersection is a parallelogram.

The radius of the inscribed circle of a triangle is $r$ , and the radius of the circumscribed circle is $R$ . Assume that $R < r \big(\sqrt{2}+1\big)$. Does this condition imply that the triangle is acute-angled?

Solve the equation $8^{2x-1}-1=343^{x-1}+\frac{3}{14} 28^x.$

Given $n$ points in a unit circle, what is the maximum possible value of the product of the distances of all point pairs?

In a championship, every team plays every other team exactly once. 3 points are awarded for winning, 0 for losing, and 1 for a draw. In the case of equal scores, the order of the teams is determined at random. The championship is in progress at the moment. Team $A$ is leading the points table. If team $A$ scores exactly $x$ points in the remaining rounds then they will win the championship. However, if $A$ scores more than $x$ points, they will not necessarily win. (It is possible for $A$ to score more than $x$ .) How many rounds remain to be played in the championship?

There are discs placed on some fields of a chessboard of $n$ columns and $k$ rows (at most one disc on each). Two discs are said to be adjacent if they lie in the same row or in the same column, and there is no other disc along the line segment connecting them. Each disc is adjacent to at most three others. What is the maximum possible number of discs on the chessboard?

A regular triangle is drawn over each side of a triangle $ABC$ on the outside. The third vertices of these triangles are $D$ , $E$ and $F$ . Given the points $D$ , $E$ and $F$ , construct the triangle $ABC$ .

Find the integer solutions of the equation $2x^3 - y^3 = 5$.

A rectangle $ABDE$ is drawn to side $AB$ of an acute triangle $ABC$ on the inside, such that point $C$ should lie on the side $DE$ . The rectangles $BCFG$ and $CAHI$ are defined in a similar way. ($A$ lies on line segment $FG$ , and $B$ lies on line segment $HI$ .) The midpoints of sides $AB$ , $BC$ , and $CA$ are $J$ , $K$ , and $L$ , respectively. Prove that the sum of the angles $GJH\sphericalangle$, $IKD\sphericalangle$ and $ELF\sphericalangle$ is $180^{\circ}$.

Determine the number of solutions of the equation $mX+4= \big|X^2-10X+21\big|$ as a function of the parameter $m$ .

Prove that $\sum_{k=1}^n (2k-1) \left[\frac{n}{k}\right] = \sum_{k=1}^n \left[\frac{n}{k}\right]^2$ for every positive integer $n$ .

Prove that if $a$ , $b$ and $c$ are the sides of a triangle of unit perimeter, then $a^2+b^2+c^2+4abc<\frac{1}{2}$.

A tetrahedron is said to be strange if the line segments connecting its vertices to the incentres of the opposite faces are concurrent. What condition is true for the edges of a strange tetrahedron?

It is a known fact that 777-headed dragons have either 9 or 13 heads on their necks. Two dragons are considered identical if they have the same number of 9-headed necks. How many different 777-headed dragons are possible?

Let $A_1$, $B_1$ and $C_1$ be the midpoints of the sides of a triangle $ABC$ . Drop a perpendicular from $A_1$ to the angle bisector drawn from vertex $A$ , from $B_1$ to the angle bisector drawn from vertex $B$ , and from $C_1$ to the angle bisector drawn from vertex $C$ . Let $A_2$ denote the intersection of the perpendiculars from $B_1$ and from $C_1$. The points $B_2$ and $C_2$ are obtained in a similar way. Show that the lines $A_1A_2$, $B_1B_2$ and $C_1C_2$ are concurrent.

Let $AB_1B_2\dots B_6$ and $AC_1C_2\dots C_6$ be regular heptagons with their vertices labelled in the same direction around the clock. Show that the lines $B_1C_1, B_2C_2, \dots, B_6C_6$ are concurrent.

Determine all real functions $f$ defined on the positive integers, such that for all positive integer $n$ , $\frac{p}{f(1)+f(2)+\dots +f(n)} =\frac{p+1}{f(n)} -\frac{p+1}{f(n+1)}$, where $p$ is a fixed positive number.

$E$ is the intersection of the diagonals of a cyclic quadrilateral $ABCD$ , and $K$ is the centre of the circumscribed circle. The intersection of the lines of sides $AB$ and $CD$ is $F$ , and the intersection of the lines of sides $BC$ and $DA$ is $G$ . The second intersection of the circumscribed circles of triangles $BFC$ and $CGD$ is $H$ . Prove that the points $K$ , $E$ and $H$ are collinear.

On the circumscribed circle of triangle $ABC$ , a point $X$ is moving along the arc $AC$ not containing vertex $B$ . Let $Y$ and $Z$ denote the points on the extensions of side $BA$ beyond $A$ and side $BC$ beyond $C$ , respectively, for which $AY=AX$ and $CZ=CX$ . What is the locus of the midpoint of line segment $YZ$ ?

Which is greater: $\log_{3}4\cdot \log_{3}6 \cdot \log_{3}8\cdot \dots \cdot \log_{3}2012 \cdot \log_{3}2014$ or $2\cdot \log_{3}3\cdot \log_{3}5 \cdot \log_{3}7\cdot \dots \cdot \log_{3} 2011 \cdot \log_{3} 2013?$

A flea is jumping along the number line. It starts at the origin, and the length of each jump is 1. The probability of any jump being in the same direction as the preceding jump is $p$ , and the probability of the opposite direction is $1-p$ . What is the probability that the flea will get back to the origin?

Prove that if opposite edges of a tetrahedron $ABCD$ are equal in length then the foot of the altitude drawn from $D$ lies on the Euler line of triangle $ABC$ .

Show that if $x$, $y$ and $z$ are rational numbers such that $x$+$y \ne z$, $z \ne$0, and ($a$-1)$^2 x$+($a$-1)$^2 y$- ( $a^2$-1 ) $z$=0, then $a$ is also a rational number.

The midpoints of the sides of the convex quadrilaterals ABCD and EFGH coincide. Prove that the two quadrilaterals have equal areas.

Find all right-angled triangles in which the measures of the sides are two-digit integers, the length of the hypotenuse is obtained by interchanging the digits of one leg, and the three two-digit numbers consist of exactly three kinds of digits, each occurring twice.

Five dice are rolled simultaneously. What is the probability that at least two identical numbers are rolled?

$E$, $F$ and $G$ are points on each side of triangle ABC and $n$ is a natural number such that $\frac{AE}{EB}=\frac{1}{n-1}$, $\frac{BF}{FC}=\frac{1}{n}$ , and $\frac{CG}{GA}=\frac{1}{n+1}$. Show that if $n \ge$5 then the area of triangle EFG is greater than the half of the area of triangle ABC .

The vertices of the parallelograms PQRL and LSTA in the plane are labelled in the same sense around the clock. The parallelograms do not have a point in common, except $L$. Prove that there exists a pentagon ABCDE (degenerated cases are allowed) in the plane such that the midpoints of the sides are $P$, $Q$, $R$, $S$, $T$, in this order.

The tangents drawn from a point $C$ lying outside a circle touch the circle at points $A$ és $B$. $M$ is a point on the shorter arc $AB$. Let $MN$, $ME$ and $MD$be the line segments drawn from $M$, perpendicular to the line segments $AB$, $BC$and $CA$, respectively. Given that $MN$=4, $MD$=2 and $AMB\sphericalangle=120^{\circ}$, find the area of triangle MNE .

We have four sacks of flour. By three measurements, the following information is obtained: the first and second sacks together are lighter than the other two sacks, the first and third together have the same mass as the other two, and the first and fourth together are heavier than the other two. Which sack is the heaviest?

In a convex quadrilateral ABCD , the line drawn through vertex $D$ parallel to line $BC$ intersects side $AB$ at its midpoint $F$. What is the area of the quadrilateral AFCD if the area of triangle ABD is 4?

Prove that $(2+3+5) \mid \big(2 \cdot 2^{5^3} + 3^{2^5} - 5^{3^2}\big)$.

Given a rectangle, construct a rectangle of half the area inside it such that the region between the two rectangles forms a frame of uniform width around the small rectangle.

In a class of 30 students, two students are absent. What is the probability that the two of them are next to each other in the alphabetical list of the names? How many students would be in a class where this probability is 0.1?

Prove that if the inscribed circle of a right-angled triangle has unit radius and the length of one leg is rational then the lengths of the other two sides are also rational.

The ideal carnival doughnut weighs 54 grams when done, and 8% is the oil absorbed during frying. It is shaped like a (centrally symmetrical) spherical segment of diameter 78 mm and height 3 cm. The volume of the dough is doubled in frying, and its mass is only increased by the oil absorbed. Before frying, the whole dough weighs 1 kg, and it is kept in a 2.5-litre bowl. Is it possible to cover the bowl with a flat lid?

Prove that it is possible to select four out of any seven integers such that their sum is divisible by 4.

On a circular table of radius 80 cm, the square tablecloth has been pulled aside, such that one corner lies exactly on the edge of the table, and the two sides that meet at the opposite vertex are tangent to the edge of the table. Determine the length of the side of the tablecloth to the nearest millimetre.

Prove that $9^m$ (where $m$ is a positive integer) can always be expressed as the sum of three positive square numbers.

$F$ is the midpoint of side $AD$ of a square ABCD . $N$ is the point lying closer to $B$ that divides side $AB$ in a 1:3 ratio. In what ratio does the circumscribed circle of triangle AFN divide the diagonal $AC$?

The sum of the first four terms of a geometric progression is 15.6, and the sum of their reciprocals is 12.48. What are these four numbers if the first term is 0.1?

The lateral faces of a regular four-sided pyramid are equilateral triangles. What is the angle enclosed by adjacent lateral faces?

The integer that consists of $m$ digits of 9 and the integer that consists of $n$ digits of 9 are multiplied together. What is the sum of the digits of the product?

What is the perimeter of the isosceles triangle that has a base of 6 cm and an inradius of 1.5 cm?

Solve the following equation on the set of pairs of integers: $x^2-3y^2+2xy-2x-10y+20=0$.

The lengths of the bases of a trapezium are 7 and 1. The trapezium is divided into two parts of equal area with a line parallel to the bases. Find the length of the segment of this parallel line that lies inside the trapezium.

All cups are different in our kitchen cupboard. One third of them has the handle broken off. How many cups have we got if the number of ways to select two cups without handles and three with handles is exactly 1200?

A sphere is cut with the plane whose distance from the centre is 2/3 of the radius. What fraction of the volume is cut off?

Three lattice points are selected at random out of the lattice points lying on the circle of equation $x^2+y^2-2x-4y-45=0$. What is the probability that the three points form a right-angled triangle?

Consider those $n$ -digit natural numbers that have no digits different from 1 and 2. Prove that there is a number among them that is divisible by $2^n$.

In a triangle, the median drawn to side $b$ is twice as long as the median drawn to side $c$ , and the two medians are perpendicular. Given that the length of the median drawn to side $a$ is 60 cm, find the perimeter of the triangle.

Solve the equation $17 \big(x^2+y^2\big)-32xy=41$ on the set of integers.

One angle of a kite is a right angle, the measure of the opposite angle is $30^\circ$, and the length of the shorter side is 10 cm. Find the length of the side of the square that has a vertex on each of three sides of the kite, and has a side parallel to the longer side of the kite.

In Flora's flower garden, tulips are grown for mothers' day. One of the flowerbeds has 53 rows with 38 flowers in each row. When Flora inspected all flowers from the first one to the last one, row by row, she observed that every other tulip had coloured streaks in it, every 19th had a broken petal, and every 53rd was not fully open yet. She also discovered that the sum of the numbers of the positions of the perfect tulips (with no coloured streaks or broken petals, fully open) was equal to nineteen times her income in forints (HUF, Hungarian currency). For how many forints did she sell a dozen of perfect tulips?

Determine the sum of the radii of the circles centred on the $y$ -axis that are tangent to the circle of equation ${(x-5)}^2+ {(y-5)}^2=25$ and to the line of equation $y=\frac{4}{3}x+6$.

A 45-decagram loaf of bread is cut in three parts. The difference between the largest and smallest pieces is 5 decagrams. What may be the masses of the pieces if the standard deviation of the masses is $\sqrt{\frac{14}{3}}$ decagrams?

What may be the digits $a$ , $b$ , $c$ in decimal notation if $\overline{aa\vphantom{b}}^2+\overline{bb}=\overline{cccc\vphantom{b}}$ ?

Find three possible natural numbers $x$ , $y$ , $z$ ($x<y<z$ ) such that $3^x+3^y+3^z=179 415$.

Solve the following equation: $\left(\frac{x-1}{2} \right)^{2} + \frac12 = \left|\frac{4x}{x+3}\right| - \frac12$.

The sides of a right-angled triangle are 3, 4, 5. Determine the line that halves both the perimeter and the area of the triangle.