KöMaL 2013

For every integer $n \ge$2 let $P$($n$) be the product of all expressions of the form $\pm\sqrt1 \pm\sqrt2 \pm\sqrt3 \pm\ldots \pm\sqrt{n}$ where the signs of the terms are chosen arbitrarily. ($a$) Prove that $P$($n$) is a positive integer. ($b$) Prove that for all $\varepsilon$>0 there exists an $n_0$ such that for every $n$>$n_0$ the largest prime divisor of $P$($n$) is smaller than $2^{2^{\varepsilon n}}$.

The circle $k_1$ is internally tangent to the circle $k$ which is externally tangent to $k_2$. The common external tangents of $k_1$ and $k_2$ are $u$ and $v$. The line $u$ is tangent to $k_1$ and $k_2$ at $P$ and $Q$, respectively, and meets $k$ at $A$ and $B$ in such a way that $B$ lies between $P$ and $Q$. Analogously, the line $v$ is tangent to $k_1$ and $k_2$ at $R$ and $S$, respectively, and meets $k$at $C$ and $D$ in such a way that $D$ lies between $R$ and $S$and $k_1$ is tangent to that arc $BD$ of $k$ which does not contain $A$ and $C$. Show that $\frac {AB\cdot AD} {AP\cdot AQ} = \frac {CB\cdot CD} {CR\cdot CS} .$

Show that for every positive integer $k$ and real number $\varepsilon$>0 there exists a polynomial $p$($x$)=$x^n$+$a_{n-1} x^{n-1}$+...+$a_1 x$+$a_0$ with real coefficients such that $p$($x$) is divisible by the polynomial ($x$-1)$^{k+1}$ and $\sum_{\ell=0}^{n-1} |a_\ell| < 1+\frac{(2+\varepsilon)k^2}{n}.$

In the plane, there are given two circles $k_1$ and $k_2$ with different radii, and a point $O$ lying outside the circles. The end-points of the tangents drawn from $O$to $k_1$ are $P$ and $Q$, the end-points of the tangents drawn from $O$ to $k_2$ are $R$ and $S$. The points $P$, $Q$, $R$, $S$ are distinct. Let $H$ be the external homothety center between $k_1$ and $k_2$. Prove that if $PR$ is not a external common tangent to the circles but passes through $H$ then $QS$ also passes through $H$.

Let $p$ be a fixed positive prime. For any polynomial $f$($x$) with integer coefficients, let $f^*$($x$)=$f$ ( (1+$x$)$^p$-1 ) . Find all polynomials $g$ with integer coefficients for which the following holds: for every positive integer $n$there is a polynomial $f$ with integer coefficients such that the coefficients of $g$-($f$-$f^*$) with degree less than $n$ are all divisible by $p^n$.

There are given $n$ random events ($n \ge$3) such that each of them has probability $\frac12$, the joint probability of each two is $\frac14$ and the joint probability of each three is $\frac18$. ($a$) Prove that the probability that none of the events occurs is at most $\frac{1}{2n}$ . ($b$) Show that there are infinitely many numbers $n$, for which the events can be specified in such a way that the probability that none of the events occurs is precisely $\frac{1}{2n}$ .

In 3-space, let $S$ be a non-degenerate conic section which is not a circle. Consider the apices of those right circular conical surfaces that contain $S$. ($a$) Show that these points lie on a conic section, uniquely determined by $S$. ($b$) Denote by $C$($S$) the conic section that contains the possible apices. Prove that $C$ ( $C$($S$) ) =$S$ for arbitrary $S$.

For some integer $n \ge$2 each pair ($i$,$j$), where 1$\le i$,$j \le n$, is written on a card. We play the following game. The $n^2$ cards are placed in an $n$×$n$table so that for every $i$ and $j$, the card ($i$,$j$) is in the $i$th row, at the $j$th position. It is allowed to exchange the cards ($i$,$j$) and ($k$,$l$) if they are in the same row or in the same column, and $i$=$k$ or $j$=$l$. Is it possible to reach that arrangement of the cards which contains (1,2) and (2,1) interchanged and all other cards are at their initial positions?

Two convex quadrilaterals have side lengths $a_1$, $a_2$, $a_3$, $a_4$, and $A_1$, $A_2$, $A_3$, $A_4$; their areas are $t$ and $T$, respectively. Show that $16tT \le \sum_{i=1}^4 a_i^2 \big(A_{i+1}^2+A_{i+2}^2+A_{i+3}^2-A_i^2\big) + 4a_1A_2a_3A_4 + 4A_1a_2A_3a_4.$ (The indices are used modulo 4.)

Find all positive integers $n$ for which there exists a sequence $a_1$<$a_2$<...<$a_n$ of positive integers with the property that the sums $a_i$+$a_j$ with 1$\le i$<$j \le n$ are distinct and among them each modulo 4 residue occurs the same number of times.

Let $a$ be a positive integer. Prove that all prime divisors of 7$a^2$($a$+1)-1 are of the form 7$k \pm$1.

Does there exist such a convex polyhedron that can be placed in a sphere with unit radius, it has at least 2013 vertices, it has no edge shorter than 1/2, and it has no triangle face?

There is given a convex quadrilateral ABCD and some points $P$, $Q$, $R$ and $S$on the line segments $AB$, $BC$, $CD$ and $DA$, respectively. The line segments $PR$and $QS$ meet at $T$. Suppose that each of the the quadrilaterals APTS , BQTP , CRTQ and DSTR have an inscribed circle. Prove that the quadrilateral ABCD also has an inscribed circle.

Color the vertices of a binary tree of depth $2^{2n}$ as follows. In the initial position let all vertices be white. Take the vertices one by one in a random order. Color the current vertex to red, except if from that vertex there starts a path of length $n$ downwards (away from the root), whose other vertices are already red. Let $p$($n$) be the probability that during the procedure, the root remains white. Determine $\lim_{n\to\infty} p(n)$.

The convex quadrilateral ABCD has an inscribed circle which is tangent to the sides $AB$ and $BC$ at $E$ and $F$, respectively. The diagonal $AC$ meets the inscribed circle at $P$ and $Q$; the point $Q$ is located between $A$ and $P$. Show that the lines $BD$, $EP$ and $FQ$ are concurrent.

Let $p$ be a positive prime number for which there is a positive integer $a$ such that $p$ divides 2$a^2$-1. Prove that there exist integers $b$ and $c$ such that $p$=2$b^2$-$c^2$.

Find all integers $k \ge$3 with the property that for every infinite sequence $P_1$,$P_2$,... of points in the plane such that no three of them are collinear, there exists a sequence $1\le i_1<i_2<\ldots<i_k$ of integers for which the points $P_{i_1},P_{i_2},\ldots,P_{i_k}$, in that order, form a convex $k$-gon.

The circles $k_0$, $k_1$, $k_2$, $k_3$ and $k_4$ lie in the plane in such a way that for $i$=1,2,3,4 the circle $k_i$ is externally tangent to $k_0$ at point $T_i$, and $k_i$ is externally tangent to $k_{i+1}$ at point $S_i$ ($k_5$=$k_1$). Let $O$ be the center of $k_0$. Let the lines $T_1 T_3$ and $T_2 T_4$ meet at $T$, and let the lines $S_1 S_3$ and $S_2 S_4$ meet at $S$. Prove that the points $O$, $T$ and $S$ are collinear.

Denote by $u_n$ the $n$th Fibonacci number ($u_1$=$u_2$=1, $u_{n+1}$=$u_n$+$u_{n-1}$). Prove that if $a$,$b$,$c$>1 are integers such that $a$ divides $u_b$, $b$ divides $u_c$ and $c$ divides $u_a$, then 5 divides $a$, $b$ and $c$, or 12 divides $a$, $b$and $c$.

Two parabolas, $\mathcal{P}_1$ and $\mathcal{P}_2$ have the same focus point. The directrix of $\mathcal{P}_1$ meets $\mathcal{P}_2$ at points $A$ and $B$. The directrix of $\mathcal{P}_2$ meets $\mathcal{P}_1$ at $C$ and $D$. Show that the points $A$, $B$, $C$ and $D$ are concyclic.

Show that for every closed polygonal disk $\mathcal{K}$ there exists a real number $\alpha$ such that for arbitrary positive integer $n$ and any points $A_1,\ldots,A_n\in\mathcal{K}$ there is a point $X\in\mathcal{K}$ satisfying $\frac{|XA_1|+\ldots+|XA_n|}{n} = \alpha$.

Let $q \ge$1 be an integer. Prove that there is an integer $C_q$ such that for every finite set $A$ of integers |$A$+$q^. A$|$\ge$($q$+1)|$A$|-$C_q$ holds. ($A$+$q^. A$ is the set of those integers that can be expressed as $a$+$qa$' with some $a$,$a$'$\in A$.)

Let ABC be a non-equilateral triangle. Consider those equilateral triangles XYZ whose vertices $X$, $Y$ and $Z$ lie on the lines $BC$, $CA$ and $AB$, respectively. Show that the locus of centers of such triangles XYZ is a pair of parallel lines which are perpendicular to the Euler line of the triangle ABC .

Let $\alpha$ be a real number. For every positive integer $q$, let $N_q(\alpha)=\min \left\{{\left|\alpha-\frac pq\right|} \colon {p \in \mathbb{Z}}\right\}$, that is, the distance from the closest fraction that can be represented with a denominator of $q$ (not necessarily cancelled to lowest terms). Show that the sequence $a_k ={\frac{1}{\log k} \sum\limits_{q=1}^{k}} N_q(\alpha)$ is convergent.

There is given a ($2^n$-1)×($2^{n+1}$-1) table with integer entries. Prove that it is possible to delete $2^n$-1 columns from the table in such a way that in each row of the remaining ($2^n$-1)×$2^n$ subtable the sum of entries is even.

There are 2013 discs in a sack, numbered from 1 to 2013. How many discs need to be drawn without replacement to be certain that there will be two among them with numbers that add up to a multiple of 7?

Find all four-digit perfect squares in which the first two digits are equal and the last two digits are also equal.

In a right-angled triangle, the segment of the perpendicular bisector of the hypotenuse between the lines of the legs is equal in length to the hypotenuse. Determine the angles of the triangle.

We have two containers with volumes of $p$ litres and of $q$ litres. It is allowed to fill up completely or empty either container at a tap, to fill up either container completely from the other one, and to pour the entire content of one into the other. Prove that if $p$>$q$ are relatively prime positive integers, and $s$is a positive integer such that $s \le p$, then it is possible to achieve that one container have exactly $s$ litres of water in it.

Prove that there exists a set of infinitely many positive integers such that no finite subset of them add up to a perfect square.

The area of a regular heptagon is 4095 units. $A$ and $B$ are two consecutive vertices, and $K$ is the centre of the heptagon. $P$ is a point lying on the circumscribed circle of triangle ABK in the interior of the heptagon. Given that $\frac{AP-PB}{AB}=\frac{2}{3}$, what may be the area of triangle ABP ?

Show that if $a$, $b$ and $c$ are positive numbers then $a^{\frac 34}+b^{\frac 34}+c^{\frac 34} >{(a+b+c)}^{\frac 34}$.

The axes of two parabolas are perpendicular to each other. Show that the intersections of the two curves lie on a circle.

The length of the base edges of a right regular pyramid of (2$k$+1) sides is 1, the centre of the base is $O$, and the length of the altitude $OB$ is $m$. The pyramid is reflected in the midpoint of $OB$. Find the volume of the intersection of the two pyramids.

The circle $k$ touches the circle $\ell$ internally at point $P$. A line $p$ passing through $P$ intersects the circles again at the points $K$ and $L$, respectively. $u$ is the tangent drawn at a point $U$ of circle $k$. One intersection of $u$ with the circle $\ell$ is $V$, and the intersection of lines $KU$ and $LV$ is $T$. Determine the locus of the point $T$ as $U$ traverses the circle $k$. (Consider both intersections of $u$ and $\ell$; if $U$=$K$, the line $KU$ is the tangent to $k$at $K$. Analogously, if $V$=$L$ then $LV$ is the tangent drawn to $\ell$ at $L$.)

A digit is written on each face of two cubes such that the number of any day of a month is obtained by placing the two cubes next to each other in an appropriate way. (One-digit numbers are written as usual, e.g. 01. A 9 may also be obtained by turning a 6 around.) In how many different ways can this be done if the relative positions of numbers on the same cube are not considered?

The radius of the circumscribed circle of an isosceles triangle of unit base is also unity. The diameter parallel to the base of the triangle cuts off a smaller triangle. Find the exact lengths of the base and legs of the smaller triangle.

Solve the equation 36$a^4$+$b^4$=9$c^4$+4$d^4$ on the set of integers.

If a token is inserted in a slot machine, it will roll a regular die and show the result. Then the player may choose to either accept the prize, which is 100 times the number rolled, and thereby end the game, or to insert a second token. In the latter case, the machine rolls the die again, and then the prize is 100 times the product of the two numbers rolled. The game then ends, there is no more choice offered. How much should a token cost so that, in the long run, the slot machine make a profit for its owner?

In a triangle ABC , tan $\alpha$=2, tan $\gamma$=1, and $b$=12. The midpoints of the sides opposite to $A$ and $B$ are $F_a$ and $F_b$, and the feet of the corresponding altitudes are $T_a$ and $T_b$, respectively. Prove that the centroid and the orthocentre of triangle ABC are collinear with the intersection of $T_a F_b$ and $F_a T_b$.

$A \ne B$ are interior points of the quadrant $XY$ centred at $O$. The parallels drawn to the line $OX$ through the points $A$, $B$ intersect the radius $OY$ at the points $A_Y$ and $B_Y$. The lines drawn parallel to line $OY$ intersect the radius $OX$ at $A_X$ and $B_X$. Determine the total area of the quadrilaterals $AA_X B_X B$and $AA_Y B_Y B$ as a function of the length $AB$.

Let $n \ge$2 be an even number, and let 0<$k$<$n$ be an integer. Find the smallest number $e$ with the following property: If a simple graph on $n$ points has at least $e$ edges then it contains $k$ pairwise edge-disjoint perfect matchings.

The altitudes of a tetrahedron ABCD all pass through the point $M$. Let $R$ denote the radius of the circumscribed sphere. Show that $MA^2$+$MB^2$+$MC^2$+$MD^2$=4$R^2$.

Given the positive numbers $a$, $b$ and $c$, determine the values of the non-negative variables $x$, $y$, $z$ such that the expression $a^2\frac{x}{y+z} + b^2\frac{y}{z+x} + c^2\frac{z}{x+y}$ is a minimum.

$C$ is an interior point of a segment $AB$ of line $e$. $k_1$, $k_2$ and $k_3$ are semicircles drawn over the line segments $AB$, $AC$ and $CB$ on the same side of line $e$. The midpoints of the arcs of the semicircles are $F_1$, $F_2$ and $F_3$. A circle touches semicircle $k_1$ on the inside at $E$, and also touches the semicircles $k_2$ and $k_3$ on the outside. Show that the midpoint $M$ of the line segment $AB$ and the points $C$, $F_1$, $F_2$, $F_3$ and $E$ are all concyclic.

Find all integers $n$ such that | 2$n^3$-6$n^2$+4$n$+3 | is a prime.

The diagonals of a symmetric trapezium are perpendicular to each other. What may be the ratio of the perimeter of the trapezium to its midline?

$f$ is a function defined on the set of natural numbers such that $f$(1)$+2^2 f$(2)$+3^2 f$(3)+...+$n^2 f$($n$)=$n^3 f$($n$) for all positive integers $n$. Given that $f$(1)=2013, find the value of $f$(2013).

Given $n$ points in the interior of a regular triangle $\mathcal{T}$ , prove that $\mathcal{T}$ can be covered with 2$n$+1 regular triangles whose sides are parallel to the sides of $\mathcal{T}$ such that they contain none of the $n$ points in their interior. Also prove that the covering is not necessarily possible with 2$n$ triangles.

A wildfire spreads at a speed of 1 km/h in all directions. When the fire has burnt a circle of radius 1 km, an excavator arrives at the edge of the fire to dig a ditch to stop the spreading of the fire. The excavator can proceed at a maximum speed of 14 km/h. Find a possible path for the excavator such that the fire should not burn a total area greater than 13 $km^2$.

In a dice game, the player rolls three dice simultaneously, and then he may roll two more times any number of his dice (0, 1, 2 or 3). The player wins the game if all three dice have the same number on top after the last roll. What is the best strategy, and with that strategy what is the probability of winning?

$k$ is the circumscribed circle of a cyclic quadrilateral ABCD . The centre of the inscribed circle of triangle ABC is $P$, and that of triangle ABD is $Q$. Let $E$ denote the midpoint of arc $BC$, and let $F$ denote the midpoint of arc $DA$ of the circle $k$. Prove that $PQ$ is parallel to $EF$.

Let $n$ be a positive integer. Prove that $\sum_{i=1}^{n} 2i\cdot \binom{2n}{n-i} =n\cdot \binom{2n}{n}$ .

Prove that the midpoint of the line segment connecting the centres of two regular polygons of the same number of sides drawn on the sides $AC$ and $BC$ of a triangle ABC on the outside coincides with the centre of the regular polygon of the same number of sides drawn on the line segment connecting the midpoints of the sides $AC$and $BC$, on the side where the vertex $C$ lies.

Solve the equation ( $x^2$+100 ) $^2$= ( $x^3$-100 ) $^3$.

In a given $n$×$n$ table, $n$-1 fields are marked. A move consists in interchanging two rows or two columns. Is it always possible to carry out an appropriate sequence of such moves, so that all marked fields terminate under the main diagonal? (The main diagonal of a table connects the top left and bottom right corners.)

Is there a positive integer that is a perfect power and has digits of 0 and 6 only in decimal notation?

$M$ and $N$ are points on the longest side, $AB$ of a triangle ABC , such that $BM$=$BC$ and $AN$=$AC$. The parallel drawn through point $M$ to the side $BC$intersects side $AC$ at $P$, and the parallel drawn through point $N$ to the side $AC$ intersects side $BC$ at $Q$. Prove that $CP$=$CQ$.

Solve the equation ( $x^2$-$x$-1 ) $^2$-$x^3$=5.

Point $P$ lies in the plane of a triangle ABC . The orthogonal projections of $P$onto the altitudes of the triangle are $A_1$, $B_1$ and $C_1$. Given that $AA_1$=$BB_1$=$CC_1$, determine the point $P$, and find the equal length of these three line segments.

200 line segments of unit length are placed in a 20×20 square. Prove that there exists a disc of unit diameter in the square such that it does not have a point in common with any of the line segments.

The function $f\colon \mathbb{R} \to \mathbb{R}$ satisfies ($x$-2)$f$($x$)-($x$+1)$f$($x$-1)=3. Evaluate $f$(2013), given that $f$(2)=5.

The centroid of a triangle ABC is $S$, and its circumcentre is $K$. The centres of the circumscribed circles of the triangles BCS , CAS and ABS are $P$, $Q$and $R$. Prove that $K$ is the centroid of triangle PQR .

There are $n$ prisoners in a prison. The guards were bored, so they invented the following game: In the courtyard, they place either a red cap or a blue cap on the head of each prisoner. The prisoners are then given some time to look at one another (each prisoner can see all caps but his own), and then each of them has to write down on a sheet of paper what colour he thinks his cap is. Those guessing correctly are allowed to walk in the courtyard the following day, too. What is the largest number $k$ for which the prisoners may follow an appropriate strategy to make it certain that at least $k$ of them are allowed to walk in the courtyard the following day?

Let $\mathcal K$ be the convex hull of the points with coordinates (1;1;1), (2;4;8), ..., ( 100;$100^2$;$100^3$ ) . How many vertices, edges and faces does $\mathcal K$ have?

The right-angled vertex of a right angled triangle is projected orthogonally onto the angle bisector of one of the acute angles. Prove that the projection lies on the midline parallel to the hypotenuse.

Some vertices of a regular nonagon are coloured red, the rest of them are coloured black. A triangle is said to be boring'' if all of its vertices are the same colour. Prove that there are two congruent boring'' triangles.

Solve the simultaneous equations below. $x$+$y$+$z$=3, $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{5}{12},$ $x^3$+$y^3$+$z^3$=45.

Is it possible for the sum of the reciprocals of 2013 different positive integers to be $a$) 2.013; $b$) 20.13?

What is the largest possible number of rays from a common starting point in the space such that they pairwise enclose obtuse angles?

Determine the minimum value of the expression $\sqrt{1-x+x^{2}}+\sqrt{1-\sqrt{3}\cdot x+x^{2}}\,$.

ABC is an isosceles right angled triangle of unit legs. $A_1$ is a point on side $AB$, $B_1$ is a point on side $BC$, and $C_1$ is a point on hypotenuse $CA$. What is the minimum possible length of $A_1 B_1$ if the triangles ABC and $A_1 B_1 C_1$ are similar?

Prove that the sum of the sines of the angles in a triangle cannot be smaller than the sum of the sines of the doubles of the angles.

Prove that every power of 2 that is larger than 4 can be represented in the form $a^2$+7$b^2$, where $a$ and $b$ are positive odd numbers.

$P$ is the point on base $AB$ of a symmetric trapezium ABCD for which $AP$-$BP$=$AC$-$BC$. The perpendicular drawn to $AB$ at $P$ intersects the lines $CD$, $AC$ and $BD$ at the points $Q$, $R$ and $S$, respectively. Let $k_1$ be the circle that touches the lines $AC$ and $BD$ at points $R$ and $S$, and let $k_2$ be the circle of diameter $PQ$. Prove that the circles $k_1$ and $k_2$ touch each other.

In this sentence, the number of numbers occurring once is $a_1$, the number of numbers occurring twice is $a_2$, ..., and the number of numbers occurring 2013 times is $a_{2013}$. Determine the values of $a_1, a_2, \ldots, a_{2013}$ such that the resulting statement is true. In how many different ways can that be done?

For what positive integers $k$ will $2^. 3^k$ be a perfect number?

Three angles of a cyclic quadrilateral are $\alpha$, 2$\alpha$ and 3$\alpha$. Determine the angles of the cyclic quadrilateral.

There are four points on the plane such that the distances of point pairs have exactly two different values, $a$ and $b$ where $a$>$b$. Find the possible values of $a$/$b$.

Solve the simultaneous equations $x^3$=5$x$+$y$, $y^3$=5$y$+$x$.

Given five points in the plane, no three of which are collinear. Show that it is possible to select three points out of them that form an obtuse-angled triangle.

The base of a regular four-sided pyramid is the unit square ABCD , its apex is $E$. $P$ is a point on base edge $AB$, and $Q$ is a point on lateral edge $EC$ such that $PQ$ is perpendicular to both $AB$ and $EC$. Furthermore, $AP$:$PB$=6:1. How long are the lateral edges?

The interior angle bisectors drawn from vertices $A$, $B$ and $C$ of triangle ABC intersect the circumscribed circle at the points $D$, $E$ and $F$, respectively. The intersections of the sides of triangles DEF and ABC , starting from $A$, towards $B$, are $G$, $H$, $I$, $J$, $K$ and $L$ in this order. Show that the triangles DGL , EHI and FKJ are all similar.

The network of roads in the city of Icosapolis corresponds to the edge graph of an icosahedron. The home of Iorgos is at one vertex of the icosahedron, while his favourite theater is situated at the opposite vertex. On his way home from the theatre after dark, he stops at every vertex he reaches, and after some hesitation decides which way to proceed. Assume that at any vertex the probability of meeting someone who will show him one possible direction to get him home along the least possible number of edges is $p$. Otherwise he will chose an edge at random. (He may as well turn back in the direction he came from.) For what value of $p$ is there a 50% chance that he will get home before getting back to the theatre?

Are there two distinct non-constant polynomials $p$ and $q$ such that $\big\{p(n)\colon n\in\mathbb{N}\big\} =\big\{q(n)\colon n\in\mathbb{N}\big\}$?

In a right-angled triangle ABC , a semicircle is drawn over the leg $AC$, on the side where the triangle lies. A tangent is drawn to the semicircle at the point $E$where it intersects the hypotenuse. The tangent intersects the leg $BC$ at point $D$. Prove that triangle EBD is isosceles.

Solve the equation $\sqrt{x+14-\sqrt{4x-8}} - \sqrt{x-1-\sqrt{4x-8}} = 3$.

Prove that if $n$+1<$k$<2$n$ then $n$ distinct lines cannot divide the plane into $k$ parts.

Find all positive integers for which $\frac{3}{\sqrt{x}} + \frac{2}{\sqrt{y}} = \frac{1}{\sqrt{2}}$.

The squares ABDE , BCFG and CAHI are drawn over the sides of a triangle ABC , on the outside. The triangles DBG , FCI and HAE are completed to form the parallelograms DBGJ , FCIK and HAEL . Prove that $AKB\sphericalangle +BLC\sphericalangle + CJA\sphericalangle =90^{\circ}$.

Determine all functions $f\colon \mathbb{R} \backslash \{0, 1\} \to \mathbb{R}$ such that $f\left(\frac{x-1}{x}\right) +f\left(\frac{1}{1-x}\right)=2-2x$ .

There are $n$ prisoners in a prison. Since the guards are bored, they invent the following game: Either a red hat or a blue hat is placed on the head of each prisoner such that no one can see the colour of the hat on their own heads. Then the prisoners are allowed to look at one another (everyone can see everyone else's hat). Finally, each of them guesses the colour of their own hat, and writes it down on a sheet of paper. If all answers are correct, the prisoners may go for a walk in the courtyard. What strategy should they agree on, so that the probability of a walk in the courtyard is a maximum?

Each of two terahedra has three edges of length $a$ and three edges of length $b$>$a$. For what values of $b$/$a$ does it follow from this information that the two tetrahedra are congruent?

Prove that a regular octagon cannot be divided into parallelograms of equal area.

Given that the events $A_2$,$A_3$,...,$A_n$ are independent and $P(A_i)=\frac{1}{2i^2}$. What is the probability that an odd number of events occur out of $A_2$,$A_3$,...,$A_n$?

Solve the following simultaneous inequalities on the set of pairs of integers: 11>2$a$-$b$, 25>2$b$-$a$, 42<3$b$-$a$ and 46<2$a$+$b$.

Prove that among any seven vertices of a regular 27-sided polygon there are four that form a trapezium.

According to the tradition of the ancient tribe of Head Counters, every year is designated either as lucky or as sinister. For example, 2013 is a lucky year since the first 2013 positive integers can be partitioned into at least two classes such that each class contains the same number of elements and the sum of the elements is also equal in each class. If such a partition does not exist, the year is said to be sinister. Which years are sinister?

Six distinct points lie on a circle. Three points are selected, and the orthocentre of their triangle is connected to the centroid of the triangle formed by the other three points. Prove that all line segments obtained in this way are concurrent.

A certain cyclic quadrilateral is also a circumscribed quadrilateral. Given three sides, construct the quadrilateral.

Represent those pairs ($a$;$b$) of real numbers in the rectangular coordinate plane for which the polynomial $x$($x$+4)+$a$ ( $y^2$-1 ) +2$by$ of two variables can be expressed as a product of two first-degree polynomials.

The sides of a triangle form a geometric progression, and its angles form an arithmetic progression. Prove that the triangle is equilateral.

Given two perpendicular skew lines and an acute angle $\alpha$, determine the number of lines that intersect both given lines and encloses an angle of $\alpha$ with each.

Let $d$($n$) denote the number of positive factors of the positive integer $n$. Determine those numbers $n$ for which $d$($n^3$)$=5^. d$($n$).

The points $D$ and $E$ lie on the line segment $AB$. In the same half plane, a regular triangle is drawn over each of the line segments $AD$, $DB$, $AE$ and $EB$. The third vertices are $F$, $G$, $H$ and $I$, respectively. Prove that if the lines $FI$ and $GH$ are not parallel, then their intersection lies on the line $AB$.

A line is drawn through each of two opposite vertices of a parallelogram such that each line intersects the extensions of the sides of the parallelogram at two points. Prove that the four points of intersection form a trapezium.

Prove that if $x_1 \ge x_2 \ge x_3 \ge x_4 \ge x_5 \ge$0, then ${(x_1+x_2+x_3+x_4+x_5)}^2\ge \frac{25}{2} \big(x_4^2+x_5^2\big)$. What is the condition for equality?

$K$, $L$ and $M$ are points of sides $AB$, $AC$ and $BC$ of triangle ABC , respectively. $KL$ is parallel to $BC$, $KL$=$LC$, and $LMB\sphericalangle = BAC\sphericalangle$. Prove that $LM$=$AK$.

Solve the equation $4^{x}-3^{x}=\mathop{\rm tg} 15^{\circ}$.

The point $D$ lies in the interior of a triangle ABC . Lines $CD$, $AD$ and $BD$intersect sides $AB$, $BC$ and $CA$ at the points $E$, $F$ and $G$, respectively. The intersection of lines $EG$ and $AF$ is $H$, and that of lines $EF$ and $BG$is $I$. Show that the lines $AB$, $FG$ and $HI$ are concurrent.

Is there a natural number $n$ for which the numbers $1^{10},2^{10},\ldots, n^{10}$ can be divided into 10 sets, such that the sum of the numbers in each set is the same?

Two points and a circle $k$ all lie on the surface of the same sphere. The circle passes through exactly one of the two points. How many circles are there on the sphere that pass through both points and are tangent to the circle $k$?

Let $\alpha$ be an irrational number. For every positive integer $q$, let $N_q(\alpha)=\min \left\{{\left|\alpha-\frac pq\right|} \colon {p \in \mathbb{Z}}\right\}$, that is, the distance from the closest fraction that can be represented with a denominator of $q$ (not necessarily cancelled to lowest terms). Show that there exists a $k$ such that $\sum\limits_{q=1}^{k} N_q(\alpha)>1$.

A square made up of an odd number of small squares is drawn on squared paper. We want to draw a closed polygon along grid lines such that it passes through every lattice point in the interior or on the boundary of the large square exactly once, and it does not leave the large square. Express the length of the polygon in terms of the length of the side of the large square.

An arbitrary interior point $P$ of a convex quadrilateral ABCD is connected to the midpoints $E$, $F$, $G$, $H$ of the sides $AB$, $BC$, $CD$, $DA$, respectively. Prove that the sum of the areas of the quadrilaterals AEPH and CGPF is equal to the sum of the areas of the quadrilaterals BFPE and DHPG . (Any of them may be a degenerate quadrilateral.)

A square of unit diagonal is cut along one of the diagonals, and the two parts are moved towards each other along the line of the other diagonal. What is the maximum possible area of the overlapping region?

Is there an arithmetic progression in which the sum of the first $n$ terms is $n$, the sum of the first 2$n$ terms is $n^2$, and the sum of the first 3$n$ terms is $n^3$?

A die is rolled three times. What is the probability that the product of the resulting numbers is 12?

For what value of the real parameter $a$ will the equation $ax^2+a^2 x+a= \frac 1a$ have two equal roots?

A kite-shaped plot of land has three interior angles of $80^o$. Its area is 900 $m^2$. What length of fencing is needed around the whole plot?

Consider all non-congruent rectangles whose side lengths are two distinct elements selected from the set $\{1; 2; \ldots; 100\}$. What is the total area of all such rectangles?

What is the remainder if the sum $2012^{2013} +2013^{2012}$ is divided by $2012^.$2013?

The real numbers $a$, $b$, $c$ in the equation $\frac{x-b-c}{a} +\frac{x-a-c}{b} +\frac{x-a-b}{c}=10$ satisfy $a$+$b$+$c$=0, abc =-48, and $bc$+$ac$+$ab$=-28. Solve the equation.

A circle is drawn over the shorter diagonal $AC$=$a$ of a parallelogram ABCD . The intersections of the circle and the parallelogram form the hexagon AIJCKL , whose sides have lengths $\frac a2$, $b$, $b$, $\frac a2$, $b$, $b$ in this order. Determine the sides and angles of the parallelogram.

For what number pairs $x$, $y$ will the numbers $x$, $x^{log x}$, $y^{log y}$, ${(xy)}^{\log (xy)}$ be consecutive terms of a geometric progression? (log denotes decimal logarithm.)

ABCD is a square of area 2 $dm^2$, and ABCDE is a right pyramid of height 3 dm. Let $H$ be the point of edge $AE$ lying closer to $A$ that divides the edge 1:2, and let $G$ be the point of edge $CE$ lying closer to $E$ that divides the edge 1:2. Let $F$ be the midpoint of edge $BE$. Find the area of the intersection of the pyramid with the plane of triangle HFG .

How many seven-digit natural numbers are there in which the digits, read from left to right, form a nondecreasing sequence (e.g. 2444689)?

A prime number $p$ is four greater than the square of an integer $n$ and the double of $p$ is one less than the cube of $n$. Find $p$.

The line segments $AA_1$, $BB_1$, $CC_1$ are the angle bisectors of a triangle ABC . Through the endpoint of the bisector of each angle, draw parallels to the lines forming the angle. Consider the segments of these six lines that lie in the interior of the triangle. Prove that the sum of their lengths cannot be longer than the perimeter of the triangle.

Prove that $a\sqrt{1-b^2}+b\sqrt{1-a^2}\le 1$.

Consider a sphere of diameter $d$, a cylinder whose diameter and height are both $d$, and a cone of base diameter $d$. What may be the height of the cone if the volumes of the three solids form an arithmetic progression in some order?

Agnes was born on 25 March and her brother Peter was born on 9 February. They would like to construct a third-degree function $f$ of integer coefficients, such that $f$(9)=2 and $f$(25)=3. Is there such a function?

One diagonal of a trapezium is 7 cm long. It divides the other diagonal into pieces of lengths 4.5 cm and 6 cm. The length of the shorter leg of the trapezium is 5 cm. Find the area of the trapezium.

Consider the line $y=\frac72 x$ . Find the distance from the line to those points of integer coordinates that are the closest to the line but do not lie on the line.

The edges of a cube are coloured in three different colours such that parallel edges are the same colour. The points dividing the edges 1:2 are marked. Two marked points of each colour are selected at random. What is the probability that they are all coplanar? (It is not required to prove that the six points of a selection are coplanar.)

Define the operation $\therefore$ as follows on the set of real numbers: $a\therefore b=(a-2)(b-2)$. Is this operation associative?

$a$, $b$, $c$, $d$, $e$ are five consecutive integers in increasing order. The dimensions of a cuboid are $a$, $b$, $c$. For what values will the diagonal of the cuboid be the hypotenuse of a right-angled triangle with legs $d$ and $e$?

For what positive integer $n$ will the number 1!+3!+...+(2$n$-1)! be a perfect square?

Bill wants to buy a glass of soda from a vending machine for 60 forints (HUF, Hungarian currency). He has five 10-forint coins and four 20-forint coins in his pocket. He pulls out coins at random. What is the probability that by pulling out four coins in a row he will get exactly 60 forints from his pocket?

Lily is making pudding by her grandmother's recipe. She has a 42×36-cm baking tin and glass bowls of diameter 10 cm. She places the bowls in the tin, right next to each other, then fills the tin with water and bakes the pudding. Her grandmother always made 12 bowls of pudding in this way, but Lily can make more. How?

Investigate which of the squares inscribed in an acute-angled triangle has maximum side.

Prove that (sin $\alpha$+1)(cos $\alpha$+1)<3 for all angles $\alpha$.

Solve the following simultaneous equations: $\frac{x^2 +2xy +3y^2}{3x^2 + 2xy +y^2} + \frac{3x^2 + 2xy +y^2}{x^2 +2xy +3y^2} =2;$ 3$x$-2$y$=1.