KöMaL 2011

Is it possible to split the set {1,2,...,2010} into pairwise disjoint, 5-element subsets in such a way that the sum of elements in each subset is divisible by 2011?

Let $f(x)=x^n+a_{n-1}x^{n-1} +\ldots+a_1x+a_0$ be a polynomial with integer coefficients, and let $d_1,\ldots,d_n$ be pairwise distinct integers. Suppose that for infinitely many prime numbers $p$ there exists an integer $k_p$ for which $f(k_p+d_1)\equiv f(k_p+d_2)\equiv\dots\equiv f(k_p+d_n)\equiv0\pmod{p}.$ Prove that there exists an integer $k_0$ such that $f(k_0+d_1)=f(k_0+d_2)=\ldots=f(k_0+d_n)=0.$

In the Cartesian plane call those regions as strips which are bounded by two parallel lines. Define the width of a strip as the side length of the inscribed squares, with sides being parallel to the coordinate axes. Prove that if finitely many strips cover the unit square 0$\le x$,$y \le$1 then the sum of widths of those strips is at least 1.

Find all real numbers $p$ for which the equation $x^3$+3$px^2$+(4$p$-1)$x$+$p$=0 has two real roots with difference 1.

Let $a$, $b$ and $n$ be positive integers and let $C$ be a finite set of integers. Suppose that every positive integer can be represented in the form $ax^n$+$by^n$+$c$with some positive integers $x$, $y$ and $c \in C$. Find the possible values of $n$.

There is given a circle $k$ on the plane, a chord $AB$ of $k$, furthermore four interior points, $C$, $D$, $E$ and $F$, on the line segment $AB$. Draw an arbitrary chord $X_1 X_2$ of $k$ through point $C$, a chord $Y_1 Y_2$ through $D$, a chord $U_1 U_2$ through $E$, finally a chord $V_1 V_2$ through $F$ in such a way that $X_1$, $Y_1$, $U_1$ and $V_1$ lie on the same side of the line $AB$, and $\frac{AX_1\cdot BX_2}{X_1X_2} =\frac{AY_2\cdot BY_1}{Y_1Y_2} =\frac{AU_1\cdot BU_2}{U_1U_2} =\frac{AV_2\cdot BV_1}{V_1V_2}$ holds. Let $Z$ be the intersection of the lines $X_1 X_2$ and $Y_1 Y_2$, and let $W$be the intersection of $U_1 U_2$ and $V_1 V_2$. Show that the lines $ZW$ obtained in this way are concurrent or they are parallel to each other.

There are given 2$n$+1 points $A_1,A_3,\ldots,A_{2n+1},A_{2n},A_{2n-2},\ldots, A_2$ ($n \ge$2) lying on a circle in this order such that $\angle A_1A_2A_3 = \angle A_2A_3A_4 = \ldots = \angle A_{2n-1}A_{2n}A_{2n+1} = \frac{90^\circ}{n}$ . The lines $A_2A_3,A_3A_4,,\ldots, A_{2n-1}A_{2n}$ split the line segment $A_1 A_{2n+1}$ into 2$n$-1 part; denote their lengths by $\ell_1,\ell_2,\ldots,\ell_{2n-1}$, respectively. Prove that $\ell_1^2-\ell_2^2+\ell_3^2-\ell_4^2+-\ldots-\ell_{2n-2}^2+\ell_{2n-1}^2 = 0$.

Prove that for every positive integer $k$ there is a positive integer $N_k$ (depending only on $k$) such that whenever $\mathcal{C}$ is a set system whose elements are at most $k$-element sets such that every two elements of $\mathcal{C}$ have an element in common, then there exists a set $A$ with at most $N_k$ elements such that every two elements of $\mathcal{C}$ and $A$ have an element in common.

Prove that there exist a real number $c$>0 such that whenever each of the numbers $a_0,a_1,\ldots,a_n$ are 1 or -1 and the polynomial ($x$-1)$^k$ divides the polynomial $a_0+a_1x+\ldots+a_nx^n$ then $k$<$c^. ln^2$($n$+1).

Determine all triples ($a$,$b$,$c$) of positive integers for which abc +1 divides $a^2$+$b^2$.

The sides of a triangle are $a$, $b$ and $c$, the lengths of the corresponding medians are $s_a$, $s_b$ and $s_c$, respectively. Prove that $\frac{s_as_b}{a^2+b^2} + \frac{s_bs_c}{b^2+c^2} + \frac{s_cs_a}{c^2+a^2} \ge \frac98$.

There is given a simple graph $G$ with vertices $v_1$,...,$v_n$. $H_1$,...,$H_n$ are sets of nonnegative integers such that for every $i$=1,2,...,$n$, the cardinality of $H_i$ is at most half of the degree of $v_i$. Prove that $G$ contains a subgraph $G$' with the same vertices such that the degree of $v_i$ in $G$' is not an element of $H_i$ for any $i$.

The positive real numbers $a$, $b$, $c$, $d$ satisfy $a$+$b$+$c$+$d$= abc + abd + acd + bcd . Prove that ${(a+b)(c+d)}+(a+d)(b+c) \ge 4\sqrt{(1+ac)(1+bd)}.$

The edges of the complete graph on $n$ vertices are labeled by the numbers $1,2,\dots,\binom{n}{2}$ in such a way that each number is used exactly once. Prove that if $n$ is sufficiently large then there exists a (possible cyclic) path of three edges such that the sum of the numbers assigned to these edges is at most 3$n$-1000. ( Kolmogorov Cup, 2009; a problem by I. Bogdanov, G. Chelnokov and K. Knop )

In the 3-dimensional hyperbolic space there are given a plane $\mathcal{P}$ and four distinct lines $a_1$, $a_2$, $r_1$, $r_2$ in such positions that $a_1$ and $a_2$ are perpendicular to $\mathcal{P}$ , $r_1$ is coplanar with $a_1$, $r_2$ is coplanar with $a_2$, finally $r_1$ and $r_2$ intersect $\mathcal{P}$ at the same angle. Rotate $r_1$ around $a_1$ and rotate $r_2$ around $a_2$; denote by $\mathcal{S}_1$ and $\mathcal{S}_2$ the two surfaces of revolution they sweep out. Show that the common points of $\mathcal{S}_1$ and $\mathcal{S}_2$ lie in a plane.

Find all prime numbers $p \ge$3 for which 1+$k$($p$-1) is prime for every integer $1\le k\le\frac{p-1}2$.

Let $A_1 A_2 A_3$ be a non-equilateral triangle, let point $M$ be its orthocenter, let $F$ be its Feuerbach point, and let $k$ be the circumcircle of the triangle. For $i$=1,2,3 denote by $k_i$ the circle that is internally tangent to $k$ and tangent to the sides $A_i A_{i+1}$ and $A_i A_{i+2}$. (The indices are considered modulo 3, i.e. $A_4$=$A_1$ and $A_5$=$A_2$.) Let $T_i$ be the point of tangency between $k$ and $k_i$. Prove that the lines $A_1 T_1$, $A_2 T_2$, $A_3 T_3$ and $MF$ are concurrent.

The elements of the set $H$ are finite sequences whose elements are from {1,2,3}. Suppose that no element of $H$ is a subsequence of any other element of $H$. Show that $H$ is finite.

We have 1000 coins, but 100 of them are counterfeit. We know the weight of the real coins and it is known that the counterfeit coins are lighter than the real coins, but they may have different weights. Using a scale we want to find a counterfeit coin. Every time we can measure the total weight of a subset of the coins; the result indicates whether that set contains a counterfeit coin or not. How many measures are needed to find a counterfeit coin for sure?

The sum of the probabilities of some (finitely many) independent events is 4. Prove that with probability higher than 1/2, the number of occurred events is congruent to 0 or 3 modulo 4.

A circle $k$ with center $O$ and four distinct fixed points $A$, $B$, $C$, $D$ lying on it are given. The circle $k$' intersects $k$ perpendicularly at $A$ and $B$. Let $X$ be a variable point on the line $OA$. Let $U$, other than $A$, be the second intersection of the circles ACX and $k$'. Let $V$, other than $A$, be the second intersection of the circles ADX and $k$'. Let $W$, other than $B$, be the second intersection of the circle BDU and the line $OB$. Finally, let $E$, other than $B$, be the second intersection of the circles BVW and $k$. Prove that the location of the point $E$ is independent from the choice of the point $X$.

Prove that whenever $a$>$b$>1 are integers such that $a$+$b$ divides $ab$+1 and $a$-$b$ divides $ab$-1 then $a<b\sqrt3$.

Show that $\frac{1}{\sin^2{\dfrac{\pi}{4k+2}}} + \frac{1}{\sin^2{\dfrac{3\pi}{4k+2}}} + \frac{1}{\sin^2{\dfrac{5\pi}{4k+2}}} + \ldots + \frac{1}{\sin^2{\dfrac{(2k-1)\pi}{4k+2}}} = 2k(k+1)$ for every positive integer $k$.

A tetrahedron $OA_1 A_2 A_3$ is given. For $i$=1,2,3 let $B_i$ be a point in the interior of the edge $OA_i$ and let $C_i$ be a point on the ray $OA_i$, beyond $A_i$. Suppose that the polyhedron bounded by the six planes $OA_{i+1} A_{i+2}$ and $B_i A_{i+1} A_{i+2}$ ($i$=1,2,3) circumscribes a sphere, and the polyhedron bounded by the planes $B_i A_{i+1} A_{i+2}$ and $C_i A_{i+1} A_{i+2}$ also circumscribes a sphere. Prove that the polyhedron bounded by the planes $OA_{i+1} A_{i+2}$ and $C_i A_{i+1} A_{i+2}$ also circumscribes a sphere.

Prove that $\prod_{i=1}^n \left(1+\frac1{x_1+\ldots+x_i}\right) + \prod_{i=1}^n \left(1-\frac1{x_i+\ldots+x_n}\right) \le n+1$ holds for arbitrary real numbers $x_1,\ldots,x_n\ge 1$.

The diagonals of a circumscribed quadrilateral ABCD meet at $E$. Show that the incenters of the triangles ABE , BCE , CDE and DAE lie on a circle.

Is it true that every sufficiently large positive integer is the sum of a Fibonacci number and a prime number?

Five applicants are to be scheduled for job interviews at five given times. On an application form, each applicant was asked to mark two out of the five times. What is the probability that each of them can be interviewed at one of the two times they have marked?

Solve the following equation: $\frac{1+x^4}{{(1+x)}^4} = \frac{3}{4}$.

In triangle ABC , the foot of the altitude from vertex $A$ is $D$, the midpoint of the altitude from vertex $B$ is $E$, and the midpoint of the altitude from vertex $C$ is $F$. Show that $EDF\sphericalangle =CAB\sphericalangle$.

$a$) Is the inequality $PA$+$PB$<$CA$+$CB$ true for all interior points $P$ of an arbitrary triangle ABC ? $b$) Is the inequality $PA$+$PB$+$PC$<$DA$+$DB$+$DC$ true for all interior points $P$ of an arbitrary tetrahedron ABCD ?

Let $a$ and $b$ be integers. Find integers $c$ and $d$ such that $an$+$c$ and $bn$+$d$ are relative primes for all integers $n$.

Given the median drawn to a leg and the angle enclosed by this median and the other leg, construct an isosceles triangle.

A regular pentagon of side $a$ is rotated about a side. The volume of the resulting solid of revolution equals that of the solid obtained by rotating a regular pentagon of side $b$ about a diagonal. Determine the ratio $a$:$b$.

Express the exact value of sin ( \pi$/2^{2011}$ ) only using digits of 1 and 2, the four basic operations and square roots.

Find the polynomial $p$ of two variables, such that $p(x+y,xy) = \sum_{k=0}^{20}x^{20-k}y^{k}?$

$P$ is an interior point of a convex polygon $\mathit K$ . Prove that is it possible to find four lines on the plane such that for each side of $\mathit K$ there is a line among the four for which this side lies in one of the open half planes and $P$ lies in the other open half plane formed by the line. Is the statement also true for only three lines instead of four?

If the numbering of the faces of an (ordinary cubical) die is correct, the sums of the numbers on the faces adjacent to any chosen face are equal. Is it possible to number the faces of a dodecahedral die 1 to 12 in the same way?

The lengths of two sides of a triangle are 12 and 18. The lengths of two sides of another, similar but not congruent, triangle are also 12 and 18. Find the lengths of the third sides of the triangles.

Given a line and a point on it, let $Z$ denote the set of those points of the line that are at integer distances from the given point. Prove that if $H$ is any three-element subset of $Z$, then it is possible to partition $Z$ into subsets congruent to $H$.

Two circles intersect at points $B$ and $C$. A line touches each of them, the points of tangency are $A_1$ and $A_2$. Prove that $\frac{A_1B}{A_1C} = \frac{A_2B}{A_2C}$ .

Regular triangles are drawn over the sides $AB$ and $BC$ of a parallelogram ABCD on the outside. Their third vertices are $E$ and $F$, respectively. Show that the sum of the angles CED and AFD is $60^o$.

Find all real numbers $p$ such that the equation $x^3$-7$x$+$p$=0 has two real roots whose difference is 1.

Four points $A$, $B$, $C$ and $D$ on a circle are selected at random, independently of each other. What is the probability that the chords $AB$ and $CD$ intersect?

Is the inequality $PA$+$PB$+$PC$<$DA$+$DB$+$DC$ true for every interior point $P$ of a regular tetrahedron ABCD ?

Prove that the following inequality is true for all positive numbers $a_1$,$a_2$,...,$a_n$. $\left(\frac{a_1}{a_2+\dots+a_n}\right)^2+\left(\frac{a_2}{a_3+\dots+a_1}\right)^2+\ldots + \left(\frac{a_n}{a_1+\dots+a_{n-1}}\right)^2\ge \frac{n}{{(n-1)}^2}.$

Find all pairs $f$($x$), $g$($x$) of polynomials of real coefficients such that $f$($x$+1)$g$($x$-1)-$g$($x$+1)$f$($x$-1)=1.

In the lottery, 5 numbers are drawn out of the numbers 1 to 90. If the numbers drawn are always listed in increasing order, which number will occur the most frequently in the second place?

Let $a$ and $b$ denote positive numbers such that $a^3$+$b^3$=1. Show that $a^2$+$ab$+$b^2$-$a$-$b$>0. (Suggested by J. Pataki, Budapest)

The lengths of the parallel sides of a symmetric trapezium are $a$ and $c$. The midpoints of the legs are $E$ and $F$. Let $G$ denote the orthogonal projection of point $E$ on the line of the leg $BC$. What is the area of the trapezium if point $C$ divides the line segment $GF$ in a 1:2 ratio? ($C$ lies closer to point $G$.)

Prove that in any set of seven different positive integers there are three numbers such that the greatest common divisor of any two of them leaves the same remainder when divided by three. (Suggested by S. Kiss, Budapest)

Given a convex pentagon, construct a line that halves its area. (Suggested by T. Horváth, (Dunakeszi)

A rectangle is divided into at least two squares of different size. Prove that it is not possible for the lengths of the sides of all squares to be Fibonacci numbers. (Suggested by G. Damásdi, (Kecskemét)

The diagonals of a cyclic quadrilateral ABCD are not perpendicular. The feet of the perpendiculars dropped from the vertices $A$, $B$, $C$, $D$ onto the diagonals not passing through them are $A$', $B$', $C$', $D$', respectively. The intersections of lines $AA$' and $DD$', $DD$' and $CC$', $CC$' and $BB$', and finally, $BB$' and $AA$' are $E$, $F$, $G$, $H$, respectively. Prove that $A$'$B$'$C$'$D$' is a cyclic quadrilateral with the centre of its circumscribed circle lying at the intersection of the line segments $EG$ and $FH$. (Suggested by B. Bíró, (Eger)

Prove that if a quadrilateral can be covered with a disc of unit radius then the product of its sides cannot be more than 4.

In a tetrahedron $A_1 A_2 A_3 A_4$, for every interior point $P$ and for any order $i$, $j$, $k$, $l$ of the numbers 1, 2, 3, 4, the following inequality is true: $PA_i$+$PA_j$+$PA_k$<$A_l A_i$+$A_l A_j$+$A_l A_k$. Does it follow that the tetrahedron is regular?

Some subsets containing at least two elements of an $n$-element set are selected. Given that any two elements of the set occur together exactly once in some selected subset, prove that the number of the subsets selected is either 1 or at least $n$.

What is the minimum possible number of convex pentagons needed to put together a convex 2011-sided polygon out of them? (Suggested by J. Mészáros, Jóka)

Let $A$ be a positive integer, and let $B$ denote the number obtained by writing the digits of $A$ in reverse order. Show that at least one of the numbers $A$+$B$ and $A$-$B$ is divisible by 11. (Suggested by J. Mészáros, Jóka)

Given four distinct points $A$, $B$, $C$, $D$ on the plane, construct two circles touching each other on the outside such that one of them passes through $A$ and $B$, the other passes through $C$ and $D$, and their point of tangency lies on the line segment $BC$.

Prove that if the product of the positive numbers $x$, $y$ and $z$ is 1, then $\frac{z^3 + y^3}{x^2+xy+y^2} + \frac{x^3 + z^3}{y^2+yz+z^2} + \frac{y^3 + x^3}{z^2+zx+x^2} \ge 2.$ (Based on the idea of J. Szoldatics, Dunakeszi)

On the same side of a given line segment $AB$, there are at most six similar triangles one side of which is $AB$. Prove that the third vertices of all these triangles are concyclic.

Let $n$>1 be a positive integer. Show that $n^3$-$n^2$ is a factor of the binomial coefficient $\binom{n^2}{n+1}$. (Suggested by G. Holló, Budapest)

A regular tetrahedron is intersected with a plane perpendicular to one of its faces in line $e$. The lines of intersection of this plane with the other three faces enclose angles of $\varphi_1$, $\varphi_2$, $\varphi_3$ with $e$. Determine the possible values of he expression $\mathop{\rm tg}\nolimits^{2}\varphi_1 +\mathop{\rm tg}\nolimits^{2}\varphi_2 +\mathop{\rm tg}\nolimits^{2}\varphi_3$.

Let $S$($n$) denote the sum of the digits for any positive integer $n$. For what integers $k$greater than 1 is there a positive real number $c_k$ such that $S$($kn$)$\ge c_k ^. S$($n$) for all positive integers $n$? (Suggested by P. Erben, Budapest)

Let $a$ and $b$ denote the semi-axes of an ellipse. Let $O$ denote the centre, and let $P_1, P_2, \ldots,P_n$ be points on the curve such that each of the angles $P_i OP_{i+1}$ is $\frac\pi n$ . Prove that if $n$>1 then $\frac{2}{OP_1^2}+\frac{2}{OP_2^2}+ \dots +\frac{2}{OP_n^2}= \frac{n}{a^2}+\frac{n}{b^2}$.

Each vertex of a solid cube is cut off. The faces of the resulting polyhedron are 8 triangles and 6 heptagons. What are the possible numbers of vertices and edges of such polyhedra?

The reciprocals of the natural numbers 2 to 2011 are written on a blackboard. In each step, two numbers $x$ and $y$ are erased and replaced with the number $\frac{xy}{xy + (1-x)(1-y)}$. This step is repeated 2009 times, until a single number remains. What may the remaining number be? (Suggested by B. Kovács, Szatmárnémeti)

Let $a \ge b \ge c$>0. Prove that $\frac{a^{2}-b^{2}}{c} +\frac{c^{2}-b^{2}}{a} +\frac{a^{2}-c^{2}}{b}\ge 3a-4b+c$ . (Suggested by J. Mészáros, Jóka)

Find all positive integers $n$ such that $2^n$-1 and $2^{n+2}$-1 are both primes, and $2^{n+1}$-1 is not divisible by 7. (Suggested by S. Kiss, Budapest)

Let $M$ denote the orthocentre of the acute-angled triangle ABC , and let $A_1$, $B_1$, $C_1$, respectively, denote the circumcentres of triangles BCM , CAM , ABM . Prove that the lines $AA_1$, $BB_1$ és $CC_1$ are concurrent.

Solve the following equation: $\frac{3x+3}{\sqrt{x}} =4+\frac{x+1}{\sqrt{x^{2}-x+1}}$. (Suggested by J. Mészáros, Jóka)

Let $D$, $E$ and $F$, respectively, be points on sides $AB$, $BC$, $CA$ of a triangle ABC such that $AD$:$DB$=$BE$:$EC$=$CF$:$FA \ne$1. The lines $AE$, $BF$, $CD$ intersect one another at points $G$, $H$, $I$, respectively. Prove that the centroids of triangles ABC and GHI coincide. (Suggested by Sz. Miklós, Herceghalom)

Each of the circles $k_1$, $k_2$ and $k_3$ passes through a point $P$, and the circles $k_i$ and $k_j$ also pass through the point $M_{i,j}$. Let $A$ be an arbitrary point of circle $k_1$. Let $k_4$ be an arbitrary circle passing through $A$ and $M_{1,2}$, and let $k_5$ be an arbitrary circle passing through $A$ and $M_{1,3}$. Show that if the other intersections of the pairs of circles $k_4$ and $k_2$, $k_5$ and $k_3$, $k_4$ and $k_5$ are $B$, $C$ and $D$, respectively, then the points $M_{2,3}$, $B$, $C$, $D$ are either concyclic or collinear.

Let $a$, $b$, $c$ denote the lengths of the sides of a triangle, and let $u$, $v$, $w$, respectively, be the distances of the centre of the incircle from the vertices opposite to the sides. Prove that $(a+b+c) \left(\frac{1}{u}+\frac{1}{v}+\frac{1}{w}\right)\le 3\left(\frac{a}{u}+\frac{b}{v}+\frac{c}{w}\right)$. (Suggested by J. Mészáros, Jóka)

Prove that $\frac{1}{\sin^2{\frac{\pi}{14}}} + \frac{1}{\sin^2{\frac{3\pi}{14}}} + \frac{1}{\sin^2{\frac{5\pi}{14}}} = 24.$ (Suggested by B. Kovács, Szatmárnémeti)

Show that if there are three collinear points among any four selected from a set of $n$ points then at least $n$-1 points are collinear.

An international conference of polypodal creatures is organized on the top of the Glass Mountain. The numbers of legs of the participants, $a_1$,...,$a_n$, all are positive even numbers. The side of the Glass Mountain is slippery. Each of the creatures gathering at the base of the mountain can only get to the top if they wear special climbing shoes on at least half their feet. At least how many shoes are needed altogether to get all participants up to the top if a shoe must always be worn on a foot when it travels up or down the mountain?

Pacworm the cheese mite is sitting in the middle of a cheese cube of 5 cm edges. He eats a tunnel through the cheese. He always proceeds 1 cm parallel to an edge of the cube and then changes direction. While turning, he makes sure that he makes a 90-degree turn and that facing in the new direction he has more than 1 cm of cheese in front. He always chooses a new direction among the possibilities with equal probability. What is the probability that after travelling 5 cm he ends up in a position where there is exactly one edge at a distance of at most 0.8 cm?

Let $a$ and $b$ be the legs of a right-angled triangle, and let $m$ be the height drawn to the hypotenuse $c$. Which line segment is longer, $a$+$b$ or $m$+$c$?

Prove that if $x$, $y$ are non-negative numbers then $x^4 + y^3 + x^2 + y + 1 > \frac{9}{2}xy .$

Regular triangles ABD , BCE , CAF are drawn to the sides of a triangle ABC on the outside. Let the midpoints of line segments $DE$, $EF$, $FD$ be $G$, $H$, $I$, respectively. Prove that $BG$=$CH$=$IA$.

Let $p$ denote a positive prime number. Solve the equation $x^3 y^3$+$x^3 y^2$-$x^2 y^3$+$x^2 y^2$-$x$+$y$=$p$+2 on the set of integers.

Given the points different from the vertices where the exterior angle bisectors of a non-isosceles triangle intersect the circumscribed circle, construct the triangle.

$m$ points are chosen on the sides of a convex $k$-sided polygon, such that the resulting point set $\mathcal{P}$ of $n$=$k$+$m$ elements is centrally symmetric. Prove that there may be at most 2$n$-3 line segments of equal length among the $\binom{n}{2}$ line segments determined by the points of the set $\mathcal{P}$ . For what point sets $\mathcal{P}$ are there 2$n$-3 line segments of equal length?

Given three pairwise skew lines, determine the possible positions of the centre of a parallelepiped that has an edge lying on each line.

Let $x$ be an integer. Prove that if $\frac{4x+1-\sqrt{8x+1}}2$ is an integer then it is a square number.

ABCD is a convex quadrilateral, such that $AB$>$BC$>$CD$>$DA$. The inscribed circles of the triangles ABD and BCD touch the diagonal $BD$ of the quadrilateral at the points $E$and $F$, respectively. Show that $EF$=$GH$. (Suggested by Sz. Miklós, Herceghalom)

According to problem B. 4283. of this journal, if a 23×23 square is dissected into 1×1, 2×2 and 3×3 squares then at least one of the pieces must be a 1×1 square. Given that there is exactly one 1×1 piece, find its possible positions in the big square. (Suggested by Z. Gyenes, Budapest)

Solve the simultaneous equations {$x$}={$x^2$}={$x^3$} (where {$y$} denotes the fractional part of the number $y$, obtained by subtracting from $y$ the greatest integer not greater than $y$).

Is there a rational number such that by cancelling some digits out of its decimal form, the digits of $\pi$ are left?

In the cyclic hexagon ABCDEF , $AB$=$BC$, $CD$=$DE$ and $EF$=$FA$. Prove that the area of the triangle BDF is half the area of the hexagon.

$D$ is an interior point of side $AB$ of an acute-angled triangle ABC . The line drawn through $D$ parallel to side $AC$ intersects side $BC$ at $E$, and the line drawn through $D$parallel to side $BC$ intersects side $AC$ at $F$. The other intersection of the circles ADF and BDE is $G$. Prove that ABEF is a cyclic quadrilateral if and only if $G$ lies on the line segment $CD$. (Suggested by Sz. Miklós, Herceghalom)

Given four points $A$, $B$, $C$ and $D$ on a line in this order, consider all pairs of touching circles such that one circle passes through $A$ and $B$ and the other passes through $C$ and $D$. Determine the locus of the points of contact.

How many solutions does the equation $\sin \log_\pi x + \log_\pi \sin x =0$ have in the interval [0,20]?

A height of a pentagon is defined as the distance of a vertex from the opposite side. Let $P$be a pentagon with all angles equal to $108^o$ and all heights different. Show that it is possible to number the consecutive vertices of $P$ in an appropriate direction around the pentagon, such that the corresponding heights satisfy the inequalities $h_1$>$h_3$>$h_4$>$h_5$>$h_2$. (Suggested by P. Kevei and V. Vígh, Szeged)

There are chessmen standing on a few fields of a 20×20 chessboard. A chessman can be removed from the board if at least one half of the fields in its row or column are vacant. What is the minimum number of chessmen needed for an appropriate arrangement such that no chessman can be removed from the board?

Consider the vertices of a regular triangle, the centre of the triangle and all points that divide its sides in a 1:2 ratio. What is the largest number of points that can be selected out the points listed above, such that no three of them form a regular triangle? (Suggested by

Consider the Fibonacci sequence defined by the recursive formula $F_1$=$F_2$=1, $F_n$=$F_{n-1}$+$F_{n-2}$. Prove that if $k$<$m$, then $\sum_{i=k}^{m} F_i F_{i+3}$ is a composite number.

Given a cyclic quadrilateral, construct a point in the interior of each side, such that they form a convex quadrilateral of the smallest possible perimeter.

The inscribed circle of triangle ABC touches side $b$ at $F$ and side $a$ at $G$. Prove that the foot of the perpendicular dropped from $A$ onto the angle bisector of angle $\beta$ lies on the line $FG$. (By conventional notation , side $b$ lies opposite to vertex $B$ and angle $\beta$ is the angle at $B$, etc.)

Prove that if $n$, $k$ are positive integers then $\binom{n}{k}\cdot(n,k)$ is divisible by $n$.

$A_1$, $B_1$ and $C_1$ are given points on sides $BC$, $CA$ and $AB$, respectively, of the acute-angled triangle ABC . Mark a point $A_2$ on the circle $AC_1 B_1$. Let $B_2$ be the intersection (different from $C_1$) of the line $C_1 A_2$ with the circle $BA_1 C_1$. Furthermore, let $C_2$ be the intersection (different from $A_1$) of the line $A_1 B_2$ and the circle $CB_1 A_1$. Prove that the points $A_2$, $B_1$ and $C_2$ are collinear.

A sack contains $k$ red balls and $n$ green balls. The balls are drawn out one by one, without replacement, until the first red ball is drawn. How many draws are needed on average?

Is it possible to place a 1.05×1.05 square inside a unit cube?

Let $p$=3$n$+1 be a prime number. Is it possible for the numbers $1^3,2^3,\ldots,n^3$ to leave pairwise different remainders when divided by $p$?

There are chessmen standing on some fields of a 20×20 chessboard. The rule is that a chessman can be removed from the board if at least a half of the fields is vacant in its row or in its column. What is the maximum possible number of chessmen on the board if all of them can be removed in a sequence of such steps?

The angles of a polygon $A_1 A_2$...$A_{2n}$ are all equal, and its sides satisfy $A_1 A_2$=$A_3 A_4$=...=$A_{2n-1} A_{2n}$ and $A_2 A_3$=$A_4 A_5$=...=$A_{2n} A_1$. Prove that the polygon is cyclic.

$a$ and $b$ are non-negative numbers such that $a^3$+$b^3$=2$ab$. Does that imply $a^2$+$b^2 \le$1+$ab$?

Lines $e_1$ and $e_2$ are perpendicular, and $P$ is a point on one of their angle bisectors, different from the intersection. Lines $f$ and $g$ pass through $P$, and intersect the lines $e_i$ at points $F_i$ and $G_i$, respectively. Determine the locus of the intersection of the lines $F_1 G_2$ and $F_2 G_1$.

Let $k$ be a positive integer. How many non-negative solutions does the equation $\left[\frac xk\right] = \left[\frac x{k+1}\right]$ have?

In a triangle ABC , the lengths of sides $AC$ and $BC$ are fixed, and the angle at vertex $C$ is variable. The midpoint of side $AC$ is $M$, the midpoint of side $BC$is $N$, and the centre of the square drawn over the side $AB$ on the outside is $O$. For what value of $ACB\sphericalangle$ will the sum of the distances $OM$ and $ON$be a maximum?

Assume that $n$ is a positive integer and $2^n$+1 is a prime. What may be the remainder of this prime when divided by 240?

The teacher in the maths club had an at most $n$th-degree polynomial in mind. The students may ask the value of the polynomial at any point on the real axis they want. What is the minimum number of points they need, in order to be able to decide in all cases whether the function is even?

The axes of two right circular cones are parallel, and their apex angles are different. Prove that their common points all lie on the surface of a sphere.

Let $n$ denote a number divisible by 3. Every third term of the sequence $n$-1, $n$-2, ..., 2, 1 is cancelled. The subsequent pairs of terms of the remaining sequence are given alternating plus and minus signs: +($n$-1), +($n$-2), -($n$-4), -($n$-5), ... . Prove that the sum of the numbers obtained in this way is $n$.

Construct the triangle ABC out of the following parts given: the midpoint of side $AB$, the foot of the altitude drawn to side $AC$, and the angle $\alpha$ lying at vertex $A$.

A die is rolled $n$ times. What is the probability that there are two equal numbers among the results obtained?

A vertical wall of a building is a square of side 20 metres. The angles subtended by the vertical sides of the square at two points on the (level) ground are measured. The results are $40^o$ and $38^o$ at one point, and $32^o$ and $40^o$ at the other point. Find the distance between the two points.

The real numbers $x$, $y$, $z$, in this order, are the first three terms of a non-constant arithmetic progression. Given that cos $x$+cos $y$+cos $z$=1 and $\sin x + \sin y + \sin z = \frac{1}{\sqrt 2}$, find the tangent of the 12th term.

Solve the equation $\sqrt{x+a_n}=x-a_n$, where $a_n=\frac{n(n+1)}{2}$ for a positive integer $n$.

The measures of four interior angles of a pentagon are $120^o$. The lengths of the four consecutive sides opposite to these angles are 2, 8, 5, 5, in this order. How long is the fifth side?

Solve the following simultaneous equations on the set of real numbers: |$x$-1|+|$x$+$y$|=6, |$y$-1|+|$x$+$y$+1|=4.

In base-$n$ notation, a number has the form 2011, and in base-($n$+3) notation it is 537. Which number is it?

The forum of this journal on the internet has exactly 1000 registered members. Each member sends an e-mail to one other member. Prove that it is possible to select a group of 334 members such that within the group no one sent an e-mail to anyone else.

Prove that the radius of the inscribed circle of a right-angled triangle is smaller than three tenths of the longer leg.

The function $f$($x$)=$x^3$+$ax^2$+$bx$+$c$ ($a, b, c \in \mathbb{R}$ ) has at least two zeros. Prove that $a^2$>3$b$.

Find the three-digit number that is twelve times the sum of its digits.

The roots of a quadratic equation are the squares of two consecutive integers. The geometric mean of the two roots is 1 greater than the difference of the roots. Write down the quadratic equation in question.

$E$ is the point dividing side $AC$ of triangle ABC in a 3:1 ratio, such that $EC$ is the shorter part. The line passing through $E$ and the midpoint $F$ of side $BC$ intersects the line $AB$ at $D$. What percentage is the area of triangle ADE of the triangle ABC ?

Define words'' as strings of at most ten characters made out of the 26 letters of the English alphabet. (A word'' may as well consist of a single letter, and two words'' are considered different if they differ in at least one letter.) Prove that the number of all possible words'' obtained in this way is divisible by 27.

A steerable airship has two engines and a given supply of fuel. If both engines are operated, the airship covers 88 kilometres per hour. If the first engine were used only, the fuel would last 25 hours longer, but the distance covered per hour would only be 45 km. If the second engine were used only, the fuel would last 16 hours longer than with two engines, and the distance covered per hour would only be 72 km. In which case can the airship travel the longest distance?

Two regular polygons are said to be matching if the double of the interior angle of one of them equals the triple of the exterior angle of the other. Find all pairs of matching polygons.

The first digit of a six-digit number is transferred to the end of the number. Then the first digit of the resulting six-digit number is, again, transferred to the end of the number. The six-digit number obtained in this way is three times the previous number and $\frac 35$ times the original number. What is the original six-digit number?

The length of one side of a triangle is 8 cm, one of the angles lying on it is $60^o$, and the radius of the inscribed circle is $\sqrt 3$ cm. How long are the other two sides?

The point $P$(8;4) divides a chord of the parabola $y^2$=4$x$ in a 1:4 ratio. Find the coordinates of the endpoints of the chord.

There are $n$ coins on the table, with their tails side facing upwards. In each step, $n$-1 coins are turned over. Is it possible to achieve that all coins have heads facing upwards?