KöMaL 2010

There is given an acute triangle ABC in the plane. Denote by $A_1$, $B_1$ and $C_1$ the perpendicular feet of an arbitrary interior point $P$ of the triangle on the sides $BC$, $CA$and $AB$, respectively. Let the inradii of the triangles PAC $_1$, $PC_1 B$, PBA $_1$, $PA_1 C$, PCB $_1$ and $PB_1 A$ be $\varrho_1$,$\varrho_2$,...,$\varrho_6$, respectively. Determine the locus of those points $P$ for which $\varrho_1$+$\varrho_3$+$\varrho_5$=$\varrho_2$+$\varrho_4$+$\varrho_6$.

Let $p$($x$) be a polynomial with integer coefficients, and let $w$ be a complex number of unit absolute value. Prove that if the number $c$=$p$($w$) is purely real, then there is a polynomial $q$($x$) with integer coefficients for which $c=q\left(w+\frac1w\right)$.

Prove that there exist positive constants $c$ and $n_0$ with the following property. If $A$ is a finite set of integers, |$A$|=$n$>$n_0$, then |$A$-$A$|-|$A$+$A$|$\le n^2$-$cn^{8/5}$.

In space, there are given three prolate spheroids $\mathcal{E}_1$, $\mathcal{E}_2$ and $\mathcal{E}_3$ (ellipsoids which can be obtained by rotating ellipses about their major axes), and a plane $\mathcal{S}$ . The three ellipsoids share one of their foci. Suppose that for each $i$=1,2,3, the surfaces $\mathcal{E}_{i+1}$ and $\mathcal{E}_{i+2}$ have exactly two common points with the plane $\mathcal{S}$ , and denote by $\ell_i$ connecting these common points. Show that the lines $\ell_1$, $\ell_2$ and $\ell_3$ are either concurrent or parallel.

Let $p$>3 be a prime. Determine the last three digits of $\sum_{i=1}^{p} \binom{i\cdot p}{p}\cdot\binom{(p-i+1)p}{p}$ in the base-$p$ numeral system.

Prove that for arbitrary complex numbers $w_1$,$w_2$,...,$w_n$ there exists a positive integer $k \le$2$n$+1 for which $\mathop{\rm Re} \big(w_1^k+w_2^k+\dots+w_n^{k}\big) \ge 0$.

In space, there are given some vectors $u_1$,$u_2$,...,$u_n$ and $v$ such that |$u_1$|$\ge$1, ..., |$u_n$|$\ge$1 and |$v$|$\le$1, and $u_1$+...+$u_n$=0. Show that |$u_1$-$v$|+...+|$u_n$-$v$|$\ge n$.

Prove that for arbitrary integers 0<$r$<$k$<$t$ there exists a positive integer $N$($r$,$k$,$t$) which satisfies the following property: whenever $G$ is an $r$-uniform hypergraph with at least $N$($r$,$k$,$t$) vertices such that there is at least one hyperedge on any $k$ vertices, then $G$contains a complete subgraph with $t$ vertices. (A hypergraph is a graph whose edges are arbitrary subsets of the vertices. The graph is called r-uniform if all edges contain exactly $r$ vertices. An $r$-uniform hypergraph is complete if any $r$ of its vertices form an edge.)

In a cyclic quadrilateral ABCD , the points $O_1$ and $O_2$ are the incenters of triangles ABC and ABD , respectively. The line $O_1 O_2$ meets $BC$ and $AD$ at $E$ and $F$, respectively. (a) Show that there exists a circle $k$ which touches the lines $BC$ and $AD$ at $E$ and $F$, respectively. (b) Prove that $k$ also touches the circumcirlce of ABCD .

Prove that for every prime $p$, there exists a colouring of the positive integers with $p$-1 colours such that the colours of the numbers { $a$,2$a$,3$a$,...,($p$-1)$a$ } are pairwise different for every positive integer $a$.

An induced subgraph $S$ of the graph $G$ is called ``dominant'' if every vertex of $G$ , outside $S$ , has a neighbor in $S$ . Does there exist such a graph which has an even number of dominant subgraphs?

Prove that there exists a real number $c$>0 with the following property: among arbitrary, pairwise distinct positive integers $a_1,a_2,\ldots,a_n$ ($n \ge$3), there are three whose least common multiple is at least $c^. n^{2.99}$.

There is given a positive integer $n$ and some straight lines in the plane such that none of the lines passes through (0,0), and every lattice point ($a$,$b$), where 0$\le a$,$b \le n$ are integers and $a$+$b$>0, is contained by at least $a$+$b$+1 of the lines. Prove that the number of the lines is at least $n$($n$+3).

Show that for arbitrary positive integers $n$, $k$ there exists a polynomial $p$($x$), with degree at most $100\sqrt{nk}$ , such that $p(0) > \big(\big|p(1)\big|+\ldots+\big|p(n)\big|\big) + \big(\big|p(-1)\big|+\ldots+\big|p(-k)\big|\big)$.

We have $n=\frac{3^k-3}2$ coins, each one indistinguishable from the other in terms of size and appearance, but only $n$-1 of them have the same weight, whilst one of them is a counterfeit, being either lighter or heavier than the rest. Using a balance we have to find the counterfeit coin. Show that by $k$ measurements it is possible to find the counterfeit coin and to determine whether it is heavier or lighter than the rest; moreover, it is possible to arrange the measurements in advance, without knowing the results of the measurements.

For what primes $p$ does there exist a cubic polynomial $f$ with integer coefficients with the property that $p$ does not divide the leading coefficient of $f$ and $f$(1),$f$(2),...,$f$($p$) give pairwise distinct remainders modulo $p$?

There are given three circles in the plane, $k_0$, $k_1$ and $k_2$, being externally tangent to each other. The center of $k_0$ is $O$ and one of its diameters is $A_1 A_2$. Denote by $B$the point of tangency between $k_1$ and $k_2$, by $C_1$ the point of tangency between $k_0$ and $k_1$ and by $C_2$ the point of tangency between $k_0$ and $k_2$. The line segments $A_1 C_2$ and $A_2 C_1$ meet at point $D$ in the interior of the circle $k_0$. Let $t_1$ and $t_2$ be the tangent lines to the circle $k_0$ at $A_1$ and $A_2$, respectively. Prove that if $t_1$ is tangent to $k_1$ and $t_2$ is tangent to $k_2$ then the line segment $OB$ passes through point $D$.

There is given a triangle ABC . For every 0<$t$<1 let $U$($t$) and $V$($t$) be the points which divide the line segments $AB$ and $BC$ in the ratio $t$:(1-$t$), respectively. Prove that there is a parabola which is tangent to all lines $U$($t$)$V$($t$).

In each of five boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$ there is initially one coin. There are two types of operation allowed: Type 1: Choose a nonempty box $B_j$ with 1$\le j \le$4. Remove one coin from $B_j$ and add two coins to $B_{j+1}$. Type 2: Choose a nonempty box $B_k$ with 1$\le k \le$3. Remove one coin from $B_k$ and exchange the contents of (possibly empty) boxes $B_{k+1}$ and $B_{k+2}$. Prove that for every integer $0$\le n \le$2^{2010}$ there is a finite sequence of such operations that results in boxes $B_1$, $B_2$, $B_3$, $B_4$ being empty and box $B_5$ containing exactly $n$ coins.

Let $m \ge$3 be a positive integer, and let $\Phi_m$($x$) be the $m$th cyclotomic polynomial, and denote by $\Psi_m$($x$) the polynomial with integer coefficients for which $x^{\varphi(m)/2} \Psi_m\left(x+\frac1x\right) = \Phi_m(x)$. Prove that for every integer $a$, any prime divisor of the number $\Psi_m$($a$) either divides $m$ or is of the form $mk \pm$1.

How many distinct subsets $A_1$,...,$A_t$ of an $n$-element set can be chosen such that $(A_i\cap A_k)\subseteq A_j$ for all 1$\le i$<$j$<$k \le t$?

Let $n \ge$3 and $k$ be positive integers. We place $n$ coins on the table with the heads sides up. Then the following operation is performed $k$ times: we choose one of the $n$ coins randomly, with equal probabilities, and flip that coin. Prove that the probability that the procedure results in all the $n$ coins lying with tails up is less than $\frac{1}{2^{n-1}}$.

Is there any infinite sequence $a_1$,$a_2$,... of nonnegative numbers for which $\sum_{n=1}^\infty a_n^2$ is finite and $\sum_{n=1}^\infty \bigg(\sum_{k=1}^\infty\frac{a_{kn}}{k}\bigg)^{2} = \infty?$

A. 521. There are given a positive integer $m$ and an infinite sequence $a_1$<$a_2$<... of positive integers such that $a_k \le mk$ holds for infinitely many indices $k$. Prove that there exist positive integers $b_1$,...,$b_m$ with the property that every integer can be written as $a_i$-$a_j$+$b_k$ where $i$,$j$ are positive integers and 1$\le k \le m$.

The vertices and side arcs of the spherical hexagon ABCDEF are located on a hemisphere. The arc $AB$ is perpendicular to the arc $BC$, the arc $AF$ is perpendicular to the arc $EF$, moreover $AB$=$AF$, $BC$=$CD$ and $DE$=$EF$. Prove that the great arcs $AD$ and $CE$ are perpendicular.

There are given a simple graph and a positive integer $n$. Show that there are two, not necessary distinct vertices $a$ and $b$ in the graph for which the number of directed walks of length $n$, starting point $a$ and end-point $b$, is even.

Two sailboats are travelling in perpendicular directions on the lake Balaton at the same speed of 10 km/h. At a given time instant, while both are approaching the intersection of their paths, the respective distances of sailboats $A$ and $B$ are 1 km and 2 km from that point. A man travelling in boat $A$ wants to swim to boat $B$ . He swims at a uniform speed of 2 km/h. When should he jump in the water in order to minimise the time spent in the water?

The three-element subsets of a seven-element set are coloured. If the intersection of two sets is empty then they have different colours. What is the minimum number of colours needed?

All possible strings of $n$ different letters are formed out of a $k$ -element alphabet. Two such strings are connected with an edge if they only differ in a single position. What is the diameter of the resulting graph?

The sides of a convex quadrilateral $ABCD$ are divided into $n\ge 2$ equal parts. On the sides $AB$ , $BC$ , $CD$ , $DA$ , let $A_k$, $B_k$, $C_k$, $D_k$ denote the $k$ th points of division, counting from the vertices $A$ , $B$ , $C$ , $D$ , respectively. For which pairs $(n,k)$ is it true that the quadrilateral $ABCD$ is a parallelogram if and only if the quadrilateral $A_kB_kC_kD_k$ is also a parallelogram?

Let $a$ , $b$ and $c$ be the sides of triangle $ABC$ , let $f_a$, $f_b$ and $f_c$ denote the lengths of the interior angle bisectors, and let $t_a$, $t_b$ and $t_c$ be the segments of the interior angle bisectors that lie inside the circumscribed circle. Show that $a^2b^2c^2= f_af_bf_ct_at_bt_c$.

Let $n$ be a positive integer. Determine the sum of all fractions $\frac{1}{xy}$ , such that the numbers $x$ and $y$ are relative primes, are not greater than $n$, and their sum is greater than $n$.

Show that the points of the curve of equation $\sqrt{X}+\sqrt{Y}=1$ lie on a parabola.

Solve the equation 8$x$(2$x^2$-1)(8$x^4$-8$x^2$+1)=1.

The planes $S_1$ and $S_2$ intersect in a line $m$. Line $e$ intersects plane $S_i$ at the point $D_i$. The angle enclosed by $e$ and $S_i$ is $\alpha_i$. Show that if $D_1 \ne D_2$, then $\alpha_1$>$\alpha_2$ holds true if and only if $D_1$ lies closer to $m$ than $D_2$.

Let $p_1$=2, and for $n \ge$1 let $p_{n+1}$ denote the smallest prime factor of the number $np_1^{1!}p_2^{2!}\ldots p_n^{n!}+1$. Prove that every prime number occurs in the sequence $p_1,p_2,\dots\,.$

Is there an $n$, such that it is possible to walk the 4×$n$ chessboard with a knight touching each field exactly once so that with a last step the knight returns to its original position? What happens if the knight is not required to return to the original position?

Show that $65^{64}$+64 is a composite number.

Given the hypotenuse of a right-angled triangle and the radius of the excircle drawn to one of the legs, construct the triangle.

Prove that if a convex polygon $\mathcal{K}$ is not a parallelogram then it is possible to select three sides, such that the triangle enclosed by the lines of the sides contains $\mathcal{K}$ .

Given that the roots $x_1$, $x_2$, $x_3$ of the equation $x^3$-($a$+2)$x^2$+(2$a$+1)$x$-$a$=0 satisfy $\frac{2}{x_1} + \frac{2}{x_2} = \frac{3}{x_3}$, solve the equation.

Two faces of a cube are ABCD and ABEF . Let $M$ and $N$ denote points on the face diagonals $AC$ and $FB$, respectively, such that $AM$=$FN$. What is the locus of the midpoint of the line segment $MN$?

Denote the centres of the escribed circles of a triangle ABC by $O_a$, $O_b$, $O_c$, the centre of the incircle by $I$, and the radius of the circumscribed circle by $R$. Let $A_1$ be the intersection of the perpendiculars dropped from point $O_b$ onto line $AB$and from point $O_c$ onto line $AC$. Show that $A_1 I$=2$R$.

Someone selected $n$ not necessarily different non-negative integers. He wrote down on a sheet of paper all of the ($2^n$-1) sums that can be formed out of the $n$ numbers. Is it possible to determine the original numbers from this information?

The centre of the regular hexagon ABCDEF in the coordinate plane is at the origin $O$, and vertex $A$ is the point (0,1). Let $\mathcal{H}_1$ and $\mathcal{H}_2$ denote the set of points inside or on the boundary of the regular triangles ACE and BDF , respectively. Determine the set of points $P$ for which the vector $\overrightarrow{OP}$ can be written in the form $\overrightarrow{OP_1}+\overrightarrow{OP_2}$, where $P_1\in \mathcal{H}_1$ and $P_2\in \mathcal{H}_2$.

Let $p$>3 be a prime number. Determine the last two digits of the number $\sum_{i=1}^{p} \binom{i\cdot p}{p}\cdot\binom{(p-i+1)p}{p}$ written in base-$p$ numeral system.

For what integers $n$>2 is it true that every convex polygon of $n$ vertices has a side such that neither of the two angles lying on the side is acute?

A 6×6×6 cube is built out of blue and red building blocks, such that every 2×2×2 part consists of exactly 3 red cubes and 5 blue cubes. Prove that it is also true for the big cube that exactly 3 vertices are red and 5 are blue.

Is there a function defined on the points of the plane, which is not identically zero but the sum of its values at the vertices of every regular pentagon in the plane is zero?

Prove that if 2$n$+1 and 3$n$+1 are square numbers for some positive integer $n$ then 5$n$+3 cannot be a prime number.

Is there a triangle in which an angle is divided into four equal parts by the altitude, median and angle bisector drawn from the vertex?

A group of 11 000 astronauts are trained for a Mars mission. Given that from any 4 of them it is possible to select a suitable crew of 3 members for the landing module, prove that it is possible to select 5 astronauts such that any 3 of them form a suitable crew. ( Kvant )

Given the triangle ABC and the line $e$, determine the point $P$ of $e$ for which $PA^2$+2$PB^2$+3$PC^2$ is a minimum.

A circle passes through vertices $B$ and $C$ of a triangle ABC . It intersects side $AB$ at $D$ and side $AC$ at $E$. The median $AF$ intersects $DE$ at $G$. Prove that $\frac{GD}{GE} = \frac{AC^2}{AB^2}$.

Solve the simultaneous equations $\cos x + \cos y + \cos z = \frac{3\sqrt{3}}{2}$, $\sin x +\sin y +\sin z =\frac{3}{2}$.

A house has the shape of a cube. A roof is built on it that consists of two isosceles triangles and two symmetrical trapeziums. The edges of the roof are all equal, and the angles enclosed by adjacent faces are all equal. Find the ratio of the edge of the roof to the edge of the cube.

Given the points $P$ and $Q$, determine the locus of the intersection of the line $e$ passing through $P$ with the plane $S_e$ through $Q$ that is perpendicular to $e$.

Solve the following simultaneous equations: $x^3$+4$y$=$y^3$+16$x$, $\frac{1+y^{2}}{1+x^{2}} =5$.

Triangle ABC has a $120^o$ angle at vertex $C$. The orthocentre of the triangle is $M$, the centre of the circumscribed circle is $O$, and the midpoint of arc ACB of the circle is $F$. Prove that $MF$=$FO$.

Find a colouring of positive integers with 7 colours such that the colours of the numbers {$a$,2$a$,3$a$,4$a$,5$a$,6$a$,7$a$} are pairwise different for every positive integer $a$.

Let $a_1$, $a_2$, $a_3$, $a_4$ denote four consecutive elements in a row of Pascal's triangle. Prove that the numbers $\frac{a_{1}}{a_{1}+a_{2}}$, $\frac{a_{2}}{a_{2}+a_{3}}$, $\frac{a_{3}}{a_{3}+a_{4}}$ form an arithmetic progression.

Prove that the perpendiculars drawn to the sides of a triangle at the points of tangency of the escribed circles are concurrent.

What is the sixth digit following the decimal point in the decimal form of $\big(\sqrt{2010}+\big[\sqrt{2010}\,\big]\big)^{100}$?

Construct the point $P$ on side $AB$ of triangle ABC for which the radii of the inscribed circles of triangles BCP and ACP are equal.

Let ABCDEF be a hexagon inscribed in a circle. Prove that $AD^. BE^. CF$=$AB^. DE^. CF$+$BC^. EF^. AD$+$CD^. FA^. BE$+$AB^. CD^. EF$+$BC^. DE^. FA$.

Is there a polynomial of degree at least two that represents a one-to-one mapping of the set of rational numbers onto itself?

The terms of sequence ($a_n$) are positive integers, such that $a_{n+1}$=$a_n ^2$+5$a_n$+1 for all $n \ge$1. Is it possible that the terms of the sequence are all composite numbers?

Given six circles in the plane that have an interior point in common. Prove that there is a circle among them that contains the centre of another in its interior.

The lengths of the sides of a parallelogram of unit area are $a$ and $b$, where $a$<$b$<2$a$. What is the area of the quadrilateral bounded by the interior angle bisectors of the parallelogram?

Solve the equation $x^{6}-x^{3}-2x^{2}-1=2(x-x^{3}+1)\sqrt{x}$ .

Show that an altitude of a triangle cannot be longer than the geometric mean of the radii of the excircles touching the two adjacent sides of the triangle.

Solve the equation $x^{3}+y^{3}+1=x^{2}y^{2}$ on the set of integers.

Solve the simultaneous equations $x$+$y$=$a$, tan $x^.$tan $y$=$b$, where $a$ and $b$ are real parameters.

Is it true that if the sum of distances of any interior point of a tetrahedron from the faces is constant, then the tetrahedron is regular?

$M$ is the midpoint of the arc $AB$ containing vertex $C$ on the circumscribed circle $k$ of triangle ABC . $J$ is the centre of the escribed circle drawn to side $AB$. The perpendicular drawn to angle bisector $CJ$ at point $J$ intersects line $AC$ at $D$ and line $BC$ at $E$. Line $MJ$ intersects circle $k$ again at $F$. Prove that the circle passing through points $D$, $E$, $F$ touches the lines $AC$ and $BC$, and also touches the circle $k$.

Someone chose $n$ not necessarily different integers, and formed sums out of them in all ($2^n$-1) possible ways. Given that 0 was not obtained, is it possible to determine the original numbers from the sums?

A pool can be filled through four different pipes. If the first and second pipes are operating, it takes 2 hours to fill the pool. Through the second and third pipes it takes 3 hours, and through the third and fourth pipes it takes 4 hours. How long does it take to fill the pool if the first and fourth pipes are operating?

A 23×23 square is divided into smaller squares of dimensions 1×1, 2×2 and 3×3. What is the minimum possible number of 1×1 squares?

Prove that a trapezium circumscribed about a circle has a diagonal that encloses an angle of at most $45^o$ with the bases.

The terms of a sequence are positive integers. The first two terms are 1 and 2. No term of the sequence is equal to the sum of two different terms. Prove that for any natural number $k$, the number of terms less than $k$ is at most $\frac{k}{3} +2$.

The leg of an isosceles right angled triangle is 36 units long. Starting at the right-angled vertex on one of the legs, an infinite sequence of regular triangles is drawn. One side of each triangle lies on the leg and the third vertex lies on the hypotenuse. The sides on the leg link together and cover the whole leg. Find the total area of the regular triangles.

The centres of the escribed circles of triangle ABC are $O_1$, $O_2$ and $O_3$. $P$ is an interior point of the triangle, different from the centre of the inscribed circle. The feet of the perpendiculars drawn from $P$ to the angle bisectors are $M_1$, $M_2$ and $M_3$. Prove that the triangles $O_1 O_2 O_3$ and $M_1 M_2 M_3$ are similar.

$A$ and $B$ are two opposite vertices of a unit cube. Find the radius of the sphere that touches the faces of the cube that meet in $A$ and the edges that meet in $B$.

The diagonals of a trapezium $A_1 A_2 A_3 A_4$ are $A_1 A_3$=$e$ and $A_2 A_4$=$f$. Let $r_i$ denote the radius of the circumscribed circle of triangle $A_j A_k A_l$, where {1,2,3,4}={$i$,$j$,$k$,$l$}. Show that $\frac{r_2+r_4}{e}=\frac{r_1+r_3}{f}$ .

Let $a$ and $b$ denote positive integers. Given that $p$ is a polynomial of integer coefficients that has both a value divisible by $a$ and a value divisible by $b$ at integer points. Prove that there is an integer point where the value of $p$ is also divisible by the least common multiple of $a$ and $b$.

Prove that the inequality below is true for all positive numbers $a$, $b$, $c$: $a^b b^c c^a \le a^a b^b c^c$.

The feet of the perpendiculars dropped from vertex $C$ of triangle ABC onto the interior angle bisectors from vertices $A$ and $B$ are $E$ and $F$, respectively. The inscribed circle of the triangle touches side $AC$ at point $D$. Prove that $EF$=$CD$.

The sum of three positive integers is 2010, and the sum of their reciprocals is $\frac{1}{58}$. What are these numbers?

What is the maximum number of edges of a regular decagonal right prism that a plane may intersect?

A 13×13 table is filled in with numbers, such that the sum of the numbers is the same in each of the 13 rows and 13 columns. What is the smallest possible number of fields in which the entries should be changed so that the 26 sums are all different?

Let $m_a$ and $m_b$ denote the lengths of the altitudes drawn to sides $a$ and $b$ of a triangle. Show that if $a$>$b$ then $a^{2010}$+$m_a ^{2010} \ge b^{2010}$+$m_b ^{2010}$.

Prove that $-\frac{1}{2}\le \frac{(x+y)(1-xy)}{(1+x^{2})(1+y^{2})}\le\frac{1}{2}$ for all real numbers $x$ and $y$.

Given a regular pentagon, find the point for which the sum of the distances from the vertices of the pentagon is minimal.

Parallelogram CDEF is inscribed in triangle ABC , with vertices $D$, $E$, $F$ lying on sides $CA$, $AB$ and $BC$, respectively. Given the length of the line segment $DF$, construct the point $E$. For what point $E$ will the length of diagonal $DF$ be minimal?

Prove that the sum of the squares of 35 consecutive positive integers is always divisible by 35.

The difference of the cubes of two consecutive positive integers is $n^2$, where $n$>0. Prove that $n$ is the sum of two perfect squares.

Soldiers are lined up along a line in the east-west direction, each of them facing north. The officer commands ``right turn''. Then they should all turn towards the east but, since they are at the very beginning of their military career, some of them get the order wrong and turn towards the west. Every soldier who now faces his neighbour concludes that they have made a mistake (disregarding the possibility of the neighbour making the mistake), and they make a 180-degree turn within 1 second. Then the process continues in the same way: every soldier who faces a neighbour makes a 180-degree turn within 1 second. Prove that the process will end in a finite number of steps.

A rectangle that is not a square is folded in two along a diagonal. Prove that the perimeter of the resulting pentagon is smaller than the perimeter of the original rectangle.

Is there a positive integer $k$, such that $\big(\dots\big((4\underbrace{!)!\big)!\dots\big)!}_{k} > \big(\dots\big((3\underbrace{!)!\big)!\dots\big)!}_{k+1}$?

What is the largest possible number of edges of a pyramid with an $n$-sided base that a plane may intersect?

Solve the equation $16^{x2}$+$y$ $+16^{y2}$+$x$ =1.

Given two points on each of two sides of triangle ABC , prove that at least one of the four triangles formed by the four points has an area that is not greater than one quarter of the area of triangle ABC .

Given the quadrilateral ABCD , show that the Feuerbach circles of triangles ABC , ABD , ACD and BCD have a point in common.

Find the smallest possible positive integer $n$ for which $3^{2n}$-1 is divisible by $2^{2010}$?

Let $a_0,a_1,\ldots,a_n$ be positive numbers, such that $a_{k+1}$-$a_k \ge$1 for all $k$=0,1,...,$n$-1. Show that $1 + \frac1{a_0} \left(1+\frac1{a_1-a_0}\right) \cdots \left(1+\frac1{a_n-a_0}\right) \le \left(1+\frac1{a_0}\right) \left(1+\frac1{a_1}\right) \cdots \left(1+\frac1{a_n}\right).$

$P$ is a given point in the interior of the acute-angled triangle ABC . The lines $AP$, $BP$ and $CP$ intersect the opposite sides at the points $A_1$, $B_1$ and $C_1$, respectively. Given that $PA_1$=$PB_1$=$PC_1$=3 and $AP$+$BP$+$CP$=43, prove that $AP^. BP^. CP$=441.

In a company, everyone knows 5 other people. (Acquaintances are mutual.) Two members of the company are appointed captains. The captains take turns selecting members for their teams, until everyone is selected. Prove that at the end of the selection process there are the same number of acquaintances within each team.

$A$, $B$, $C$, $D$, $E$ and $F$ are a group of six people. $n$ bars of chocolate given to the group in the following way: Everyone gets at least one, $A$ gets less than $B$, $B$ gets less than $C$, $C$ gets less than $D$, $D$ gets less than $E$, and finally, $F$ gets the most. The members of the group know these conditions, they know the value of $n$, and of course, they know how many bars of chocolate they were given themselves. They have no other information available for them. What is the smallest possible value of $n$ for which it is possible to give them the bars of chocolate so that no one can tell how many bars of chocolate everyone has?

The radii of three concentric circles are 1, 2, and 3 units. A point is marked on each circle such that they form a regular triangle. What may be the length of the side of the triangle?

Given that $r$ is a positive rational number and $r^r$ is also rational, prove that $r$ is an integer.

Let $E$ denote the point closer to vertex $B$ that divides side $BC$ of a square ABCD in a 1:4 ratio. Let $F$ be the point obtained by reflecting about $C$ the point closer to $D$ that divides side $CD$ in a 1:2 ratio. Prove that the lines $AE$ and $BF$ intersect on the circumscribed circle of the square ABCD .

Solve the following simultaneous equations: $\frac{1}{\sqrt{1-x^2}} + \frac{1}{\sqrt{1-y^2}} =\frac{35}{12}$, $\frac{x}{\sqrt{1-x^2}} - \frac{y}{\sqrt{1-y^2}} =\frac{7}{12}$.

Let $P$ and $Q$ be arbitrary points on the edges $AB$ and $CD$ of a given tetrahedron ABCD , respectively. Determine the locus of the midpoint of the line segment $PQ$.

A triangle is drawn on a rectangular sheet of paper. Unfortunately, the vertices are off the sheet, but there is a segment of each side that lies on the sheet. Given that the orthocentre of the triangle is on the sheet, find the orthocentre by doing construction on the sheet.

Write down the numbers $x_k=\big[k\sqrt2\,\big]$ ($k=1,2,\ldots$) in a row, and underneath, in a second row list the integers 0<$y_1$<$y_2$<... that do not occur among the numbers $x_k$. Express the difference $y_k$-$x_k$ as a function of $k$.

Prove that the following inequality is true in every triangle: $\frac{b}{\sin \left(\gamma + \dfrac{\alpha}{3}\right)} + \frac{c}{\sin \left(\beta + \dfrac{\alpha}{3}\right)} > \frac{2}{3} \cdot \frac{a}{\sin \left(\dfrac{\alpha}{3}\right)}$.

If the sum of its digits is added to a certain three-digit number, the resulting number consists of identical digits. If the sum of the digits is subtracted from the original number, the result will also consist of identical digits. What is the original number?

In a right-angled triangle ABC , there is a $30^o$ angle at vertex $B$. $D$ is the centre of the square drawn on the outside to the hypotenuse $BC$. Find angle $\angle$ ADB .

A group of 5 little girls and 7 little boys in the kindergarten are playing at weddings. They choose a bride and groom, a registrar, two bridesmaids, one best man for the bride and one for the groom, one witness for the bride and one for the groom. Three of the girls each have a brother in the group, too, and there are no other pairs of brothers or sisters. In how many different ways can the selection be made if a sister and brother cannot play the bride and groom, and the registrar cannot be the brother or sister of either of the newly weds. (Bridesmaids are girls and best men are boys. There is no restriction on the sex of the witnesses.)

A cylinder lying on a horizontal plane is held fixed by placing a prism against each side. (The prisms touch the lateral surface of the cylinder along an upper edge.) What is the radius of the cylinder if the heights of the prisms are 9 cm and 2 cm, and their separation is 23 cm?

The members of a small group of representatives in the parliament of Neverland take part in the work of four committees. Every member of the group works in two committees, and any two committees have one member in common from the group. How many representatives are there in the group?

$P$ is a point on side $AC$, and $Q$ is a point on side $BC$ of triangle ABC . The line through $P$, parallel to $BC$ intersects $AB$ at $K$, and the line through $Q$, parallel to $A$C intersects $AB$ at $L$. Prove that if $PQ$ is parallel to $AB$ then $AK$=$BL$.

A rhombus is constructed out of four metal rods of 20-cm length hinged at the vertices. The longer diagonal is originally 32 cm long. The rhombus is slightly compressed along the longer diagonal. As a result, the other diagonal gets longer by 1.2 as much as the longer one gets shorter. What are the new lengths of the diagonals?

Six cards are numbered 1 to 6 and shuffled. Three cards are drawn in a row, without replacement. What is the probability that the resulting sequence is increasing?

The degree of a polynomial $p$($x$) is at most four. It has zeros and also minima at $x_1$=-3 and $x_2$=5. Given that the polynomial $q$($x$)=$p$($x$-1) is even and has a local maximum value of 256, find the polynomial $p$($x$).

For integers $a$ and $b$, let $a$o$b$ denote the number one greater than the non-smaller one of the two numbers, and let $a$*$b$ denote the number one greater than the non-greater one of the two numbers. Solve the equation ($x$o2010)*2011=$x$+2.

Extend each diagonal of a square in one direction by a length equal to the side of the square. How many isosceles triangles are determined by the endpoints of the extensions and the vertices of the square?

Solve the equation ($ax^2$+$bx$+14)$^2$+($bx^2$+$ax$+8)$^2$=0 on the set of integers. $a$ and $b$are integers.

ABC is an equilateral triangle in the plane. Consider the plane figure formed by the points for which the distance from vertex $A$ is not greater than the side of the triangle and the distances from vertices $B$ and $C$ are not smaller than the side of the triangle. By what factor is the area of the figure greater than the area of the triangle?

In a gift shop, there are 160 Easter greating cards stacked on the counter. A customer separates the stack of cards in two (not necessarily equal) parts, each containing at least two cards. He buys one card of one of the two stacks. The next customer follows the same procedure: he divides one of the stacks in two (according to the above conditions), buys one card from one part. Then comes the next customer, and so on. Is it possible in this way to obtain stacks of cards on the counter that consist of four cards each?

$x$ and $y$ are real numbers such that $x$+3$y$=12 and $x \ge$2$y \ge$0. What values may $x$+2$y$have?

In the lottery, five out of the numbers 1 to 90 are drawn. Ticket holders mark five numbers on each ticket. A man bought two tickets and marked ten different numbers on them altogether. Given that four out of his ten numbers were drawn, what is the probability that he had a ticket with $a$) four numbers drawn? $b$) two numbers drawn?

Vertices $B$ and $C$ of a triangle are connected to the points that divide the opposite sides into three equal parts. The connecting line segments determine a quadrilateral. Prove that the quadrilateral has a diagonal parallel to side $BC$.

Solve the equation $\log_{2010} (2009 x)=\log_{2009} (2010 x)$.

A regular hexagon is rotated about its axes of symmetry. Find the ratio of the surface areas of the resulting solids.

In a mathematics competition, there were three problems. 56 participants solved at least one problem. 2 participants solved all problems. Out of those solving the second problems, 10 more solved the third problem than the first one. The number of those solving both of the first and second problems was 10 larger than the number of those solving the third problem only. All participants solving the first and third problems solved the second problem, too. There were 14 participants altogether who solved the first problem only or the second problem only. How many participants solved the third problem?

How many 9-digit numbers (in decimal system) divisible by 11 are there in which every digit occurs except zero?

Find the isosceles triangle of minimum area circumscribed about a semicircle, such that its base lies on the line of the diameter of the semicircle and its legs are tangent to the semicircle.

Define the function $\mathop{\rm lac} (x)$ on the set of real numbers as follows: $\mathop{\rm lac} (x)=x, {\rm if \ } x\in [2n; 2n+1], { \rm \ where \ } n\in\mathbb{Z}, {\rm or} -x+4n+3, {\rm if \ } x\in ]2n+1; 2n+2[, {\rm \ where \ } n\in\mathbb{Z}.$ Solve the equation $\mathop{\rm lac} (2x^2 + x + 4)=\mathop{\rm lac} (x^2 + 7x -1)$ on the set of real numbers.

There are four unit spheres inside a larger sphere, such that each of them touches the large sphere and the other three unit spheres. What fraction of the volume of the large sphere is filled by the four unit spheres altogether?

The lawn of a stadium is bounded by two parallel line segments of length 100 metres joined by two 100 metre-long semicircles. By what factor is the area of a circle of perimeter 400 m larger than the area of the lawn?

The digits of a 2010-digit number divisible by nine are added. The digits of the resulting number are also added, and this process is repeated once more. What may be the final result?

Solve the equation $x$+$y$=$x^2$-$xy$+$y^2$, where $x$ and $y$ are integers.

Determine the range of the function given by the following rule of assignment: $f(x)=\frac{{(x+a)}^2}{(a-b)(a-c)} + \frac{{(x+b)}^2}{(b-a)(b-c)} + \frac{{(x+c)}^2}{(c-a)(c-b)}$, where $a$, $b$ and $c$ are different real numbers.

The diagonals of a convex quadrilateral ABCD intersect at point $M$. Extend diagonal $AC$beyond $A$ by the length of $MC$, and extend diagonal $BD$ beyond $B$ by the length of $MD$ to get the points $E$ and $F$. Prove that $EF$ is parallel to a midline of the quadrilateral.

Three regular dice are rolled. The three numbers obtained on top are listed in an arbitrary order. Then the list is continued with the numbers obtained on the bottom, in the same order. The resulting six-digit number is divided by 111, 7 is subtracted from the ratio, and the difference is divided by 9. Prove that the result is a three-digit number whose digits are the numbers rolled.

Let $\alpha$($n$) denote the measure of the interior angles of a regular $n$-sided polygon. What is $n$ if $\alpha$($n$+3)-$\alpha$($n$)=$\alpha$($n$)-$\alpha$($n$-2)?

A fair coin is tossed ten times in a row. Every time a head is tossed, a digit of 2 is written down. When a tail is tossed, a digit of 3 is written down. What is the probability that the resulting ten-digit number is divisible by $a$) 3, $b$) 4?

Prove that $\frac{2 \cos 40^\circ - \cos 20^\circ}{{\sin 20}^\circ}=\sqrt 3$.

Consider the circles drawn on two adjacent sides of a square of side 2 units. Find the radius of the circle that touches one of the sides and one circle from the inside and the other circle from the outside.

Currently there are six different kinds of banknotes in Hungary: 500, 1000, 2000, 5000, 10000 and 20000 forint notes. How many different sums of money can be paid with three notes?

A natural number $n$ is chosen between two consecutive square numbers. The smaller square is obtained by subtracting $k$from $n$, and the larger one is obtained by adding $l$ to $n$. Prove that the number $n$-$kl$ is a perfect square.

$T$ is the foot of the perpendicular drawn from vertex $A$ of an acute-angled triangle ABC to side $BC$. The feet of the perpendiculars drawn from $T$ to the sides $AB$ and $AC$ are $P$ and $Q$. Prove that the quadrilateral BPQC is cyclic.