KöMaL 2012

Show that there exist infinitely many pairs ($a$,$b$) of positive integers with the property that $a$+$b$ divides $ab$+1, $a$-$b$ divides $ab$-1, $b$>1 and $a>b\sqrt3-1$.

Prove that for an arbitrary sequence $a_1,a_2,\ldots$ of nonnegative real numbers and $\varepsilon$>0 there exist infinitely many positive integers $n$ for which $n^2 \big(4a_n(1-a_{n-1})-1 \big) \le \frac14+\varepsilon$.

Suppose that in a simple graph $G$ with $n$ vertices the minimal degree $\delta$($G$) is at least 3$n$/4. Prove that for any 2-coloring of the edges of $G$, there is a connected subgraph with at least $\delta$($G$)+1 vertices whose edges have the same color.

The circumcenter of the cyclic quadrilateral ABCD is $O$. The second intersection point of the circles ABO and CDO , other than $O$, is $P$, which lies in the interior of the triangle DAO . Choose a point $Q$ on the extension of $OP$ beyond $P$, and a point $R$ on the extension of $OP$ beyond $O$. Prove that $\angle$ QAP =$\angle$ OBR holds if and only if $\angle$ PDQ =$\angle$ RCO .

Prove that for arbitrary real numbers $a_1,\ldots,a_n$ there exist a real $t$ such that $\sum_{i=1}^n \big|\sin(t-a_i)\big| \le \ctg\frac\pi{2n}.$

Show that the positive integers can be coloured with three colours in such a way that the equation $x$+$y$=$z^2$ has no solution ($x$,$y$,$z$) consisting of distinct numbers with the same colour.

Prove that there exists a constant $C$>0 for which the following statement holds: if $n$ is a positive integer and $A_1,A_2,\ldots,A_N\subset\{1,2,\ldots,n\}$ are sets such that every two of them has at least two, and every three of them has at most three elements in common, then $N$<$Cn^2$.

The incircle of triangle ABC is $k$. The circle $k_A$ touches $k$ and the segments $AB$ and $AC$ at at $A$', $A_B$ and $A_C$ respectively. The circles $k_B$, $k_C$ and the points $B$', $C$' are defined analogously. The second intersection point of the circles $A$'$B$'$A_B$ and $A$'$C$'$A_C$, other than $A$', is $K$. The line $A$'$K$ meets $k$ at $R$, other than $A$'. Prove that $R$ lies on the radical axis of the circles $k_B$ and $k_C$.

Given a right circular cone with vertex $O$, and a fixed point $P$ in the base plane of the cone. Draw a line $x$ through $P$ which intersects the base circle of the cone at two points, $X_1$ and $X_2$. Show that $\tan\frac{\angle POX_1}2 \cdot \tan\frac{\angle POX_2}2$ does not depend on the choice of the line $x$.

Show that $\frac{a^3b}{{(3a+b)}^p} + \frac{b^3c}{{(3b+c)}^p} +\frac{c^3a}{{(3c+a)}^p} \ge \frac{a^2bc}{{(2a+b+c)}^p} + \frac{b^2ca}{{(2b+c+a)}^p} + \frac{c^2ab}{{(2c+a+b)}^p}$ holds for all positive numbers $a$, $b$, $c$, $p$.

Let $f$($k$)$=2^k$+1 for arbitrary positive integer $k$. Is there any positive integer $n$ which divides $f$ ( $f$($n$) ) , but does not divide $f$ ( $f$ ( $f$($n$) ) ) ?

Let 1$\le p$<2 be a real number. Prove that ($x$+$y$)$^p$+($z$+$v$)$^p$+($x$+$z$)$^p$+($y$+$v$)$^p \le x^p$+$y^p$+$z^p$+$v^p$+($x$+$y$+$z$+$v$)$^p$ for all nonnegative real numbers $x$, $y$, $z$ and $v$.

Let $k$ be the incircle in the triangle ABC , which is tangent to the sides $AB$, $BC$, $CA$ at the points $C_0$, $A_0$ and $B_0$, respectively. The angle bisector starting at $A$ meets $k$ at $A_1$ and $A_2$, the angle bisector starting at $B$meets $k$ at $B_1$ and $B_2$; $AA_1$<$AA_2$ and $BB_1$<$BB_2$. The circle $k_1 \ne k$ is tangent externally to the side $CA$ at $B_0$ and it is tangent to the line $AB$. The circle $k_2 \ne k$ is tangent externally to the side $BC$ at $A_0$, and it is tangent to the line $AB$. The circle $k_3$ is tangent to $k$ at $A_1$, and it is tangent to $k_1$ at point $P$. The circle $k_4$ is tangent to $k$ at $B_1$, and it is tangent to $k_2$ at point $Q$. Prove that the radical axis between the circles $A_1 A_2 P$ and $B_1 B_2 Q$ is the angle bisector starting at $C$.

The positive integers are coloured with a finite number of colours. A function $f$from the set of positive integers to itself has the following two properties: ($a$) if $x \le y$, then $f$($x$)$\le f$($y$); and ($b$) if $x$, $y$, and $z$ are (not necessarily distinct) positive integers of the same colour and $x$+$y$=$z$, then $f$($x$)+$f$($y$)=$f$($z$). Does it follow that the function $\frac{f(x)}x$ is bounded from above?

($a$) Prove that if $n \ge$2 and the product of the positive real numbers $a_2,a_3,\ldots,a_n$ is 1 then ${(1+a_2)}^2{(1+a_3)}^3 \cdots {(1+a_n)}^n > \frac{n^n{(n-1)}^{n-1}}{4^{n-1}}$. ($b$) Show an example for an integer $n \ge$2 and positive real numbers $a_2,a_3,\ldots,a_n$ having product 1 that satisfy ${(1+a_2)}^2{(1+a_3)}^3\ldots{(1+a_n)}^n < 1.000001 \cdot \frac{n^n{(n-1)}^{n-1}}{4^{n-1}}$.

($a$) Find all pairs ($a$,$b$) of relatively prime positive integers $a$, $b$ such that $b$ divides $a^2$-3 and $a$ divides $b^2$-3. ($b$) Find all pairs ($a$,$b$) of relatively prime positive integers $a$, $b$ such that $b$ divides $a^2$-5 and $a$divides $b^2$-5.

Given a triangle ABC and a line $\ell$ through its incenter. Denote by $A$', $B$' and $C$' the mirror images of $A$, $B$ and $C$ about $\ell$, respectively. Let the lines through $A$', $B$' and $C$', parallel to $\ell$, meet the lines $BC$, $CA$ and $AB$ at $P$, $Q$ and $R$, respectively. Prove that the points $P$, $Q$ and $R$ lie on a line and this line is tangent to the incircle.

Call a set $A\subset\mathbb{Z}$ interesting if 2$y$-$x \in A$ for all pairs $x$,$y \in A$ with $x$<$y$. Let $a_1$<$a_2$<...<$a_k$ be positive integers ($k \ge$2) whose greatest common divisor is 1. Prove that if $A$ is an interesting set and $\{0,a_1,\ldots,a_k\}\subset A$ then $a_1$+$a_k$-3$\in A$.

Given a triangle ABC . For an arbitrary interior point $X$ of the triangle denote by $A_1$($X$) the point intersection of the lines $AX$ and $BC$, denote by $B_1$($X$) the point intersection of the lines $BX$ and $CA$, and denote by $C_1$($X$) the point intersection of the lines $CX$ and $AB$. Construct such a point $P$ in the interior of the triangle for which each of the quadrilaterals $AC_1$($P$)$PB_1$($P$), $BA_1$($P$)$PC_1$($P$) and $CB_1$($P$)$PA_1$($P$) has an inscribed circle.

The side lengths of a triangle are $a$, $b$, and $c$. Prove that $\sqrt{\frac{(2a+c)(2b+c)}{(3a+b)(3b+a)}} + \sqrt{\frac{(2b+a)(2c+a)}{(3b+c)(3c+b)}} + \sqrt{\frac{(2c+b)(2a+b)}{(3c+a)(3a+c)}} < \frac52.$

Two circles $k_1$ and $k_2$ with centres $O_1$ and $O_2$, respectively, intersect perpendicularly at $P$ and $Q$. Their external homothety center is $H$. The line $t$is tangent to $k_1$ at $T_1$ and tangent to $k_2$ at $T_2$. Let $X$ be a point in the interior of the two circles such that $HX$=$HP$=$HQ$, and let $X$' be the reflection of $X$ about $t$. Let the circle $XX$'$T_2$ and the shorter arc $PQ$ of $k_1$ meet at $U_1$, and let the circle $XX$'$T_1$ and the shorter arc $PQ$ of $k_2$ meet at $U_2$. Finally, let the lines $O_1 U_1$ and $O_2 U_2$ meet at $V$. Show that $VU_1$=$VU_2$.

Let $D$={0,1,2,...,9} be the set of decimal digits, and let $R \subset D$×$D$be a set of ordered pairs of digits. An infinite sequence ($a_1$,$a_2$,$a_3$,...) of digits is said to be compatible with R if ($a_j$,$a_{j+1}$)$\in R$ for all positive integer $j$. Determine the smallest positive integer $K$ with the property that if an arbitrary set $R \subset D$×$D$ is compatible with at least $K$ distinct digit sequences then $R$ is compatible with infinitely many digit sequences.

Let $n \ge$2 and let $p(x)=x^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0$ be a polynomial with real coefficients. Prove that if for some positive integer $k$ the polynomial ($x$-1)$^{k+1}$ divides $p$($x$) then $\sum_{\ell=0}^{n-1} |a_\ell| > 1+\frac{2k^2}{n}$ .

Prove that if $S\subset\{1,\ldots,n\}$ and $|S|>\frac{n}3$, then it is possible to select at most four , not necessarily distinct elements from $S$, whose sum is a power of 2.

Find all positive integers $n$, nonzero reals $c_1,c_2,\ldots,c_n$ and real $t$ for which there exists a finite, nonempty set $\{A_1,A_2,\ldots,A_n\}$ of points in the plane $S$ and a nonconstant function $f\colon S\to\mathbb{R}$ such that $\sum_{i=1}^n c_if\big(\varphi(A_i)\big)=t$ for every similarity transformation $\varphi$ of the plane $S$.

Prove that if the edges of a $k$-chromatic graph are colored with two colors then there is a tree in the graph, containing $k$ vertices, whose edges have the same colors.

When the age of the ship is the same as the captain's age now, the captain will be exactly 32 years older than the ship was when the captain was half as old as the ship is now. How old is the captain?

The area of a symmetrical trapezium $\mathcal T$ is 4 units, it is not a rectangle, and it can be divided into four congruent trapezia that are similar to $\mathcal T$ . Determine the sides and angles of $\mathcal T$ .

There are 98 sticks lying on the table, their lengths are 1,2,3,...,98 units. Ann and Bill play the following game: With Ann starting the game, they take turns removing one stick of their choice. The game ends when there remain exactly three sticks on the table. Ann wins if the three sticks can form a triangle. Otherwise, Bill wins. Which player has a winning strategy?

Let $B$ be an interior point of a line segment $AC$. The Thales circles of the line segments $AB$, $BC$ and $AC$ are $k_1$, $k_2$ and $k_3$, respectively, with radii of $r_1$, $r_2$ and $r_3$. A common exterior tangent of circles $k_1$ and $k_2$ cuts the circle $k_3$ into two circular segments. The radius of the circle inscribed in the smaller circular segment is $r_4$. Prove that $r_1 ^. r_2$=$r_3 ^. r_4$.

Given a vertex, the orthocentre and the centroid, construct the triangle.

Solve the equation $\sin x+\frac 12 \cos x= \sin^2(x+{45}^\circ)$. ( 3 points )

Regular triangles ABD , BCE and CAF are drawn over the sides of triangle ABC on the outside. Let $G$, $H$ and $I$ denote the midpoints of line segments $DE$, $EF$ and $FD$, respectively. Prove that the sum of the angles AHB , BIC and CGA is $180^o$.

Jack Potter is an addicted gambler. Yesterday, he threw 20 000 forints (HUF, Hungarian currency) in a one-armed bandit machine without his family knowing about it. To make things worse, he took that money from the amount that the family reserved for food. To avoid the affair being revealed, today he is taking the remaining 40 000 forints of the family, too, and goes back to the casino to play roulette. Since he does not want to risk too much, he is betting 1000 forints on red or on black in each game. If he wins, which has a probability of 18/37, then he gets another 1000 forints. Otherwise he loses his bet. He stops playing either when he has gathered a total of 60 000 forints -- in that case he can put all the money back in its place -- or when he has lost all the money. What is the probability that Jack Potter will manage to gather the 60 000 forints?

The lengths of the edges of a cuboid are 7, 14 and 21 units. Which pair of skew face diagonals lying on adjacent faces have the smallest distance?

Let $t$ denote a fixed integer. Show that for every odd prime number $p$, there exists a positive integer $n$, such that (3-7$t$)$2^n$+(18$t$-9)$3^n$+(6-10$t$)$4^n$ is divisible by $p$.

There are 99 sticks lying on a table, their lengths are 1,2,3,...,99 units. Andrea and Bill play the following game: they take turns removing one stick of their choice. Andrea starts the game. The game ends when there are exactly three sticks remaining on the table. If it is possible to make a triangle out of the three sticks then Andrea wins. Otherwise, Bill is the winner. Who has a winning strategy?

Given the lines of the medians of a triangle and one point lying on a side, construct the triangle.

Velo City Transportation Company are planning to operate a bus service along a ``uniformly populated'' straight road of length $\ell$. How should the $n$ bus stops be positioned along the road, so that the customers need to walk as little as possible?

Solve the equation $x^{2}-8(x+3)\sqrt{x-1}+22x-7=0$.

In a tetrahedron ABCD , opposite edges are perpendicular, and face BCD is an acute-angled triangle. The foot of the altitude from vertex $A$ is $T$. Find the shortest path $AT$ on the surface of the tetrahedron.

Show that if $\alpha$, $\beta$ and $\gamma$ are the angles of a triangle then (sin $\alpha$+sin $\beta$+sin $\gamma$)$^2$>9sin $\alpha$sin $\beta$sin $\gamma$.

A triangle is inscribed in a unit circle. What may be the distance between the centres of two escribed circles of the triangle?

$A_1 B_1 C_1$ and $A_2 B_2 C_2$ are two triangles such that their sides $A_1 B_1$ and $A_2 B_2$, $B_1 C_1$ and $B_2 C_2$, as well as $A_1 C_1$ and $A_2 C_2$ are parallel. Vertex $A_1$ is connected to vertices $B_2$ and $C_2$, vertex $B_1$ is connected to $C_2$ and $A_2$, and vertex $C_1$ is connected to $A_2$ and $B_2$. Given that the areas of the original triangles are $T_1$ and $T_2$, what may be the area of the hexagon formed by the midpoints of the connecting line segments obtained in this way?

We play the following game of tossing a coin: We make a bet before each toss. The bet may be at most as large as all the money we have. Then we make a guess on the result. If our guess is right, we will receive the double of the bet, otherwise we will lose it. Determine those values of $k$>1 for which there exists an integer $n$with a strategy such that the probability of receiving $k$ times our initial amount of money in at most $n$ games is at least $\frac{1}{k}$ .

Are there positive integers $n$ and $k$ such that $\big(5+3\sqrt{2}\,\big)^{n}= \big(3+5\sqrt{2}\,\big)^{k}$?

There is a 98×98 chessboard on the monitor of a computer. With the mouse, you can select a rectangle bounded by lattice lines of the chessboard. If you click on the rectangle, each chessboard field inside it will change to the opposite colour. What is the minimum number of clicks needed to make the whole chessboard have the same colour.

Solve the equation (1+$x$)$^8$+ ( 1+$x^2$ ) $^4$=82$x^4$.

Prove that every natural number not divisible by 10 can be multiplied by an appropriate natural number, such that the product is a palindromic number in decimal notation.

$T$ is the foot of the altitude drawn from vertex $A$ of an acute-angled triangle ABC . The midpoint of side $BC$ is $F$. The centres of the squares drawn over the sides $AB$ and $AC$ on the outside are $K$ and $L$, respectively. Prove that KTFL is a cyclic quadrilateral.

Let $x$, $y$, $z$ denote positive integers, such that $\frac{6841x-1}{9973}+\frac{9973y-1}{6841}=z$ . Prove that $\frac{x}{9973}+\frac{y}{6841}>1$.

Given the centres of the circumscribed circle and two escribed circles of a triangle, construct the triangle.

The angle bisectors of a triangle ABC intersect the opposite sides at the points $A_1$, $B_1$ and $C_1$, respectively. For what triangles is it true that $\overrightarrow{AA_1} + \overrightarrow{BB_1}+ \overrightarrow{CC_1} =\mathbf{0}$?

Given two different points $B$ and $C$ in the plane, determine the locus of those points $A$ for which the altitude drawn from vertex $A$ of the triangle ABC is the geometric mean of the line segments $BC$+$AC$ and $BC$-$AC$.

On a winter's day, an absent minded mathematician went on a walk with his old poodle along a long straight alley. He was so absorbed in his thoughts that when he finally thought of his dog, he had to realize that it was not near him. In the snowfall, he did not see further than 5 metres. He did not see the dog in front of him, he did not see it behind either, and he did not know which direction it had gone. After a little thinking, he went to look for his poodle. The old poodle can only walk half as fast as its master. The mathematician chose a searching strategy, such that the following condition should hold for the smallest possible value of the constant $c$: if the dog is at a distance $x$ from him then he needs to cover a distance of at most $cx$ to find it. What is this smallest value of $c$?

The areas of the faces of a tetrahedron are $a$, $b$, $c$, $d$. The angle enclosed by the faces of areas $a$ and $b$ is $\gamma$, the angle of $b$ and $c$is $\alpha$, and the angle of $c$ and $a$ is $\beta$. Prove that $d^2$=$a^2$+$b^2$+$c^2$-2$ab^.$cos $\gamma$-2$bc^.$cos $\alpha$-2$ca^.$cos $\beta$.

Ann, Bill and Cecilia play the following game. They take turns choosing an integer between 1 and 10, and adding that number to the sum of the numbers that have been said so far. (For simplicity, in each turn, the player states the sum.) The player who is first able to say 100 wins the game. Prove that the two girls, with the appropriate strategy, are able to achieve that one of them is the winner.

$a_1$,$a_2$,...,$a_n$ és $b_1$,$b_2$,...,$b_k$ are two given sequences with positive integer terms with $a_i \le k$ and $b_j \le n$. Show that there are integers 0$\le i_1$<$i_2 \le n$ and 0$\le j_1$<$j_2 \le k$ such that $a_{i_1+1}+\ldots +a_{i_2}=b_{j_1+1}+\dots +b_{j_2}$.

The points $E$ and $G$, respectively, are obtained by rotating vertex $D$ of a square ABCD about $A$ and vertex $B$ (lying opposite $D$) about $C$ outwards, through the same acute angle $\alpha$. The intersection of lines $AE$ and $BG$is $F$. Line $GD$ intersects the circles GCB and BDE again at points $L$and $M$, respectively. Prove that the points $A$, $B$, $F$, $L$, $M$ are cyclic.

A convex solid has six square faces and eight regular hexagonal faces. Given that no two square faces have a vertex in common, and the solid has an edge of unit length, determine the volume of the solid.

How many squares are formed by the lattice points in an $n$×$n$ square lattice?

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

The escribed circle drawn to side $AC$ of a triangle ABC touches the lines of sides $BC$, $AC$ and $AB$ at the points $A_1$, $B_1$ and $C_1$, respectively. Let $F$ denote the midpoint of the line segment $A_1 B_1$. Prove that $\angle B_1 C_1 C$=$\angle A_1 C_1 F$.

In how many zeros does the number $4^{56}$ $+6^{54}$ end in decimal notation?

$A$, $B$ and $C$ are three distinct given points lying on the line $e$ in this order. What is the locus of the points $P$ for which the inscribed circle of triangle ACP touches line $e$ at $B$?

Consider a convex $n$-sided polygon that has no four vertices lying on the same circle. Three vertices are selected in every possible way, and the circle passing through the three points is drawn. Such a circle is called thin if the vertices of the polygon (except the vertices of the triangle) all lie outside the circle. The circle is called fat if all vertices of the polygon lie on the closed disc bounded by the circle. Are there more thin circles or more fat circles?

Let $t$>0 be a real number, and let $T_i$ denote the sum $t^{i}+\frac{1}{t^{i}}$ (for all positive integers $i$). Prove that $T_{1}T_{2}T_{4}\ldots T_{2^{k-1}}=T_{1}+T_{3}+T_{5}+\ldots +T_{2^{k}-3}+T_{2^{k}-1}$ for all positive integers $k$.

With three planes, a cuboid $T$ is divided into eight smaller cuboids. The small cuboids are coloured black and white such that adjacent pieces have different colours. Given that the sum of the volumes of the white cuboids equals the sum of the volumes of the black cuboids, show that one of the planes halves the volume of $T$.

In a parallelogram ABCD , $AB$>$BC$. Construct that point $P$ in the interior of the parallelogram for which $\angle$ APD +$\angle$ BPC $=180^o$ and $\angle$ PAB +$\angle$ PDA $=90^o$.

Consider a finite set of positive numbers such that no number is greater than the sum of the other numbers. Prove that it is possible to partition the numbers into two sets such that the sum of the numbers in one set is at most twice the sum of the numbers in the other set.

Let $f$ be a real function defined on the set of positive real numbers such that $f\big(\sqrt{xy}\,\big) = f\left(\frac{x+y}{2}\right)$ for all $x$,$y$>0. Prove that $f$ is a constant function.

How long is the shortest line segment that halves the area of the triangle with sides 3, 4 and 5?

Lines $a$ and $b$ are given lines in the plane, $A$ is a given point not lying on $a$, and $B$ is a given point not lying on $b$. If point $O$ does not coincide with $A$ or $B$, then there exists a transformation composed of a rotation about $O$and a central enlargement from $O$, such that $A$ is mapped to $B$. What is the locus of those points in the plane for which this transformation maps the intersection of lines $OA$ and $a$ to the intersection of lines $OB$ and $b$?

For $x$>1, let $A$($x$) denote the sum of reciprocals of the positive square-free numbers less than $x$, and let $B$($x$) denote the sum of reciprocals of the non-square-free positive numbers less than $x$. Prove that $A$($x$)>$B$($x$).

Regular triangles ABD , BCE and CAF are drawn over the sides of a triangle ABC , on the outside. Let $G$ and $H$ denote the midpoints of the line segments $BD$ and $BE$, respectively, and let the midpoint of triangle CAF be $I$. Prove that the lines $AH$, $CG$ and $BI$ are concurrent.

Let $p \ge$2 be a real number. Prove that ($x$+$y$)$^p$+($z$+$v$)$^p$+($x$+$z$)$^p$+($y$+$v$)$^p \le x^p$+$y^p$+$z^p$+$v^p$+($x$+$y$+$z$+$v$)$^p$ for all non-negative real numbers $x$, $y$, $z$and $v$.

In the Four-square Round Forest, trees form a regular triangular lattice. Is it possible to build a fence around a rectangular part of the forest such that the vertices of the rectangle are lattice points and the number of lattice points on the boundary of the rectangle is the same as in the interior?

What is the sum of the digits in the cube of the number that consists of $n$ digits of 9?

The line segment $A_0 A_{10}$ is divided into 10 equal parts. The dividing points are $A_{1}, A_{2}, \ldots, A_{9}$ in this order. The third vertex of the regular triangle drawn over the line segment $A_8 A_{10}$ is $B$. Show that $\angle BA_0 A_{10}$+$\angle BA_2 A_{10}$+$\angle BA_3 A_{10}$+$\angle BA_4 A_{10} =60^o$.

Is there a positive integer $n$ such that each root of the equation ( 4$n^2$-1 ) $x^2$-4$n^2 x$+$n^2$=0 can be represented as a finite decimal?

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

Consider two circular discs that have no common points. The circle drawn on the line segment connecting their centres as diameter intersects each of the common exterior tangents at two points. Prove that the diagonals of the quadrilateral formed by the four points of intersection are the common interior tangents.

The interior angle bisectors of triangle ABC intersect the opposite sides at $A_1$, $B_1$, $C_1$. Prove that the area of the triangle $A_1 B_1 C_1$ cannot be greater than one fourth of the area of triangle ABC .

Find those polynomial functions of integer coefficients that assign a positive prime number to every Fibonacci number. (The sequence $F_n$ of Fibonacci numbers is defined by the recurrence relation $F_n$=$F_{n-1}$+$F_{n-2}$ with seed values $F_0$=0, $F_1$=1.)

Prove that the sum of the squares of seven consecutive integers cannot be a perfect square.

Consider the cubic equation $px^3$-$qx^2$-$rx$+$s$=0 where $p$, $q$, $r$, $s$ are positive numbers such that $ps$=$qr$. Prove that the equation has two different real roots. On what condition are there three different roots?

The points $K$, $L$, $M$ and $N$, respectively, lie on sides $AB$, $BC$, $CD$ and $DA$ of a square ABCD . Given that $\angle$ KLA =$\angle$ LAM =$\angle$ AMN $=45^o$, prove that $KL^2$+$AM^2$=$LA^2$+$MN^2$.

Prove that for every convex polyhedron with five faces there exists a plane that does not pass through any vertex but intersects each face.

Prove that there are infinitely many ways to express 169 as a sum of the squares of two rational numbers.

In an acute triangle ABC , $\alpha$<$\beta$ (with conventional notations). Let $R$and $P$ be the feet of the altitudes drawn from vertices $A$ and $C$, respectively. Let $Q$ denote a point of line $AB$, different from $P$, such that $AP^. BQ$=$AQ^. BP$. Prove that line $RB$ bisects the angle PRQ .

Prove that if $\alpha$, $\beta$ and $\gamma$ are the angles of an acute-angled triangle then $\mathop{\rm tg}\nolimits^3 \alpha +\mathop{\rm tg}\nolimits^3 \beta +\mathop{\rm tg}\nolimits^3 \gamma \ge 9\sqrt{3}$.

In an isosceles triangle ABC , $D$ and $E$ denote points on the legs $AB$ and $AC$, respectively, such that $AD$=$BC$=$EC$. What may be the measure of the angle at vertex $A$ if triangle ADE is isosceles?

The escribed circle drawn to side $AB$ of a triangle ABC touches the lines of sides $AB$, $BC$ and $CA$ at the points $E$, $F$, $G$, respectively. The intersection of lines $AF$ and $BG$ is $H$. The inscribed circle of the triangle formed by the midlines of triangle ABC touches the side parallel to $AB$ at point $N$. Prove that the points $E$, $H$ and $N$ are collinear.

Find all polynomials $f$ of integer coefficients such that for every positive integer $n$ the prime factors of the numbers $n$ and $f$($n$) coincide.

In Elastica, only natural numbers less than ten thousand are known. These numbers are written down in a flexible way. The rule for Elastican numbers is as follows: each number is represented in the lowest possible base such that the resulting number have at most four digits. Unfortunately, the numbers represented in this way are not always possible to decode uniquely: For example, the four-digit Elastican number 1101 may mean 13 as well as 37. Is there a three-digit Elastican number that represents more than one natural number?

On a white sheet of squared paper, 40 small squares are coloured red. Prove that it is possible to select 10 red squares such that no two of them have a point in common.

Prove that the equation $1+2+2^{2}+\ldots +2^{x}=y^{z}$ has no solution of positive integers $x$, $y$, $z$ where $z$>1.

What is the largest possible fraction of a triangle that can be covered by a square with all its vertices on the perimeter of the triangle?

Let $a$ and $b$ be positive integers. How many non-negative integers $n$ are there such that $\left[\frac{n}{ab}\right] + \left[\frac{n+b}{ab}\right] + \left[\frac{n+2b}{ab}\right] + \ldots + \left[\frac{n+(a-1)b}{ab}\right] = \left[\frac{n}{ab}\right] + \left[\frac{n+a}{ab}\right] + \left[\frac{n+2a}{ab}\right] + \ldots + \left[\frac{n+(b-1)a}{ab}\right]$? ([$x$] denotes the greatest integer not greater than $x$.)

Is there a plane that does not pass through any vertex of a regular octahedron but intersects all the faces?

Show that 168 cannot be expressed as the sum of the squares of two rational numbers.

Given the points $P_1$, $P_2$, $P_3$ and the distances $e_1$, $e_2$, $e_3$, construct a circle such that the length of the tangent drawn from $P_i$ is $e_i$.

$P$ and $Q$ are interior points on side $AC$ of a non-isosceles triangle ABC such that $\angle ABP = \angle QBC< \frac 12\cdot \angle ABC$ . The interior angle bisectors drawn from $A$ and $C$ intersect the line segment $BP$ at points $K$and $L$, and the line segment $BQ$ at points $M$ and $N$, respectively. Prove that the lines $AC$, $KN$ and $LM$ are concurrent.

The names of 100 convicts are placed in 100 numbered drawers in some order. Then the convicts are brought one by one from their cells in a random order, and each of them is allowed to pull out 50 drawers one by one. If he finds his own name in a drawer then he is led to a separate room. Otherwise all 100 convicts are executed immediately. Finally, if each of them succeeds, they are all set free. Show that, knowing the rules, the prisoners can follow a strategy that results in a larger than 30% probability of going free.

Prove that if the natural numbers $a$ and $b$ only differ in the order of their digits then the sum of the digits in the numbers 5$a$ and 5$b$ is the same.

Let ($n$,$k$) denote the greatest common divisor of the positive integers $n$ and $k$, and let [$n$,$k$] denote their least common multiple. Show that, for all positive integers $a$, $b$, $c$, the greatest common divisor of the numbers [$a$,$b$], [$b$,$c$], [$c$,$a$] equals the least common multiple of the numbers ($a$,$b$), ($b$,$c$), ($c$,$a$).

$F$ is the midpoint of base $BC$ of the isosceles triangle ABC . $D$ is an interior point of the line segment $BF$. The perpendicular dropped from point $D$onto $BC$ intersects side $AB$ at $M$, and the line through point $D$, parallel to $AB$ intersects side $AC$ at $P$. Express the ratio of the areas of triangles AMP and ABC in terms of $k$=$BD$:$BC$.

Given the parallelogram ABCD and points $F$ and $G$, such that $AF$=$FC$, $BG$=$GD$ and given that triangles AFC and BGD are similar, prove that the line $FG$ is perpendicular to a side of the parallelogram.

An absent minded professor is travelling in a train along a subway line having infinite length in both directions. He would like to get from Alpha station to Tau station. At every station, the probability that he looks up from his notes is the same $p$>0 (independently of each other). If he looks up and observes that he has reached his destination, then he gets off. If he can see that he has passed his destination, then he also gets off and gets on the train travelling in the opposite direction. If he looks up and sees that he has not reached his destination yet, he stays in his train. If he does not look up, he will stay on the train wherever he is. What is the probability that he will get off at Tau station after a while?

The sequence ($a_n$) is defined as follows: $a_1$=$a_2$=1, $a_3$=2, $a_{n+3} =\frac{a_{n+2}a_{n+1}+n!}{a_n}$ ($n \ge$1). Prove that all terms of the sequence are integers.