KöMaL 2008

The perimeter of a certain convex hexagon is $p$. The midpoints of the sides form an inscribed hexagon such that all its angles are $120^o$ and its perimeter is $p_1$. Prove that $p_1\le\frac{\sqrt3}2p.$

Given a continuous function $f\colon \mathbb{R}\to\mathbb{R}$ such that for any real number $a$>0, the sequence $f(a),f(2a),f(3a),\ldots$ converges to 0. Show that $\lim\limits_{x\to\infty}f(x)=0$.

Prove that for every prime number $p$ and integer $r$ there exists a nonnegative integer $n$for which $\binom{2n}{n}\equiv r\pmod{p}$ .

The lengths of tangents drawn from the points $A$, $B$ and $C$ to a sphere $s$ are $a$, $b$and $c$, respectively. $T$ is a point on the sphere, not co-planar with $A$, $B$ and $C$, such that $\frac{AT}{a}=\frac{BT}{b}=\frac{CT}{c}$ . Prove that the sphere passing through the points $A$, $B$, $C$, $T$ touches $s$.

Prove that the inequality $\sum\limits_{i=1}^n\ \sum\limits_{j=1}^n \frac{a_ia_j}{i+j}\ge 0$ is true for all real numbers $a_1$,$a_2$,...,$a_n$.

The numbers 1,2,...,$N$ are colored with 3 colors such that each color is used at most $\frac{N}{2}$ times. Let $A$ be set of all (ordered) 4-tuples ($x$,$y$,$z$,$w$), consisting of such numbers, such that $x+y+z+w\equiv 0 \pmod{N}$ and $x$, $y$, $z$, $w$ have the same color. Similarly, let $B$ be the set of all (ordered) 4-tuples ($x$,$y$,$z$,$w$) such that $x+y+z+w\equiv 0 \pmod{N}$ , the numbers $x$, $y$ have the same color, $z$ and $w$ have the same color, but these two colors are distinct. Prove that |$A$|$\le$|$B$|.

In the convex quadrilateral ABCD , denote by $r_A$, $r_B$, $r_C$ and $r_D$ the inradii of triangles BCD , CDA , DAB and ABC , respectively. Prove that the quadrilateral ABCD is cyclic if and only if $r_A$+$r_C$=$r_B$+$r_D$.

The function $f\colon \mathbb{R}\to \mathbb{R}$ satisfies the equation $f$ ( $x$+$f$($y$) ) +$f$ ( $f$($x$)+$y$ ) =2$x$+2$y$. Prove that $f$(2$x$)=2$f$($x$) for all real numbers $x$.

Let $\mathcal{U}\subset P(S)$ be an upset and let $\mathcal{D}\subset P(S)$ be a downset on a set $S$ of $n$ elements, i.e. ($a$) for every $X\in\mathcal{U}$ we have $Y\in\mathcal{U}$ for all $X \subset Y \subset S$; ($b$) for every $X\in\mathcal{D}$ and $Y \subset X$, $Y\in\mathcal{D}$ . Prove that $|\mathcal{U}| \cdot |\mathcal{D}| \ge 2^n \cdot |\mathcal{U}\cap\mathcal{D}|$.

A simple graph on 2$n$ points has $n^2$+1 edges. Prove that the edges form at least $n$triangles.

$n$ points are selected at random on a sphere, independently of each other. What is the probability that the convex hull of the selected points contains the center of the sphere?

Is there any polynomial $p$ having degree at least 1 over the reals that satisfies $p^2$($x$)-1=$p$($x^2$+1) for all real numbers $x$?

Let $H$ be a set with $n$ elements and let each of the families $\mathcal S$ and $\mathcal T$ consist of $p$ subsets of $H$ such that these 2$p$ subsets are pairwise distinct. Suppose that for every $A\in \mathcal S$ and $B\in \mathcal T$ , the sets $A$ and $B$ have at least one common element. Show that $\frac{p}{2^n}<\frac{3-\sqrt{5}}{2}$.

The point $D$ lies in the triangle ABC such that the circles inscribed in triangles ABD , BCD , and CAD pairwise touch each other. On lines $BC$, $CA$, $AB$, $AD$, $BD$, $CD$, denote the touching points by $A_1$, $B_1$, $C_1$, $A_2$, $B_2$, $C_2$, respectively. Let lines $B_1 C_2$ and $B_2 C_1$ meet at $E$, and let lines $A_1 C_2$ and $A_2 C_1$ meet at $F$. Show that the lines $AF$, $BE$, and $C_1 D$ are concurrent.

Prove that, for every odd prime $p$, $\sum_{i=1}^{p-1}{2^i i^{p-2}}- \sum_{i=1}^{\frac{p-1}{2}}{i^{p-2}}$ is divisible by $p$.

In space, $n$+1 points $P_1$,$P_2$,...,$P_n$ and $Q$ are given, $n \ge$4, no four of which are in the same plane. It is known that for each triple of distinct points $P_i$, $P_j$ and $P_k$ one can find a point $P_l$ such that $Q$ is interior to the tetrahedron $P_i P_j P_k P_l$. Show that $n$ must be even.

Denote by $F_n$ the $n$th Fibonacci number ($F_0$=0, $F_1$=1, $F_{k+1}$=$F_k$+$F_{k-1}$). Show that one can find a positive integer $n$, having at least 1000 distinct prime divisors, such that $n$ divides $F_n$.

If $p$($x$) is a polynomial with degree less than 2$n$, show that $|p(n)| \le 2\sqrt{n} \cdot \max\big( |p(0)|, |p(1)|, \ldots, |p(n-1)|, |p(n+1)|, |p(n+2)|, \ldots, |p(2n)| \big).$

The exsphere of the tetrahedron ABCD , corresponding to face ABC , touches the face ABC at $P$. The exsphere, corresponding to face ABD , touches ABD at $Q$. Show that $ACB\sphericalangle + AQB\sphericalangle = ADB\sphericalangle + APB\sphericalangle$.

Let $p$ be an odd prime and let 1<$a$<$p$ be an integer. Prove that $p$ divides $\sum_{k=0}^{p-2} {(-1)}^ka^{k^2}$ if and only if there exists an odd positive integer $k$ for which $a^k$ is congruent to 1 modulo $p$.

Let $H$ a set and denote by $P$($H$) the family of all subsets of $H$. Suppose that $f$ and $g$are $P$($H$)$\to P$($H$) maps such that $f$($X$)$\subset f$($Y$)$\subset H$ and $g$($X$)$\subset g$($Y$)$\subset H$whenever $X \subset Y \subset H$. Prove that there exist sets $A$,$B \subset H$ for which $f$($A$)=$H$$B$and$g$($B$)=$H$\$A$.

Do there exist a set of cirlces in the plane such that every line intersects at least 1, but at most 100 of them?

Let ABCD be a circumscribed trapezoid such that the lines $AD$ and $BC$ intersect at point $R$. Denote by $I$ the incenter of the trapezoid, and let the incircle touch the sides $AB$ and $CD$ at points $P$ and $Q$, respectively. Let the line through $P$, which is perpendicular to $PR$, meet the lines angle bisector $AI$ and $BI$ at points $A_1$ and $B_1$, respectively. Similarly, let the line through $Q$, perpendicular to $QR$, meet $CI$ and $DI$at $C_1$ and $D_1$, respectively. Show that $A_1 D_1$=$B_1 C_1$.

We are given two triangles. Their side lengths are $a$,$b$,$c$ and $A$,$B$,$C$ respectively, the areas are $t$ and $T$, respectively. Prove that -$a^2 A^2$+$a^2 B^2$+$a^2 C^2$+$b^2 A^2$-$b^2 B^2$+$b^2 C^2$+$c^2 A^2$+$c^2 B^2$-$c^2 C^2 \ge$16$tT$.

Let 0$\le k \le n$ and $m \ge$2 be integers. Consider the $k$-element subsets of $\{1,2,\ldots,n\}$; in every such subset, compute the residue of the sum of elements, modulo $m$. Prove that if the $m$ residues are distributed uniformly - i.e. every residue occurs exactly $\frac{\binom{n}{k}}{m}$ times - then $n \ge m$.

A path $P_1$,$P_2$,...,$P_k$ in a simple graph is said to be a fan if the degree of every point $P_i$ is $i$. What is the maximum number of points that a fan may have in a simple graph on $n$ points?

The sequence $a_1, a_2, \ldots$ of integers has infinitely many positive terms and infinitely many negative terms. For every $n$, the remainders of $a_1, a_2, \ldots, a_n$ divided by $n$are pairwise different. How many times does the number 2008 occur in the sequence?

The radius of a circle inscribed in a triangle is $r$. The tangents drawn to the circle parallel to the sides cut three small triangles off the original triangle. Prove that the sum of the radii of the inscribed circles of the small triangles is also $r$.

Prove that every number not greater than $n$! can be expressed as a sum of at most $n$different factors of the number $n$!.

The orthocentre of an acute-angled triangle is $M$, the centre of its circumscribed circle is $O$, and its sides are $a$<$b$<$c$. The line of side $c$, the line of the altitude drawn to side $b$, and the line $MO$ form a triangle that is similar to the original triangle but with opposite orientation. Find the angles of the triangle.

The angles of a triangle are $\alpha$, $\beta$, $\gamma$. What is the largest possible value of sin $\alpha$sin $\beta$cos \gamma$+sin^2 \gamma$?

$O$ is a given point in the plane. Give an example for congruent triangles $A$ and $B$, such that every circle centred at $O$ has parts of the equal areas inside the triangles $A$and $B$.

Determine all functions $f\colon \mathbb{R}\to\mathbb{R}$ for which $f$($x$)+$f$ ( $x$+$f$($y$) ) =2$x$+$y$ for all real values of $x$ and $y$.

Given the ABC and PQR triangles such that vertex $A$ of triangle ABC bisects side $QR$. Vertex $P$ bisects side $BC$. Line $QR$ bisects angle BAC , and line $BC$ bisects angle QPR . Prove that $AB$+$AC$=$PQ$+$PR$.

Is it possible to draw $a$) 13, $b$) 14 circles on the plane, such that each of them is tangent to exactly three others?

Al and Bill take turns choosing cards one by one from the deck of 81 SET cards (see the article on page http://galileo.stmarys-ca.edu/bdavis/set.pdf ) and placing them on the table. When SET first appears on the table, the player placing the last card loses the game. If Al begins, which player has a winning strategy?

Given five lattice points on the coordinate plane, show that there is a pair among them, such that the midpoint of the line segment formed by them is also a lattice point.

Let $G$ be a simple graph in which the degree of every vertex is at least 2. Show that $G$ has a vertex, such that every edge drawn from that vertex belongs to a circuit.

Prove that if both midlines of a quadrilateral halve its area then the quadrilateral is a parallelogram.

Erect a perpendicular to each of the sides $AB$, $BC$, $CA$ of a triangle of perimeter $k$ at the vertices $A$, $B$, $C$, respectively. Let $K$ denote the perimeter of the new triangle obtained from the perpendiculars. Prove that $\frac{K}{k}= \cot\alpha+ \cot\beta+ \cot\gamma$, where $\alpha$, $\beta$ and $\gamma$ are the angles of the triangle ABC .

Let $A$, $B$, $C$, $a$, $b$, $c$ denote positive integers, $a^. b^. c$>1. Prove that there exists a positive integer $n$, such that $A^. a^n$+$B^. b^n$+$C^. c^n$ is a composite number.

The lengths of the tangents drawn from the points $A$ and $B$ to a circle $k$ are $a$ and $b$, respectively. $T$ is a point of the circle not collinear with $A$ and $B$ for which $\frac{AT}{a}=\frac{BT}{b}$ . Prove that the circle passing through the points $A$, $B$, $T$touches $k$.

The positive integers $a$ and $b$, written in decimal system, can be obtained from each other by rearranging their digits. Prove that $a$) the sum of the digits of 2$a$ equals that of 2$b$; $b$) if both $a$ and $b$ are even, then the sum of the digits of $\frac{a}{2}$ equals that of $\frac{b}{2}$.

Prove that for every positive integer $n$, $\min_{k \in \mathbb{N}} \left[k + \frac{n}{k}\right] = \big[\sqrt{4n + 1}\,\big]$.

Let $S$($n$) denote the sum of the digits of the natural number $n$. Prove that there are infinitely many natural numbers $n$ not ending in 0, such that $S$($n^2$)=$S$($n$).

Find all right-angled triangles, such that the sides are integers, and the sum of the legs is obtained by adding 6 to the hypotenuse.

$C$ is a given point in the plane, and there is a given circle in the plane. If $AB$ is a diameter of the circle, what is the locus of the orthocentres of triangles ABC ?

Find the maximum of the function $f(x)=\sqrt{x-2} + \sqrt{2x-7} + \sqrt{18-3x}$ .

Lisa and Sam are playing the following game: Lisa marks a point $L_1$ on side $AB$ of a given triangle ABC . Then Sam chooses a point $S$ on side $BC$. Finally, Lisa marks a point $L_2$ on side $CA$. Lisa is trying to make the area of triangle $L_1 SL_2$ as large as possible, and Sam is trying to make it as small as possible. What will be the area of triangle $L_1 SL_2$ if both of them play as well as possible?

Prove that if the vertices of a nonagon are all lattice points then it has three vertices, such that the centroid of the triangle formed by them is also a lattice point.

$a$ and $n$ are positive integers, such that $n$>1 and $n\mid a^n-1$. Prove that ($a$-1;$n$)>1.

Let $G$ be a simple graph in which the longest path consists of $k \ge$3 edges and the degree of each vertex is at least $k$/2. Show that every edge of $G$ belongs to a circuit. (A path is a sequence of linking edges that visits each vertex at most once.)

Three points are selected at random on a circle, independently of each other. What is the probability that the resulting triangle is acute-angled?

Prove that natural numbers can be coloured in 2008 colours, such that each colour occurs, and whenever 3$a$+5$b$=7$c$ is true, there are two numbers among $a$, $b$ and $c$ that have the same colour.

The diagram shows a plane figure that cannot cover a unit semicircle but two congruent copies of the figure can cover a unit circle. Is there a convex figure that has the same property?

Each of the digits 0 to 9 is written in 10 fields of a 10×10 table. Is it possible that every row and every column has at most 4 different digits in it?

Find all sequences ($a_n$) of positive integers, such that ($a_i$,$a_j$)=($i$,$j$) for all $i \ne j$.

Prove that if a symmetrical trapezium has an inscribed circle then its height is the geometric mean of the bases.

Four points are selected at random on a sphere, independently of each other. What is the probability that the resulting tetrahedron contains the center of the sphere?

Prove that if the lengths of the sides of a triangle are 2, 3 and 4 then it has angles $\alpha$ and $\beta$, such that $2$\alpha$+3$\beta$=180^o$.

$P$ is an interior point of side $B_1 B_2$ of an acute-angled triangle $AB_1 B_2$. Let $Q$denote the reflection of $P$ about point $A$. Let $D_i$ denote the perpendicular projection of $P$ onto the line segment $AB_i$, and let $F_i$ be the midpoint of the line segment $D_i B_i$. Prove that if $QD_1$ is perpendicular to $PF_1$, then $QD_2$ is perpendicular to $PF_2$.

Solve the equation $x^4$-7$x^3$+13$x^2$-7$x$+1=0.

The angle bisector $f$ drawn from vertex $A$ of an acute-angled triangle ABC intersects side $BC$ at $D$ and the circumscribed circle at $E$. The altitude drawn from $C$ intersects $f$ at $M$ and the circumscribed circle at $Q$. The altitude drawn from $B$ intersects $f$ at $N$ and the circumscribed circle at $P$. Prove that $\frac{BD}{DC} = \frac{PE \cdot AN \cdot ME}{QE \cdot CM \cdot NP}.$

Prove that $\sum_{k=0}^m \binom mk \binom{t+k}{m} =\sum_{k=0}^m\binom mk\binom tk\cdot2^k$ for arbitrary positive integers $m$, $t$.

Are there four positive integers, such that the greatest common divisor of each pair of them is greater than 1, but the greatest common divisor of each three is 1?

Prove that if $n \ge$6 then every triangle can be dissected into $n$ isosceles triangles.

Is it possible to leave one out of the numbers $1, 2, \ldots, 2008$ so that the remaining 2007 numbers can be listed in an order $a_1, a_2, \ldots, a_{2007}$, in which the differences ${|a_1-a_2|}, {|a_2-a_3|}, \ldots, {|a_{2006}-a_{2007}|}, {|a_{2007}-a_1|}$ are all different?

Given infinitely many cuboids with edges parallel to the axes of the coordinate space, one vertex at the origin and all vertices at points of non-negative integer coordinates. Is it always possible to select two, one of which contains the other?

The points of tangency of the inscribed circle on the sides of a triangle ABC are $E$, $F$and $G$. The distances of an arbitrary point $P$ of the inscribed circle from the sides are $a$, $b$ and $c$, and its distances from the lines $FG$, $EG$ and $EF$ are $e$, $f$and $g$. Prove that abc = efg .

Solve the following equation on the set of integers: $2^{\frac{x-y}{y}} - \frac{3}{2}y = 1$.

Given a side of a triangle and the two points on it where the lines dividing the opposite angle into three equal parts intersect the side, construct the triangle.

A digit is written in each field of a 10×10 table. Every digit occurs exactly 10 times. Prove that there is a row or column that contains at least 4 different digits.

Determine the number of sections the plane is divided by the sides of a regular $n$-gon.

Assume xyz =8. Prove that $\frac{1}{\sqrt{x^2+1}}+\frac{1}{\sqrt{y^2+1}}+\frac{1}{\sqrt{z^2+1}}>1$.

Let $a$ and $b$ be positive integers. Is it possible that $a^2$+4$b$ and $b^2$+4$a$ are both perfect squares?

On sides $BC$ and $CD$ of a given rectangle ABCD , construct the points $P$ and $Q$, such that ABPQ forms a kite that is symmetrical to the diagonal $AP$.

Find possible numbers $a$, $b$, $c$, such that ($n$+3)$^2$=$a^.$($n$+2)$^2$+$b^.$($n$+1)$^2$+$c^. n^2$ for every positive integer $n$.

Point $A$ moves along one arm of an angle of vertex $C$ and point $B$ moves along the other arm so that $CA$+$CB$=1 remains valid. Prove that there exists a point that lies on the perpendicular bisectors of $AB$ in all positions.

Determine those quadrilaterals ABCD , such that $PA^2$+$PC^2$=$PB^2$+$PD^2$ for all points $P$ of the plane.

Solve the following simultaneous equations on the set of positive numbers: $x^4$+$y^4$-$x^2 y^2$=13, $x^2$-$y^2$+2$xy$=1.

The angle bisectors of a triangle intersect the circumscribed circle at the points $P$, $Q$, $R$. Given the points $P$, $Q$, $R$, construct the triangle.

Prove that $\big(\sqrt{2}+1\big)^{\frac{1}{100}}+ \big(\sqrt{2}-1\big)^{\frac{1}{100}}$ is an irrational number.

The hexagon ABCDEF and the triangle PQR lie in the same plane. The quadrilaterals ABRQ , CDPR , EFQP are all rectangles. $a$) Prove that the perpendicular bisectors of the sides $BC$, $DE$, $FA$ are concurrent. $b$) Show that there exists a triangle $P$'$Q$'$R$', too, for which the quadrilaterals BCR '$Q$', DEP '$R$', FAQ '$P$' are rectangles.

Let $n$ be a positive integer, and let $a_1$,$a_2$,...,$a_n$ be pairwise different integers. Prove that the polynomial $(x-a_1)(x-a_2) \ldots (x-a_n)-1$ cannot be expressed as a product of two polynomials of integer coefficients whose degrees are at least one.

A billiards ball at corner $A$ of a rectangular billiards table ABCD is struck. It starts moving along the angle bisector. It rebounds from the sides $CD$, $BC$ and finally $AB$, and collides head-on with a ball that is at the centre of the rectangle. In what other direction is it possible to strike the ball at point $A$, so that it rebounds from three different sides and then hits the ball at the centre head-on?

Prove that if $a$, $b$, $c$, $d$ are integers and $a$+$b$+$c$+$d$=0 then 2($a^4$+$b^4$+$c^4$+$d^4$)+8 abcd is a square number.

For what positive integers $k$ does 1 occur as a term of the sequence ($a_n$), given that $a_1$=$k$ and $a_{n+1}=\frac{a_n}{2}$ if $a_n$ is even, and $a_{n+1}$=$a_n$+5 if $a_n$ is odd?

Solve the equation $cos^n$x$-sin^n x$=1 for all positive integers $n$.

Three lines passing through a point $P$ are cut by two other lines at the points $A$, $B$, $C$and $A$', $B$', $C$', respectively. Prove that $\frac{AA'}{A'P}\overrightarrow{BC} +\frac{BB'}{B'P}\overrightarrow{CA} +\frac{CC'}{C'P}\overrightarrow{AB}=\mathbf{0}$, where the ratios are considered directed.

Is it possible to obtain a closed disc as a union of two disjoint and congruent point sets? (A closed disc includes the points of the circumference of the circle, too.)

Prove that the graph of the function $x\mapsto \frac{9x+7}{3x+12}$ is symmetric to the point (-4;3).

Define a leaf as a figure obtained by the intersection of a finite number of isometric closed discs, and a side of a leaf as the intersection of the leaf with the circumference of any circle out of the ones forming the leaf. Prove that the sides of a leaf are connected.

Cards are drawn one by one from a deck of French cards until the hand contains a poker. Given that the first card drawn is an ace, what is the probability that the process terminates with an ace poker?

7 fields of a 5×5 chessboard are red, and 18 fields are blue. Two of the red fields lie along the edge of the board. Line segments separating two red fields are also coloured red, and line segments separating blue fields are coloured blue. All the rest of the lines, including the edges of the board, are black. There are 35 black line segments altogether. How many red line segments are there?

Find the locus of points on the plane such that the sum of squares of their distances from two given points is constant.

We determine the square root of each 4-digit number, and if the result is not an integer, we round it to the nearest integer. Decide whether we have rounded up or down more numbers.

Given two points in the interior of a $45^o$ angle, construct an isosceles triangle that has its base on one arm of the angle, its apex on the other arm, and each leg passing through one of the given points.

From the endpoints of a chord $AB$ of a circle, drop perpendiculars onto a tangent drawn at a point $P$ of the circle, different from $A$ and $B$. Drop a perpendicular from $P$ onto the chord $AB$. Prove that the perpendicular dropped onto the chord is the geometric mean of the perpendiculars dropped onto the tangent.

Solve the equation $x^5$-$x^3 y^2$-$x^3 y$-$x^2 y^3$+$y^2$=0 on the set of integers.

$P$ and $Q$ are two different points on the diagonal $AC$ of a parallelogram ABCD . The line through $P$ parallel to $AB$ intersects the sides $BC$ and $AD$ at $K$ and $L$, and the line through $Q$ parallel to $BC$ intersects the sides $AB$ and $CD$ at $M$ and $N$. Show that the areas of the triangles PNM and QKL are equal.

Sequence ($a_n$) is defined by the recursion $a_0$=0, $a_1$=1 and $a_n$=2$a_{n-1}$+$a_{n-2}$ for $n$>1. Prove that if $2^k\mid n$ , then $2^k\mid a_n$.

Consider a finite number of line segments on a line. Prove that if they are rearranged so that the midpoints of any two line segments are moved closer to each other then the total length of the union of the line segments will not increase.

A mouse is placed in one corner cell of a 3×3×3 lattice of cubes, and a piece of cheese is placed in the central cell. The mouse moves about the lattice in a random way, entering an adjacent cell in each step. In how many steps is it expected to find the cheese?

Annie and Bonnie are playing the following game on a 2008×2008 chessboard: Annie selects a few fields of the board but does not tell Bonnie which. Then she writes in each field the number of selected fields that it is connected to by an edge or vertex. Is that enough information for Bonnie to tell which fields were selected? What is the answer if they play on a 2009×2009 board?

$P$ and $Q$ are points on the sides $BC$ and $CD$ of a rectangle ABCD , such that the triangle APQ is equilateral. Prove that $t_{AQD}$+$t_{ABP}$=$t_{PCQ}$.

Let $t_3(a_1,\ldots,a_k)$ denote the number of three term arithmetic progressions that can be selected from the terms of a sequence $a_1$<$a_2$<...<$a_k$. Prove that $t_3(a_1,\ldots,a_k)\le t_3(1,2,\ldots,k)$.

$M$ is a point in the interior of a square ABCD , such that DCM $\angle$= MAC \angle$=25^o$. Find the measure of the angle ABM .

Given a convex quadrilateral, construct a rhombus with vertices lying on the sides of the quadrilateral, and sides parallel to the diagonals of the quadrilateral.

Let $n$ be a positive integer. Prove that $\sum_{0\le k<n/2} \binom{n}{2k+1} 13^k$ is divisible by $2^{n-1}$.

Solve the following equation: $\sqrt{2}\cdot (\sin x+\cos x)= \tan x+ \cot x$ .

Two altitudes of an acute-angled triangle ABC are $BE$ and $CF$. Line $EF$ intersects the circumscribed circle at $P$ and $Q$. Prove that $AP$=$AQ$.

$E$ is the midpoint of the arc $BC$ on the circumscribed circle of a cyclic quadrilateral ABCD . The midpoint of arc $DA$ is $F$. Let $P$ and $Q$ be the centres of the inscribed circles of triangles ABC and ABD , respectively. Show that $PQ$ is parallel to $EF$.

Prove that ($a^2$+2)($b^2$+2)($c^2$+2)$\ge$3($a$+$b$+$c$)$^2$ is true for all real numbers $a$, $b$, $c$.

Show that the expression $\frac{x}{y^3-1}+\frac{y}{1-x^3}+\frac{2(x-y)}{x^2y^2+3}$ has a constant value for all $x$ and $y$ for which it is meaningful and $x$+$y$=1.

A lecture hall has 24 lamps. Each lamp may operate 4 light bulbs at most. When the maintenance staff has installed four bulbs in some of the lamps, it became clear that the number of bulbs available was not enough. Then they continued with three bulbs per lamp, then two bulbs per lamp, and finished the job with one bulb per lamp. How many bulbs were missing if there were twice as many lamps with one bulb as with four bulbs, and there remained no bulbs at all for half as many lamps as those with three bulbs?

The length of the hypotenuse of a right-angled triangle is $c$, and its area is $t=\frac{c^2}{8}$. Find the exact measures of the angles of the triangle.

The whole numbers are written down up to a certain number $n$ divisible by 50, and then the multiples of 50 are cancelled. Prove that the sum of the remaining numbers is a perfect square.

The base edges and the lateral edges of a truncated pyramid of square base are all 4 units long. The sides of the top face are 2. What is the maximum possible distance between two vertices of the truncated pyramid?

A game uses 11 red, 7 blue and 20 green counters. The bank returns two green counters for one red and one blue, two blue counters for one red and one green, and two red counters for one blue and one green. In making these trades, the player is trying to achieve that he has counters of a single colour. Which colour?

Find all positive integers, such that the result is 2007 if the sum of the digits is subtracted from them.

How many triangles of different shapes are there in which the measures of the angles in degrees are all integers?

ABC is a given triangle. Find that line passing through the vertex $C$ where the perimeter of triangle ABD is greater than the perimeter of the triangle ABC for all points $D$ of the line that are different from $C$.

Three rods of negligible thickness are fixed to each other in a common point at one end, in pairwise perpendicular positions. The lengths of the rods are 1, 2 and 3. The rigid structure is placed on the table with the free ends of the rods touching the table. Determine the height of the common endpoint above the plane of the table.

Is it possible to dissect a regular triangle into $a$) 2007, $b$) 2008 regular triangles?

Find all positive integers that have as many factors divisible by six as factors not divisible by six.

Three sides of a quadrilateral are $a=4\sqrt3$, $b$=9 and $c=\sqrt3$. Sides $a$ and $b$enclose an angle of $30^o$, and sides $b$ and $c$ enclose an angle of $90^o$. What angle do the diagonals enclose?

Solve the following equation on the set of pairs of integers: ($x$+2)$^4$-$x^4$=$y^3$.

A rectangle of sides 15 cm and 20 cm is rotated about a diagonal. What is the volume of the resulting solid of revolution?

Prove if $n$ is a positive integer then $2^{4n}$-1 or $2^{4n}$+1 is divisible by 17.

There are two lamps on the ceiling of a rectangular classroom of length 10 m. They emit conical light beam of apex angles of $90^o$. One is situated in the middle of the ceiling, and illuminates a circle of diameter 6 m on the floor. The lampshade of the other lamp is turned so that the illuminated region on the floor is long enough to draw a line segment of length 10 m on it in the direction parallel to the length of the room, but no light falls on the walls at the ends of the room. Find the distance between the two lamps.

In an arithmetic progression of common difference $d$, $a_1$=1 and $a_n$=81. In a geometric progression of common ratio $q$, $b_1$=1 and $b_n$=81. Given that $\frac qd=0.15$, find all such sequences.

The area of a sector of a circle is 100. what is the radius if the perimeter is a minimum?

Rick, Dan and Alan play table tennis with two players at one side of the table and the third one at the other side. Dan and Alan playing together beat Rick three times as often as Rick beats them, Dan wins the game against Rick and Alan as often as he loses it, and Alan wins the game against Rick and Dan twice as often as he loses it. Last time they played six games during the afternoon, two in each arrangement of players. What is the probability that Rick won at least once?

On 1 March, 2008, the Hungarian National Bank withdrew 1 and 2-forint (HUF) coins from circulation. At the cash desk, prices of individual items are not rounded, but the total is rounded to the nearest five. An information leaflet formulates the rule of rounding as follows: Totals ending in 1 or 2 are rounded down to 0; totals ending in 3 or 4 are rounded up to 5; totals ending in 6 or 7 are rounded down to 5; totals ending in 8 or 9 are rounded up to 0. Martin buys two croissants in the little shop on the corner every morning. In a few days, he saves exactly the price of a croissant. Given that a croissant costs more than 10 forints, what may be the price of a croissant so that he can save that amount in the shortest possible time?

For what values of the constant $c$ does the equation $x^2- 2\left|x+\frac14\right|+c=0$ have exactly three distinct real roots?

A rabbit is sitting at a point $A$, at a distance of 160 m from a straight railway track. The perpendicular projection of $A$ onto the track is $T$. A train is approaching $T$ at a speed of 30 m/s. The distance of the front of the train from point $T$ is 300 m initially. The rabbit can run at 15 m/s. Can he cross the track in any direction before the train comes?

If one leg of a right-angled triangle is increased by 5 and the other leg is decreased by 5, its area will increase by 5. How will the area of the square drawn over the hypotenuse change?

The midpoint of the base $AB$ of an isosceles triangle ABC is $F$, and its orthocentre is $M$. Given that the centroid of the triangle lies on the inscribed circle and $FM=\sqrt6$, what may be the lengths of the sides of the triangle?

In what base of notation is the multiplication $166^.$56=8590 correct?

Buildings on one side of a street are numbered with consecutive even numbers. The sum of the numbers in a block is 78. Given that there are at least 5 buildings in the block, what may be the number of the fourth building?

The radius of a given circle $k$ is 5 units. $P$ is a given point outside the circle. Construct a chord that is 8 units long and subtends a right angle at point $P$.

Given that a right-angled trapezium has an inscribed circle, show that the length of the right-angled leg is the harmonic mean of the lengths of the bases.

Peter's favourite CD has eleven songs on it. He likes the eighth one best. When he inserts the CD in the player, he presses a button to start the first song and then he needs to press another button seven times to reach the favourite song. If he sets the player to ``random'', the player shuffles the 11 songs and plays them in that random order. What is the probability that it takes fewer presses of buttons to reach the favourite song in this way?

Solve the equation 10$x$-5=9[$x$] on the set of real numbers (where [$x$] stands for the greatest integer not greater than $x$).

The number of sides of a polygon is doubled. For what polygons will the sum of the angles of the new polygon be a multiple of the sum of the angles of the original polygon?

The sides of a rectangle are $a$ and $b$. Two congruent circles are placed in a rectangle so that they have no interior point in common. What is the ratio of the sides of the rectangle if the diameter of the largest possible circles is $d=\frac25a+\frac16b$ ?

In a certain board game played with a die, our token is four fields from home. If we roll at least four, we get home. If we roll a three, we can be certain that the next roll will take us home. What is the probability that we need more than two rolls to get home?

The diameter of one base of a truncated cone is 100 mm. If the diameter is increased by 21%, leaving the height and the size of the other base unchanged, the volume of the truncated cone will also increase by 21%. What is the diameter of the other base?

A rectangular sheet of paper is folded along a diagonal. The four corners now form a trapezium with three sides of equal length. How long is the shorter side of the original rectangle if the longer side is 12 cm?

During the first half of the year, the price of gas was raised three times. The percentage increases were 5%, 6% and 10%. Experts predicted a total rise by 1/3 of the price during the whole year. If they are right, what further percentage rise is to be expected this year?

$M$ is the orthocentre of the isosceles triangle ABC of base $AC$. Given that $AC$=$BM$, find the angles of the triangle.

Solve the following equation on the set of real numbers: $\sin^2 (x+y)-\cos^2 (x-y)=1$.

In last year's football Champions' League, it happened for the first time ever that four teams of the same country, England (Arsenal, Chelsea, Liverpool and Manchester United), qualified for the best eight. The eight teams were paired up at random, and the winner from each pair qualified for the best four. $a$) Some English football supporters wished that the English teams would not be paired together, to make it possible for all four of them to qualify for the semi-finals. What was the probability of such a pairing? $b$) Other English supporters wished that the four English teams would form two pairs, to make sure that two of them would qualify for the semi-finals. What was the probability of such a pairing? $c$) In reality, two English teams were paired together and the other two teams got non-English opponents. What was the probability of such a pairing?

Hungarian licence plates of vehicles contain (three letters and) three digits. The digits do not need to be different. Someone observed that on the average, the licence plates of nearly 3 vehicles out of 10 contain identical digits. Is he right?

Solve the following simultaneous equations: 3$x^2$-$xy$=1, 9$xy$+$y^2$=22.

The arms of a pair of compasses need to be opened through an angle twice as wide to draw a circle of radius 6.5 cm as for a circle of radius 3.3. How long are the arms of the compasses?

In a survey carried out with a group of students, it turned out that 2/3 of the boys and 3/4 of the girls like raspberries. It also turned out that the same number of boys and girls liked raspberries. What fraction of the students taking part in the survey like raspberries?

3% of the 800 inhabitants of the planet Awr have two heads. Half the remaining inhabitants have three heads, and the other half have only one. A shipment of fruits is arriving from the neighbouring planet Bwr: one delicious cwr for each head. How many, exactly?

Steve has a four-digit number in mind that Alex has to guess. Alex chooses a four-digit number, and Steve tells him how many digits it contains that stand in the appropriate position. [For example, if the number to find out is 1234 and the guess is 6231, then 2 digits are correct: 2 and 3. (The 1 does not count, since that is not in the right position.)] Alex had nine guesses: 2186, 5127, 6924, 4351, 5916, 8253, 4521, 6384, 8517. In each of the nine numbers, he guessed exactly one digit right, that is, each of the nine numbers contains exactly one digit that stands in the right position. What is the number that Steve has in mind?

The ratio of the difference, sum and product of two numbers is 1:7:24. What are the two numbers?

Street lamps in a village are placed at equal intervals along the road. I observed that during 1 minute I passed as many lamp posts as 1/12 of my speed in km/h. Given that I was moving at a uniform speed and I started and finished counting when I was midway between two posts, calculate the separation between lamp posts.

Jack and Jill start simultaneously at the bottom of the escalator. The escalator is moving uniformly upwards, and the children are also walking uniformly up the escalator without skipping any stair. Jill takes 40 seconds to get to the top, and in doing so she walks 40 steps. Jack takes 50 seconds and the number of his steps is 20. Then the escalator is stopped owing to technical problems and passengers need to walk up the stairs. How many steps will they take (without skipping any stair)?

Jack loves playing the following memory game: There are 18 kinds of picture cards, two copies of each. The 36 cards are shuffled and all layed on the table face down. The game consists of turns. In each turn, two cards are turned face up. If they are a pair of identical cards, they are removed from the table. If not, they are turned face down again. The game continues until he has removed all cards from the table. Since Jack has excellent memory, he can always remember the position of any card that he has seen. With this condition, is it possible for a game to consist of 17 turns? Of 18 turns? Of 35 turns? If so, give an example for a possible game. If not, explain why. (Assume that Jack is always trying to finish the game in the lowest number of turns, that is, he will not turn up cards again that he has already seen.)

Add all positive integers for which the quotient and remainder are equal if the number is divided by 2008. What is the result?

A team of eight members is paid 224 000 forints (Hungarian currency, HUF) for doing a certain job. The leader of the team (one of the eight members) gets 7 000 forints more than the average pay per member. Given that all the other members get the same amount, calculate how the sum is divided by the members of the team.

Eight points are marked on a circle, at equal distances between adjacent points. Consider all scalene (non-isosceles) triangles formed by the eight points. What may be the angles of the triangle?

Billy and Sophie are making rings out of paper strips and link them together to form chains. Each of them uses a 21×30 cm sheet of paper for making the chain. Billy cuts up the sheet parallel to the 21-cm sides, and a Sophie cuts it parallel to the 30-cm sides. Both of them make strips 1 cm wide, curve them into rings and glue them with 0-cm overlaps. The rings link together to form a chain without any deformation. Which chain is longer? (In calculating the perimeter of a circle, use 3.14 for the value of $\pi$, and disregard the thickness of the paper.)

In honour to the Scottish mathematician Dudley Langford, numbers of the following property are called DudLa numbers: Every digit of the number occurs at least twice, and between any pair of identical digits there is as many other digits as the value of the digit in question. For example, 723 121 327 is a DudLa number since there is 1 other digit between two 1's, 2 other digits between two 2's, 3 other digits between two 3's and 7 other digits between two 7's. Find as many 7-digit DudLa numbers as you can.

Consider all possible seven-digit numbers formed by using each of the digits 1, 2, 3, 4, 5, 6, 7 once. $a$) Show that the difference of any pair of such seven-digit numbers is divisible by 9. $b$) How many pairs of different numbers can be selected out of them, such that one is a factor of the other?

Four boys and four girls went dancing. In the first four dances, each boy danced exactly once with each girl. (They remained with the same partner during each dance, and only switched between the dances.) Charles danced the Viennese waltz with Fanny and Bernard danced it with Helen. Albert danced tango with Gaby and David with Fanny. Gaby danced mambo with Charles and Emily with David. Who were the partners dancing the first dance, the English waltz together?

Four classes of 8th-grade students took part in a mathematics competition organized by a school. Each team scored a different number of points. A mathematics teacher added up the scores of two teams in every possible way and revealed that the largest sum was 128, and the smallest sum was 82. He also disclosed that the team in the second place was only 4 points ahead of the team in the third place. What was the difference between the scores of the teams in the first and four places?

A knight is to be transferred from C5 to H2 on the chessboard with the smallest possible number of moves. How many steps does this way consist of? Find 12 several minimal ways. (A move means two squares in one direction and then one square at an angle of $90^o$.)

$a$) Is it possible for the sum of four consecutive odd numbers to be a four-digit number with all digits identical? $b$) Is it possible for the sum of five consecutive odd numbers to be a five-digit number with all digits identical? $c$) Is it possible for the sum of eight consecutive even numbers to be an eight-digit number with all digits identical?

On what condition will the sum of the squares of six consecutive natural numbers be divisible by 7?

From the second term onwards, the $n$th term of a sequence of numbers is obtained by adding $n$ to the reciprocal of the preceding term of the sequence. The fifth term is $\frac{225}{43}$. Find the fourth and sixth terms.

Jack represented on a pie chart what he bought on his money in October. He spent 1500 forints (HUF, Hungarian currency) on lunch, 1000 forints on presents, 1250 forints on movie tickets, 500 forints on hiring films on DVD. His spendings on chewing gum were represented by a 20-degree sector of the pie chart. He had no other expenses in addition to those mentioned above. How much did he spend on chewing gum in October?

The numbers 1 to 9 are written in a 3×3 table in some order. The six three-digit numbers obtained by reading the rows left to right and the columns top to bottom are all added. What are the two smallest possible sums obtained in this way? For example, in the arrangement shown by the Figure, the sum obtained is 531+296+748+527+394+168=2664. 5 3 1 2 9 6 7 4 8

A box contains red, blue and green marbles. At least twelve marbles need to be drawn from the box to make it certain that there will be a red one among them, and at least seventeen need to be drawn to make sure that there is a red one and also a green one among them. At least seven need to be drawn to be certain that there is a marble among them that is not blue. At least how many marbles need to be drawn to make sure that at least two of them are green?

In a certain town, each inhabitant is either truthful or a liar, and each of them is either sane or insane. Truthful people say what they think and liars say the opposite of what they think. Sane people think what is true, and insane people think the opposite of the truth. Four inhabitants of the town said the following: Andy: I am insane. Mandy: I am truthful. Sandy: I am a liar. Dandy: I am sane. Andy: Sandy is truthful. Mandy: Dandy is insane. Sandy: Mandy is a liar. Dendi: Sandy is sane. Determine who is truthful and who is a liar, who is sane and who is insane.

The measures of the sides of two rectangles in centimetres are whole numbers. One of the areas is 18 $cm^2$, and the other is 7 $cm^2$. The difference of the perimeters is 6 cm. Find the lengths of the sides of the rectangles.

Sarah and Kate were solving a maths problem that involved the multiplication of two positive integers. The last digit of one of the numbers was blurred in print. Sarah thought it was an 8, and thus the result of her multiplication differed by 904 from the right answer. Kate thought it was a 3, and she got a product of $60 116$. What may have been the multiplication in the problem?

Ten teams took part in a football championship. Every team played every other team exactly once. In each game, the winner team gets 3 points and the losers get 0. In the case of a tie, each team gets 1 point. At the end of the championship, the total of the scores of the teams was 119. Is it true that there was at least one team that tied at least four times?

A 11×11 cm square can be tiled with two different sets of 5 rectangles whose sides in centimetres are integers, with each length from 1 to 10 cm occurring exactly once. (Tiling means that the rectangles are not allowed to reach over the edge of the square, and they cannot even partly overlap.) Find both tilings.

The base $AB$ of an isosceles triangle ABC is extended beyond vertex $B$ by the length of the leg. The endpoint obtained is $C_1$. A perpendicular is erected onto the base $AB$ at vertex $A$. $C_2$ is the point on the perpendicular that lies in the same half plane as vertex $C$, at a distance from $A$ that equals the length of the leg. Given that the points $C_1$, $C$ and $C_2$ are collinear, find the angles of triangle ABC .

In a mathematics lesson, the teacher gave the students five kinds of problems to practise. There were three problems of each kind. Students got 1 mark for solving a single problem of a kind, 4 marks per problem if they solved two of the same kind, and 9 marks per problem if they solved all three of a kind. The students worked on the problems and received marks in teams. At the end of the lesson, every team had a different number of marks, but all their numbers of marks were found to be divisible by 3. What was the maximum possible number of teams?

In a vote, 55% of the boys and 5% of the girls voted Yes''. Thus the majority voted Yes''. How many boys may have voted ``No'' at least?

Ann, Beth and Cathy all filled out the same test of six true-or-false questions. Ann's answers (in this order) were F, F, T, T, T, T; Beth's answers were (in this order) T, F, F, T, T, T, and Cathy's answers (in this order) were T, T, F, F, T, T. Ann only got two answers wrong and Beth got only two of them right. How many correct answers may Cathy have got?