KöMaL 2007

In a right triangle ABC , point $F$ is the perpendicular foot of $C$ on hypotenuse $AB$. Let point $Q$ on $AB$ be the perpendicular foot of a certain point $P$ on leg $AC$. The line through $B$ which is perpendicular to $BP$ intersects line $CF$ at point $R$. Show that the intersection of circles with diameters $AC$ and $FQ$, different from $F$, lies in the interior of segment $PR$.

$n \ge$3 people runs several rounds of race without any tie. Denote by $f$($n$) the smallest possible number of rounds such that for any three people, all their six possible orders occur. Prove that ln $n$<$f$($n$)<20ln $n$.

Show that for all pair of reals 0<$a$<$b$ there exists a positive integer $n$ such that $a < \frac{\varphi(n+2)}{\varphi(n)} < b.$

The diagonals of a cyclic quadrilateral ABCD meet at $X$. The circles ABX and CDX meet again at $Y$. Take point $Z$ such that triangles BZC and AYD are similar. Show that if BZCY is convex, then it has an inscribed circle.

Find all functions $f\colon (0,\infty)\to\mathbb{R}$ such that | $f$($x$)-$f$($y$) | $\le$|$x$-$y$| for all $x$,$y$>0 and there exist positive constants $a$, $b$, $c$ satisfying $f(ax)+f\left(\frac{b}{x}\right) =f(cx)$ for all $x$.

Find positive constants $\alpha$ and $c$ such that $\bigg| \sum_{k=1}^N \bigg\{\frac{k^2}{N}\bigg\} - \frac{N}{2} \bigg| < cN^{1-\alpha}$ holds for all positive integer $N$.

Let $x_1$,$x_2$,...,$x_n$,$x_{n+1}$ be positive real numbers with $x_1$+$x_2$+...+$x_n$=$x_{n+1}$. Prove that $\sum_{i=1}^n\sqrt{x_i(x_{n+1}-x_i)} \le \sqrt{\sum_{i=1}^n x_{n+1}(x_{n+1}-x_i)}.$

Find all positive integers $n$ for which the numbers $1,2,3,\ldots,2n$ can be colored with $n$colors such that each color is used exactly twice and each of the numbers $1,2,\ldots,n$ appear exactly once as the difference between two numbers with the same color.

Given a convex quadrilateral ABCD and a point $P$ in its interior such that $AP$=$CP$, $\measuredangle ABC=\measuredangle APD$ and $\measuredangle CDA=\measuredangle CPB$ . Prove that $DA^. AB^. BP$=$BC^. CD^. DP$.

Let $n \ge$2 and let $a_1,a_2,\ldots,a_n, x_1,x_2,\ldots,x_n$ be positive real numbers such that $a_1$+...+$a_n$=$x_1$+...+$x_n$=1. Prove that $2\sum_{1\le i<j\le n}x_ix_j \le \frac{n-2}{n-1} + \sum_{i=1}^n \frac{a_ix_i^2}{1-a_i}\,.$

For arbitrary positive numbers $a$, $b$, $c$, compute the radius of the greatest circle contained by the ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1.$

Given a convex lattice polygon such that both $x$- and $y$-coordinates of each vertex lie in the interval [0,$n$]. Show that the number of vertices is less than 100$n^{2/3}$.

Find all pairs $f$($x$), $g$($x$) of polynomials with integer coefficients satisfying $f$ ( $g$($x$) ) =$x^{2007}$+2$x$+1.

Let $n \ge$2 and let $u_1=1,u_2,\ldots,u_n$ be arbitrary complex numbers with absolute values at most 1 and let $f$($x$)=($x$-$u_1$)($x$-$u_2$)...($x$-$u_n$). Prove that polynomial $f$'($x$) has a root with a non-negative real part.

Triangle ABC is not isosceles. The incenter is $I$, the excenter is $O$. The incircle touches the sides $BC$, $CA$, $AB$ at points $D$, $E$, $F$, respectively. Lines $FD$ and $AC$intersect at $P$, lines $DE$ and $AB$ intersect at $Q$. The midpoints of segments $EP$ amd $FQ$ are $M$ and $N$, respectively. Prove that $MN$ and $OI$ are perpendicular.

Find all integer $a$ for which there exists distinct positive integers $x$ and $y$ such that ( axy +1) divides ($ax^2$+1)$^2$.

Prove that if $a$, $b$, $c$ are real numbers such that $a^2$+$b^2$+$c^2$=1 then $a+b+c \le 2abc+\sqrt2$.

Points $A$, $B$, $C$ lie inside the convex hexagon MNPQRS , such that the triangles ABC , NAM , PQB and CRS are similar. Let $X$, $Y$, $Z$ be the midpoints of the line segments $NP$, $QR$, $SM$, respectively, and let $G$, $K$, $I$ be the centroids of the triangles ABC , MPR and NQS , respectively. Prove that ($a$) If triangle ABC is equilateral then triangle GKI is equilateral; ($b$) Triangles ABC and XYZ are similar if and only if triangle ABC is equilateral.

Prove $(a+b+c)\left(\frac1a+\frac1b+\frac1c\right) \ge 6\left(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\right)$ whenever 1$\le a$,$b$,$c \le$2.

Prove that $\big| \big\{n\sqrt2\,\big\} - \big\{n\sqrt3\,\big\}\big| > \frac1{20n^3}$ for every positive integer $n$.

Let $p$>3 be a prime and $1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{p} = \frac{a}{b}.$ Prove that $p^4$ divides $ap$-$b$.

On the circumcircle of the equilateral triangle ABC , choose an arbitrary point $P$ on arc $AB$. Mark the points $Q$ and $R$ on arc $BC$ and the points $S$ and $T$ on arc $CA$ for which $AP$=$BQ$=$CR$=$CS$=$AT$ holds. Let lines $BQ$ and $PR$ meet at $U$ and let lines $AS$ and $PT$ meet at $V$. Finally, let lines $AU$ and $BV$ meet at $W$. Show that each two of lines $AU$, $BV$and $PW$ enclose an angle of $60^o$.

Prove that $\frac{a_1^2}{1-a_1} + \frac{a_2^2}{a_1-a_2} + \frac{a_3^2}{a_2-a_3} + \ldots + \frac{a_n^2}{a_{n-1}-a_n} > \frac12(a_1+2a_2+3a_3+\ldots+na_n)-1$ whenever $1>a_1>a_2>\ldots>a_n>0$.

Given an isosceles triangle ABC such that the midpoint of base $BC$ is $M$, points $D$ and $E$ lie on sides $AB$ and $AC$, respectively, and $DE$ is parallel to $BC$. Choose two points, $P$ and $Q$ on the extensions of segment $BC$ beyond $B$ and $C$, respectively, such that $\frac 1{MP}+\frac 1{MQ} =\frac 1{MB}$ . Let $PD$ and $QE$ meet at $R$. What is the locus of point $R$?

For an arbitrary sequence of numbers $A=(a_0,a_1,a_2,\ldots)$, let $SA=(a_0,a_0+a_1, a_0+a_1+a_2,\ldots)$ be the sequence of partial sums of the series $a_0+a_1+a_2+\ldots$. Is there any sequence $A$, not constant zero, for which the sequences $A$, $SA$, SSA , SSSA ,... are all convergent?

The $n$-element subsets of a set of size $n^2$+$n$-1 are distributed into two groups. Prove that one of those groups contains $n$ pairwise disjoint sets.

In how many different ways can one pay 1000 forints (Hungarian currency, HUF) in 100, 20 and 5-forint coins, exactly 100 of them altogether? (The order does not matter.)

A counter is travelling from the lower left corner to the upper right corner of the chessboard. It moves either one field to the right or one field up in each step. How many routes are there that pass through at least one of the four fields at the centre?

The lengths of the legs in a right-angled triangle ABC are $AC$=3 and $BC$=4. The point $A$is shifted in a direction parallel to $BC$ to the point $A_1$, then the point $B$ is shifted in a direction parallel to the line $A_1 C$ to the point $B_1$, and finally $C$ is shifted in a direction parallel to $A_1 B_1$ to the point $C_1$. The resulting triangle $A_1 B_1 C_1$ is right-angled at $B_1$ and its leg $A_1 B_1$ is 1 unit long. How long is the leg $B_1 C_1$?

On the sides $AB$ and $AC$ of an acute-angled triangle ABC as diameters, draw semicircles outside the triangle. The lines of the altitudes drawn from the opposite sides intersect the semicircles at $M$ and $N$. Prove that $AM$=$AN$.

In the chess tournament of a school, everyone played everyone else exactly once. Every player gained the same number of points playing against girls as playing against boys. Prove that the number of the participants is a perfect square. (1 point is due for winning, 0,5 for a draw, and 0 for losing.)

Construct with ruler and compass, a triangle given is an angle and the lengths of the altitude and the median drawn from the vertex of that angle.

Solve the following inequality: $\sqrt{x + \sqrt{x}} - \sqrt{x - \sqrt{x}} > \frac{4}{3} \sqrt{\frac{x}{x + \sqrt{x}}}.$

Determine the locus of the centres of all regular triangles circumscribed about a given acute triangle. (The triangle $G$ is said to be circumscribed about the triangle $H$ if the vertices of $H$ lie on the sides of $G$.)

Prove that the harmonic mean of the lengths of the diagonals of a regular heptagon of unit sides is 2.

Let $n$ be a positive integer. Prove that there exist $n$ integers such that their sum is 0 and their product is $n$ if and only if $n$ is divisible by 4.

In how many different ways is it possible to place 14 bishops on a chessboard so that they do not attack each other?

Given a triangle ABC . Find the locus of the points $P$ in the plane for which $AP^2$+$BP^2$=$CP^2$?

$N$ and $k$ are positive integers. We have counted the different ways of expressing $N$ in the form $a$+$b$+$c$, where 1$\le a$,$b$,$c \le k$, and the order of the terms also matters. Is it possible that we got 2007?

Through an interior point of a triangle of sides $a$, $b$, $c$ a parallel line is drawn to each side. Given that the segments of these lines intercepted by the triangle have the same length, find that length.

Solve the following simultaneous equations on the set of positive real numbers: $x^2$+$xy$+$y^2$=2, $y^2$+$yz$+$z^2$=5, $z^2$+$xz$+$x^2$=3.

Prove that the following relationship is true in every triangle: 2$R$($s_a$+$s_b$+$s_c$)$\ge a^2$+$b^2$+$c^2$. (where $a$, $b$, $c$ are the sides, $s_a$, $s_b$, $s_c$ are the lengths of the medians and $R$ is the radius of the circumscribed circle.)

The vertices of triangle ABC are labelled in counterclockwise order. The angles at the vertices $A$, $B$, and $C$ are $\alpha$, $\beta$ and $\gamma$, respectively. The vertex $B$ is rotated about the point $A$ through the angle $\alpha$ in clockwise direction. The point $B_1$ obtained is rotated about $B$ through the angle $\beta$ in clockwise direction. Finally, the point $B_2$ obtained is rotated about $C$ in clockwise direction to the point $B_3$. Given the points $B$, $B_3$ and $O$ the middle point of the incentre of the triangle ABC , construct the triangle. Investigate how the solution depends on the positions of the three given points.

The real numbers $a_i$, $b_i$ ($i=1, 2, \ldots, n$ ) satisfy the following inequalities: $0\le a_n\le a_{n-1}\le \ldots\le a_2\le a_1$, and $a_1\cdot a_2\cdot \ldots\cdot a_k\le b_1\cdot b_2\cdot \ldots\cdot b_k$, if 1$\le k \le n$. Prove that $a_1+\ldots+a_n\le b_1+\ldots+b_n$.

100 freshly graduated mathematicians are looking for jobs. They consult two headhunter companies that have the same 100 positions to offer. Each company makes an offer to each applicant, a different one to each of them. Each applicant chooses from the two offers and, luckily, all positions are filled. Three months later, however, each of them decides to change for the job offered by the other company. (If it is the same as his actual job, then he decides to stay.) Show that all positions will be filled again.

A few players are fighting a paintball battle. At a certain time instant, their distances from each other are all different. Then everyone shoots at the person who is closest to them. Is it possible for the paths of the projectiles to intersect each other?

A square is dissected into acute-angled triangles. What is the minimum number of pieces?

The lawn in our garden consists of $n$ blades of glass. We want to cut the grass so that each blade of glass becomes equal in length but the total length of them stays the same. We are allowed to make $n$-1 cuts on the grass blades altogether, and we may fix the cut pieces to each other and to the remaining pieces. Is it always possible to complete the grass cutting in this way?

Let $s$($m$) denote the number of digits of $m$ for any positive integer $m$. Find all positive integers $n$, such that no digit of it is zero and $s$($n^2$)$=2^{s(n)}$.

Let $n \ge$4 be an integer, and let $a_1,a_2,\ldots,a_n$ denote nonnegative real numbers. Prove that ${(a_1+a_2+a_3)}^2 {(a_2+a_3+a_4)}^2\cdot \ldots \cdot {(a_{n-1}+a_n+a_1)}^2{(a_n+a_1+a_2)}^2$ $\ge 2^n{(a_1+a_2)}^2{(a_2+a_3)}^2\cdot \ldots \cdot {(a_{n-1}+a_n)}^2{(a_n+a_1)}^2.$ Discuss also the case of equality.

The midpoints of the sides of a convex pentagon are $F_1$, $F_2$, $F_3$, $F_4$, $F_5$, in this order. (The vertex $A$ is between $F_5$ and $F_1$.) Let $P$ be the point in the plane for which the quadrilateral $PF_2 F_3 F_4$ is a parallelogram. Prove that the quadrilateral $PF_5 AF_1$ is also a parallelogram.

$a$, $b$, $c$ are positive numbers, such that $a^2$+$b^2$+$c^2$+ abc =4. Prove that $a$+$b$+$c \le$3.

Prove that in a right-angled triangle the intersections of the bisectors of the acute angles with the circumscribed circle are collinear with the points where the inscribed circle touches the legs.

The isosceles triangle ABC has a right angle at $C$. Let $P$ be an arbitrary point of side $BC$, and let $G$ denote the orthogonal projection of the point $C$ onto $AP$. Let $H$ be the point of the line segment $AP$ for which $AH$=$CG$. Find the angle subtended by the line segment $GH$ at the midpoint of $AB$.

A few players are fighting a paintball battle. At a certain time instant, the distances between the players are pairwise different. Everyone shoots at the player closest to him. At most how many players may shoot at the same player?

In a sequence of real numbers, the sum every 5 consecutive terms is positive, and the sum of every 7 consecutive terms is negative. How long may be the sequence?

Find all non-negative integers $n$ for which there are non-negative numbers $a$ and $b$, such that $n^2$=$a$+$b$ és $n^3$=$a^2$+$b^2$.

$A$, $B$, $C$, $D$ are collinear points in this order. Draw the regular triangles ABE and CDF on one side of the line. Let $G$ denote the intersection of the circles ACE and BDF on the same side of the line ABCD . Prove that $AGD\sphericalangle =120^\circ$.

We have a rectangular sheet of paper. We want to make polygons of 20 sides from it according to the following rule: In each step, we choose a piece of paper (initially, this must be the original sheet) and cut it in two along a straight line. Continuing the procedure what is the minimum number of cuts needed to get one hundred pieces 20-sided polygons?

Prove that if the product of the real numbers $x$, $y$, $z$ is 1, then $x^4$+$y^4$+$z^4$+$x^2 y^2$+$y^2 z^2$+$z^2 x^2 \ge$2($x$+$y$+$z$).

The vertex $P$ of a rectangular block with edges $a$, $b$, $c$ is selected. Consider the plane passing through the vertices adjacent to $P$. Denote the distance of the plane from $P$is $m$. Prove that $\frac{1}{m^2} = \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}.$

A regular dodecahedron is inscribed in a unit sphere and vectors are drawn from the centre of the sphere to the vertices. What is the sum of these vectors?

Find the smallest possible value of $x^2$+$y^2$, given that $x$ and $y$ are real numbers, $x \ne$0 and $xy$($x^2$-$y^2$)=$x^2$+$y^2$.

Xenia, Yvette and Zoe are friends. On a certain day, each of them makes a one-hour visit in the same internet café that is open from 12 noon to 8 pm. Each visits at a random time, independently of the others. What is the probability that they all meet there?

Prove that if $n$>1 is an integer then there exists a sequence of integers $a_1$=$n$<$a_2$<$a_3$<..., such that $a_1 ^2$+...+$a_k ^2$ is divisible by the number $a_1$+...+$a_k$ for all values of $k$.

$P_1,P_2,\ldots,P_{2007}$ are given points in the plane, no three of which are collinear. Show that for all 1$\le i \le$2007 there is an even number of triangles $P_j P_k P_l$ that contain $P_i$ in the interior. (British competition problem)

$A$' is the reflection of the vertex $A$ of an equilateral triangle ABC about the opposite side. A line passing through $A$' intersects the lines $AB$ and $AC$ at the points $C$' and $B$', respectively. What is the locus of the intersection of lines $BB$' and $CC$'?

For every positive integer $n$, let $a_n$ denote the number of ways $n$ is obtained as a sum of terms that are all 1, 3 or 4. The order of the terms also matters. Prove that $a_{2006} a_{2007} a_{2008}$ is a perfect cube.

$a$, $b$, $c$ are the sides of a triangle, such that $a$+$b$=2$c$. Prove that the centres of the inscribed and circumscribed circles are concyclic with the midpoints of the sides $a$ and $b$.

Let $A$, $B$, $C$, $D$ denote four distinct points in the space. The line segment $PQ$ is ``moved'' from $AB$ to $DC$ so that in each time instant $P$ divides the line segment $AD$ in the same ratio as $Q$ divides $BC$. Then the line segment $PQ$ sweeps out a surface. Prove that the same surface will be obtained if a line segment $RS$ is moved in the same way from $AD$ to $BC$.

Prove that ${\rm cot} 70^{\circ} + 4 \cos 70^{\circ} = \sqrt{3}.$

The angle bisectors drawn from the vertices $A$ and $B$ of a triangle ABC intersect the opposite sides at $A_1$ and $B_1$, respectively. $P$ is the intersection of the ray $A_1 B_1$ with the circumscribed circle of the triangle. Prove that $\frac{1}{PA} = \frac{1}{PB} + \frac{1}{PC}.$

$p$($x$) and $q$($x$) are polynomials of real coefficients that have no real roots in common, and $p$ ( $q$($x$) ) =$q$ ( $p$($x$) ) for all $x$. Prove that the polynomials $p$ ( $p$($x$) ) and $q$ ( $q$($x$) ) have no common real roots either.

Assoc City has 2007 inhabitants. They like joining clubs. The following two resolutions are passed to regulate club life: (1) every club must have exactly 101 members, and (2) no two clubs may have exactly the same set of members. In the senate of the city, each club is represented by a member, and it is against the law for two clubs to have the same representative. The committee of law are planning to impose a limitation on the number of clubs, so that however they are formed according to the rules above, it should always be possible to delegate representatives. What is the maximum number of clubs that may be allowed by the committee?

Alice has 8 keys on a ring. The keys look all alike, and both sides of them are the same. In order to be able to distinguish them, Alice puts a coloured cap on each key. How many colours does she need?

Is it possible to colour the positive rational numbers in red and blue so that ($a$) the sum of red numbers is red and the sum of blue numbers is blue; ($b$) the product of red numbers is red and the product of blue numbers is blue?

$A$', $B$' and $C$', respectively, are the reflections of the vertices of a triangle ABC about the centre of the circumscribed circle. Prove that the sum of the areas of the triangles $A$'$BC$, $AB$'$C$, and ABC ' equals the area of the triangle ABC .

The diagonals of a convex quadrilateral enclose an angle of $45^o$. Drop a perpendicular from each vertex of the quadrilateral onto the line of the diagonal connecting the two adjacent vertices. Find the ratio of the area of the quadrilateral formed by the feet of the perpendiculars to the area of the original quadrilateral.

In every group of 9 out of a company of 12 people, there are 5 people who pairwise know each other. Prove that there are 6 members of the company who pairwise know each other, too.

The circle $k_2$ touches the circle $k_1$ on the inside at point $A$. The tangent drawn to circle $k_2$ at a point $D$ different from $A$ intersects the circle $k_1$ at the points $B$and $C$. Prove that $AD$ bisects the angle BAC .

$P$ is a given point on the tangent drawn at point $B$ of a circle of diameter $AB$. The point of tangency of the other tangent from $P$ is $C$. $T$ is the orthogonal projection of $C$ on the line $AB$. Prove that $AP$ bisects the line segment $CT$.

Prove that $\frac{1}{3^2} + \frac{1}{5^2} + \dots + \frac{1}{{(2n+1)}^2} < \frac{1}{4}.$ for every positive integer $n$.

Prove that if each of the numbers $a_1$,$a_2$,...,$a_n$ is at least 1, then ($a_1$+1)($a_2$+1)$^. ...^.$($a_n$+1) \ge$2^{n-1}$($a_1$+$a_2$+...+$a_n$-$n$+2).

Is there a base of notation of numbers in which the rule for divisibility by 9 is the same as the rule for divisibility by 4 in decimal notation, the rule for divisibility by 4 is the same as the rule for divisibility by 9 in decimal notation, and divisibility by 7 can be decided by simply looking at the last digit?

Construct two lines through a given point of one side of a triangle that divide the area of the triangle into three equal parts.

How many numbers can be selected out of the first 1000 positive integers at most, so that the sum of no pair of selected numbers is divisible by their difference.

A regular triangle is drawn on the side $BC$ of a triangle ABC towards the outside, and one on the side $CA$ towards the inside. The third vertices of the regular triangles are $A^*$ and $B^*$, respectively. The reflection of vertex $C$ in the line $AB$ is $C$'. Prove that the points $A^*$, $B^*$ és $C$' are collinear.

Solve the following equation: $\frac{x^2 + 1}{x^2 + 11} = \frac{1}{6}\sqrt{\frac{11x - 6}{6-x}}.$

The radii of two spheres are 5 and 3 units, and the distance of the centres is 4 units. Calculate the volume of their intersection.

$AB$ is a given line segment in the plane. Consider all arbitrary points $C$ for which the triangle ABC is not isosceles. The exterior angle bisector of angle $C$ intersects the line $AB$ at $D$, and its intersection with the tangent drawn at point $A$ to the circle ADC is $P$. Determine the locus of all possible points $P$.

Let $n$ be an integer greater than 1. Prove that the equation $\frac{x^n}{n!}+\frac{x^{n-1}}{(n-1)!}+\ldots +\frac{x}{1!}+1=0$ has no rational solution.

2007 natural numbers are written on the circumference of a circle. Is it possible that if the larger of each pair of two adjacent numbers is divided by the smaller one, the result is always a prime number?

All the participants in a dance course are married couples. Someone noticed that the difference between the heights of the two members of each couple is at most 10 cm. The dance instructor asks both the men and the women to stand in a line in increasing order of heights. Dancing couples are made by this rule: The tallest woman dances with the tallest man, and so on. Prove that the differences between the members of every dancing couple are also at most 10 cm.

The midpoint of one side of a triangle is $F$, and the points dividing another side into three equal parts are $H_1$ and $H_2$. The third side is divided into $n$ equal parts by the points $N_1, N_2, \ldots, N_{n-1}$. Consider all triangles $FH_i N_j$ where $i$=1,2, $j=1,\ldots,n-1$. Show that for any triangle selected out of these triangles there is exactly one other triangle that has the same area.

Solve the following equation: 2cos 5$x$+2cos 4$x$+2cos 3$x$+2cos 2$x$+2cos $x$+1=0.

The circles $k_1$ and $k_2$ touch each other on the outside at $P$. One of their common tangents not passing through $P$ is $e$, which touches $k_1$ and $k_2$ at $A$ and $B$, respectively. The line $f$ is parallel to $e$, touches $k_1$ at $C$ and intersects $k_2$ at $D$ and $E$. Prove that the common chord of the circumcircles of the triangles ABC and ADE contains $P$.

Let $f$($n$) denote the sum of the digits of the positive integer $n$ in decimal notation. What may $f$(3$a$) be if $f$($a$)=100 and $f$(124$a$)=700?

The reflections of an interior point $P$ of triangle ABC about the midpoints of the sides $BC$, $CA$ and $AB$ are $A$', $B$' and $C$', respectively. Show that the lines $AA$', $BB$', $CC$' are concurrent.

Ann és Bill take turns writing digits of 0 or 1 from left to right next to each other. Ann starts. The game ends when each of them has written down exactly 2007 digits. The resulting sequence of zeros and ones is read as a number written in binary notation. Ann wins if the result is the sum of two perfect squares and Bill wins if it is not. Which of them has a winning strategy?

$a$, $b$, $c$ are positive real numbers, and $ab$+$bc$+$ca$=1. Prove that $\frac{1-a^2}{1+a^2}+\frac{1-b^2}{1+b^2}+\frac{1-c^2}{1+c^2}\le\frac{3}{2}.$

The lengths of the medians of a triangle are $a$, $b$, $c$. Prove that for all positive numbers $\alpha$, $\beta$, $\gamma$ it is possible to construct a triangle with sides $\alpha a$+$\beta b$+$\gamma c$, $\gamma a$+$\alpha b$+$\beta c$, $\beta a$+$\gamma b$+$\alpha c$.

A 2007×2008 chessboard is covered by a few 2×2 and 1×4 dominoes without overlaps. One 2×2 domino of the set of dominoes used is replaced with a 1×4 domino. Prove that it is not possible to cover the chessboard with the modified set.

For what pairwise different positive integers is the value of $\frac{a}{a+1}+\frac{b}{b+1}+\frac{c}{c+1}+\frac{d}{d+1}$ an integer?

Prove that the average distance of a point in the interior of a regular polygon from the side of the polygon equals the radius of the inscribed circle.

An equilateral triangle of unit side is moved between the arms of a $120^o$ angle XOY so that vertex $A$ lies on the arm $OX$, vertex $B$ lies on the arm $OY$, and the line $AB$separates the points $C$ and $O$. Determine the locus of vertex $C$.

Solve the following simultaneous equations: $a\sqrt a+b\sqrt b =183,$ $a\sqrt b+b\sqrt a =182.$

A finite number of circular discs cover an area $T$ on the plane together. Prove that it is possible to select a few discs that do not overlap and cover an area of at least $T$/9 together.

The vertices of a regular polygon of 2$n$ sides are $A_1, A_2, \ldots, A_{2n}$ in this order. Prove that $\frac{1}{A_1A_2^2} + \frac{1}{A_1A_n^2} = \frac{4}{A_1A_3^2}.$

$a$, $b$, $c$ are positive real numbers, and $ab+bc+ca=\frac{1}{3}$. Prove that $\frac{a}{a^2-bc+1}+\frac{b}{b^2-ca+1}+\frac{c}{c^2-ab+1}\ge\frac{1}{a+b+c}.$

Show that the equation $x^3$-$x^2$-2$x$+1=0 has two real roots $a$ and $b$, such that $a$-$ab$=1.

Prove that every factor of $\underbrace{111\ldots1}_{5^n}$ ends in a 1 for all positive integers $n$.

Which six-digit number will be multiplied by three if its first digit is reduced by three and a digit of three is attached to the end?

Solve the equation $\left(\frac{x}{2}\right)^{2} =\left(\frac{x+1}{3}\right)^{3}$.

The legs of a right-angled triangle are $a$ and $b$. The line segment of length $d$ connecting the vertex of the right angle to a point of the hypotenuse encloses an angle $\delta$ with the leg $a$. Prove that $\frac{1}{d}= \frac{\cos\delta}{a}+ \frac{\sin\delta}{b}.$

A trough with regular triangular cross section is filled to the brim with water. Through what angle should it be tilted in order to make one fifth of the water flow over? The triangular ends of the trough should move in their vertical planes.

A roll of a die is said to be successful if the score is at least three. Which is more probable: Rolling at least once successfully out of two or at least twice out of four?

The charge of money withdrawal from an ATM is made up of two terms. There is a flat charge independent of the amount withdrawn and another part that is proportional to it. For $40 000$ forints (Hungarian currency, HUF) withdrawn the charge is 221 forints, and for $100 000$ it is 485 forints. What is the charge for $85 000$ forints?

The wall of a building is decorated by a square with its circumscribed circle drawn. There is a semicircle drawn outwards on each side of the square as a diameter, so that the circular arcs form four crescent shaped figures. What is the length of the side of the square if the area of such a crescent is 1 $m^2$?

How many eight-digit numbers are there in which every digit occurs the same number of times as the value of the digit? (Example: $33 414 434$.)

The measures of the volume and the surface area of a cube with edges of 6 unit lengths are both 216. Find all square based prisms of integer edges that have this property.

Find all pairs of natural numbers whose product is five times their difference.

What maximum number of sides may a convex polygon have in which the interior angles form an arithmetic progression wit a common difference of d$=1^o$?

Prove that if $x$, $y$, $z$ are positive real numbers and xyz =1, then the values of the expressions $\frac{1}{1+x+xy}, \frac{y}{1+y+yz}, \frac{xz}{1+z+xz}$ cannot all be greater than $\frac{1}{3}$.

We have sixteen painted Easter eggs. Three of them are red. We place ten eggs in a larger box and six in a smaller box in a random arrangement. What is the probability that there will be a red egg in each box?

A hemispherical soup bowl has a volume of 8 litres. How much soup will fill the bowl to half its height?

The vertices of a triangle ABC in the coordinate plane are $A$(0,4), $B$(3,0), $C$($c$,6). The area of the triangle is 7. Find $c$, given that 0<$c$<3.

The acute angle of a rhombus of unit sides is $60^o$. How many circles are there, such that the vertices of the rhombus are equidistant from the circumference of the circle? What are the radii of the circles?

The length of the sides of the regular triangle ABC is 6 cm. A bug starts at the vertex $C$of the triangle and crawls uniformly towards vertex $A$ at a speed of 4 mm/s. At the same time, another bug starts from vertex $B$ and crawls towards vertex $C$ at a speed of 3 mm/s. How long does it take for the distance between the bugs to reach the smallest value, and what is the smallest distance?

For what values of the real parameter $v$ do the following simultaneous equations have no solution? $x$+$y$+$z$=$v$, $x$+$vy$+$z$=$v$, $x$+$y$+$v^2 z$=$v^2$.

75% of a certain three-digit number of different digits consists of the same digits as the original number but with no digit in the same place. Which number is it?

The area of a rectangle ABCD is $100\sqrt5$. Let $P$ denote the point on side $AB$, closer to $A$, that divides $AB$ in a 1:4 ratio. Given that the line segment $PD$ is perpendicular to diagonal $AC$, calculate the perimeter of the rectangle.

A circle $K$ is touched on the inside by a circle $k$ of half the radius. Construct a chord in $K$, perpendicular to the line connecting the centres of the circles that is divided into three equal parts by its intersections with $k$.

Find the largest value of the expression $\sqrt{x-2}+2\sqrt{3-x}.$

Ann writes down two arbitrary natural numbers that consist of the same digits in different orders. She subtracts the smaller number from the greater one, and multiplies the difference by an arbitrary natural number. Then she deletes a non-zero digit from the product. She tells the remaining number to Bill. With a little thinking, Bill can guess the deleted digit. How?

The sides of a right-angled triangle form an arithmetic progression. Determine the ratios of the sides. Prove that the radius of the inscribed circle is the common difference of the arithmetic progression.

A group of ten people went to the cinema. They bought tickets to 5 consecutive seats in each of two different rows. Al and Ben wanted to sit next to each other, while Sue and Ann did not want to sit next to each other. How many different seatings were possible?

The lengths of the sides of a triangle are whole numbers. One of them is 7, and the opposite angle is $60^o$. What may be the area of the triangle?

Nine integers that add up to 90 are written on the perimeter of a circle. Prove that there are four adjacent numbers that add up to at least 40.

Find the positive integers $n$ for which $n^3$+1 and $n^2$-1 are both divisible by 101.

Is there a right-angled triangle in which the lengths $a$, $b$, $c$ of the sides are integers, ($a$,$b$,$c$)=1 and the length of one median is 7.5?

The centre of the inscribed circle of triangle ABC is $O$, and the centre of the escribed circle drawn to side $BC$ is $K$. On what condition will the quadrilateral BKCO be a kite? On what condition will it be a rectangle?

According to the trainer of a football team, the probability of a player to score a goal with a penalty kick is 95%. What is the probability that exactly three misses the goal out of five players?

Which prime numbers can be expressed as a sum of two positive composite numbers?

In 2006, the income of an insurance company increased by 25% and its expenses increased by 15% relative to the previous year. The profit ($\hbox{income}-\hbox{expenses}$ ) of the company increased by 40%. What percentage of the income was taken by the expenses in 2006?

Solve the following simultaneous equations: $x$+2$y$+3$z$+4$v$=$a$, $y$+2$z$+3$v$+4$x$=$b$, $z$+2$v$+3$x$+4$y$=$c$, $v$+2$x$+3$y$+4$z$=$d$, where $a$, $b$, $c$, $d$ are given real numbers.

Connect the midpoints of the sides of a rectangle to the opposite faces. The eight lines obtained in this way form an octagon. What fraction of the whole area of the rectangle is the area of the octagon?

A right-angled triangle is cut into a triangle and a quadrilateral by the perpendicular bisector of its hypotenuse. The ratio of the diagonals of the quadrilateral is $\big(1+\sqrt3\,\big):2\sqrt2$. Find the acute angles of the right-angled triangle.

A field is bounded on one side by a 50-metre-long wall. Mehemed wants to surround the largest possible rectangular area with an electric fence for his grazing his cows. How can he achieve that if he has 44 metres of wire that he can fix to the ground with poles at $a$) 1-metre, $b$) 2-metre intervals. What is the area of the surrounded pasture in each case?

The average of Steve's mathematics marks was between 4 and 5 before the last maths test of the first semester was given back to the class. (Marks in Hungarian schools are 1 to 5, 5 means excellent and 1 means fail. Teachers sometimes give half marks such as 3 and a half, too, to indicate that it is between a 3 and a 4, but final marks are whole numbers.) The teacher said to Steve, ``16 of you took this test. I also gave half marks. The range of your marks is 2, the mode is 4.5, and the median is 4. The average of the marks is the worst possible average that a group of 16 may have with these conditions. If you can tell me whether you may or may not have got a 5 on this test, then you will get a 5 for the first semester.'' Steve did get his 5. What was his answer to the question?

Find all integer solutions of the equation $x^2$+12=$y^4$.

The lengths of the parallel sides of a cyclic trapezium are $a$=10, $c$=15, the radius of the circumscribed circle is $r$=10. What may be the length of the legs? What is the area of the trapezium?

The areas of the rectangles obtained by intersecting a certain cuboid with a plane passing through two parallel edges may be $t_1$=60, $t_2=4\sqrt{153}$, or $t_3=12\sqrt{10}$. Calculate the volume and surface area of the cuboid.

In a football championship, Peter's favourite team scored a total of three goals in the first four matches played. Other teams scored a total of two goals against the team. Given that a team gets 3 points for winning, 1 for a draw and none if they lose the game, how many points may Peter's favourite team have after the four matches played?

In Dragon Castle, 14-headed baby dragon is sleeping while mummy dragon is making pancakes. She lays them on a plate and leaves home. One head wakes up, eats 1/11 of the pancakes and falls asleep again. Then another head wakes up, eats 1/13 of the remaining pancakes and falls asleep again. Next, three heads wake up at the same time, eat 1/14 of the pancakes each, and finally they wake up the rest of the heads, too. Those who have eaten, do not get any more pancakes, the remaining 9 heads eat them all. (They do not necessarily eat the same number.) How many pancakes may have been on the plate, given that every head ate a whole number of pancakes, and if they had shared them equally, no one would have got more than 143?

In a game, two dice are rolled and there are 9 cards numbered 1 to 9. The total of the numbers rolled is to be put together as a sum of number cards (or a single card). Each card can only be used once. Used cards are set aside and the sum of the next two numbers rolled is put together out of the remaining cards. And so on while it is possible. $a$) What is the minimum number of rolls may in a sequence of rolls where all number cards are used up? Give an example, too. $b$) In the worst case, what is the smallest number of rolls that may end the game?

How many four-digit numbers are there that consist of different digits, end in 5 and are divisible by each of their digits?

Steve has three classmates named Smith, four named Black, one named White and two named Tailor. Among these boys, there are four with the first name of Jack, two called Alex, three called Bob and one Martin. Each boy has only one first name, and no two boys in the class have exactly the same name. What are the full names of these classmates of Steve?

When more than one files are selected on the computer for copying at the same time, and the process of copying is started, there are two progress bars displayed. The upper one shows what percentage of the file just being copied is done and the lower one shows what percentage of the whole copying task is done (including the files copied before the current file). Suppose there are three files copied in this way. When the first file is being copied and the upper progress bar reads 35%, the lower bar reads 14%. When the upper bar reads 24% during the copying of the second file, the lower bar reads 46%. What does the upper progress bar read when the lower one reads 72%? (The given data are exact values, not approximations.)

Let us make 3×3 Latin squares out of a deck of French cards. (There is a card in each field of the Latin square.) Number cards are worth the value printed on them; jacks, queens, kings and aces are worth 11, 12, 13 and 1, respectively. In a Latin square, the sum of the numbers is the same in each row, column and diagonal. Let us call this equal sum the ``magic number'' of the square. $a$) What is the largest possible magic number that can be achieved if all the nine cards used are clubs? $b$) Is there a square whose magic number is 37 if any 9 cards may be used?

Find all five-digit numbers $\overline{abcde}$ , such that there are exactly $a$ zeros, $b$ones, $c$ twos, $d$ threes and $e$ fours among the digits. (Different letters do not necessarily denote different digits.)

There are two tribes living in Binumeria. The difference between the two tribes is that they write numbers in notations using different bases. A member of each tribe was asked how many members their tribe had. Each of them answered $10 011$. How many inhabitants are there in Binumeria, given that each tribe consists of more than 100 but less than 1000 members?

$a$) Find a solid bounded by polygonal faces that has 12 vertices and 18 edges. $b$) Is there a solid bounded by polygonal faces that has 12 vertices and 16 edges?

An arbitrary point is selected on each side of a rectangle. On what condition will the area of the quadrilateral formed by the four points be the half of the area of the rectangle?

Draw a line segment $AB$ on the plane and investigate the number of points on each line of the plane that form an isosceles triangle with the line segment $AB$. Define that number as the isosceles number'' of the line. Find all possible isosceles numbers''.

Three bunnies are sitting in the grass. There is a heap of carrots in front of each of them, 36 carrots altogether. Suppose they simultaneously pass some carrots to one another: the first bunny passes one third of his carrots to the second one, the second one passes one fourth of his carrots to the third one, and the third bunny passes one fifth of his carrots to the first one. As a result, each bunny will have the same amount of carrots as initially. How many carrots does each of them have?

``Maci sajt'' is a brand name of processed cheese that comes in a round box containing eight individually wrapped sectors that just fit in the box. Stevie ate two sectors for breakfast, and arranged the remaining ones in the box as shown in the figure. His mom did not like that. She said that Stevie was squeezing the cheese pieces. Stevie insisted that there was no squeezing, the pieces just fit. Who was right?

Sebastian and Vincent were building towers out of wooden building blocks. They both used identical prisms of square base. Sebastian laid the blocks on each other along their faces of smaller area, while Vincent laid them on each other along their faces of larger area. Sebastian's tower of four blocks turned out to have the same height as Vincent's tower of six blocks. Find the height of the towers, given that the sum of their base areas is 160 $cm^2$.