KöMaL 2006

Point $P$ lies in the interior of the acute triangle ABC . The circumcenters of triangles ABC , BCP , CAP and ABP are $O$, $A_1$, $B_1$ and $C_1$, respectively. Prove that $\frac{\mathrm{area}\,(\Delta A_1B_1O)}{\mathrm{area}\,(\Delta ABP)} = \frac{\mathrm{area}\,(\Delta B_1C_1O)}{\mathrm{area}\,(\Delta BCP)} = \frac{\mathrm{area}\,(\Delta C_1A_1O)}{\mathrm{area}\,(\Delta CAP)}.$

Find all functions $f\colon \mathbb{R}\to\mathbb{R}$ such that ( $f$($x$)+$f$($y$) ) ( $f$($z$)+1 ) =$f$($xz$-$y$)+$f$($x$+$yz$) for all real $x$, $y$, $z$.

Construct a sequence $a_1$,$a_2$,...,$a_N$ of positive reals such that $n_1 a_{n0}$ +$n_2 a_{n1}$ +...+$n_k a_{nk-1}$ >2.7($a_1$+$a_2$+...+$a_N$) for arbitrary integers 1=$n_0$<$n_1$<...<$n_k$=$N$.

The inradius of a tetrahedron is $r$. There exists three spheres which touch the four face planes such that all touching points are outside the surface of the tetrahedron; the radii of those spheres are $r_1$, $r_2$ and $r_3$. Show that $r_1$+$r_2$+$r_3$>9$r$.

Prove that if $n$>1 is an integer such that $3^n +4^n$ is divisible by $n$ then $n$ is divisible by 7.

Let $a_1$,$a_2$,...,$a_N$ be nonnegative reals, not all 0. Prove that there exists a sequence 1=$n_0$<$n_1$<...<$n_k$=$N$+1 of integers such that $n_1 a_{n0}$ +$n_2 a_{n1}$ +...+$n_k a_{nk-1}$ <3($a_1$+$a_2$+...+$a_N$).

Let 1<$a$<2 be a real number. ($a$) Show that there exists a unique sequence $x_1$,$x_2$,... of positive integers satisfying $x_{i+1} \ge x_i ^2$ for all indices $i$ and $\left(1+\frac1{x_1}\right)\left(1+\frac1{x_2}\right)\dots = a.$ ($b$) Prove that inequality $x_{i+1}$>$x_i ^2$ holds for infinitely many indices if and only if $a$ is irrational.

The elements of families $\mathcal{A}$ and $\mathcal{B}$ are finite sets, each having at most $n$ elements. Suppose that for an arbitrary finite set $F$ there exist $X\in\mathcal{A}$ and $Y\in\mathcal{B}$ such that $X \cap Y \cap F$=Ø. Prove that there exist $X\in\mathcal{A}$ and $Y\in\mathcal{B}$ such that $X \cap Y$=Ø.

Given $n$ unit circle ($n \ge$3) in the plane, their centers are $O_1$,$O_2$,...,$O_n$. Any line of the plane intersects at most two of them. Show that $\sum_{1\le i<j\le n}\frac1{O_iO_j}\le\frac{(n-1)\pi}4.$

Given $n$ 3-element subset of a certain set $S$. Prove that if $n$=6 then the elements of $S$can be colored with two colors such that each subset contains both colors. Is the same true for $n$=7?

Prove that $\frac{x}{y+2z+3u}+ \frac{y}{z+2u+3x}+ \frac{z}{u+2x+3y} +\frac{u}{x+2y+3z} \ge \frac{2}{3}.$ for all real numbers $x$, $y$, $z$, $u$.

ABCD is a tetrahedron. The insphere touches the face ABC at $H$. The exsphere opposite $D$(which also touches the face ABC and the three planes containing the other faces) touches the face ABC at $O$. If $O$ is the circumcenter of ABC , show that $H$ is the orthocenter of ABC .

Two players play the following game: Let $n$ be a fixed integer greater than 1. Starting from number $k$=2, each player has two possible moves: either replace the number $k$ by $k$+1 or by 2$k$. The player who is forced to write a number greater than $n$ loses the game. Which player has a winning strategy for which $n$?

Find all positive integers $n$ for which the equation $a + b + c + d = n\sqrt{abcd}$ has a solution in positive integers.

The real numbers $a$, $b$, $c$, $x$, $y$, $z$ satisfy $a \ge b \ge c$>0 and $x \ge y \ge z$>0. Prove that $\frac{a^2x^2}{(by+cz)(bz+cy)} + \frac{b^2y^2}{(cz+ax)(cx+az)} + \frac{c^2z^2}{(ax+by)(ay+bx)} \ge \frac34.$

Assign to each side $b$ of a convex polygon $P$ the maximum area of a triangle that has $b$ as a side and is contained in $P$. Show that the sum of the areas assigned to the sides of $P$ is at most three times the area of $P$.

In pyramid ABCDE , the angle between plane ACE and faces ABCD , ABE , BCE , CDE and DAE are all $45^o$. Prove that $AB^2$+$AD^2$=$BC^2$+$CD^2$.

The positive reals $a_1,a_2,\ldots,a_n$ and $b_1 \le b_2 \le^{...} \le b_n$ satisfy $a_1$+$a_2 +^{...}$+$a_k \le b_1$+$b_2 +^{...}$+$b_k$ for all 1$\le k \le n$. Prove that $\frac1{a_1}+\frac1{a_2}+\cdots+\frac1{a_n} \ge \frac1{b_1}+\frac1{b_2}+\cdots+\frac1{b_n}.$

For a positive integer $m$, let $s$($m$) be the sum of the digits of $m$. For $n \ge$2, let $f$($n$) be the minimal $k$ for which there exists a set $S$ of $n$ positive integers such that $s\left(\sum\limits_{x\in X}x\right)=k$ for any nonempty subset $X \subset S$. Prove that there are constants 0<$C_1$<$C_2$ with $C_1 log_{10} n \le f$($n$)$\le C_2 log_{10} n$.

The angle bisectors of triangle ABC intersect sides $BC$, $CA$ and $AB$ at points $A_1$, $B_1$ and $C_1$, respectively. On line $A_1 B_1$, denote by $F$ the perpendicular foot point of $C_1$. Prove that line $FC_1$ bisects angle AFB .

Let $O$ be the point of intersection of the diagonals $AC$, $BD$ of a convex quadrilateral ABCD . Let $G_1$ and $G_2$ be the centroids of triangles OAB and OCD , respectively. Let $H_1$ and $H_2$ be the orthocenters of triangles OBC and ODA , respectively. Prove that $G_1 G_2$ is perpendicular to $H_1 H_2$.

Prove $(xy+yz+zx) \left(\frac 1{{(x+y)}^2}+\frac1{{(y+z)}^2}+\frac1{{(z+x)}^2}\right) \ge\frac94$ for all positive reals $x$, $y$, $z$.

Let $a$>$b$ and $n$ be arbitrary positive integers. Prove that $n$ divides $\varphi$($a^n$-$b^n$).

The angle $A$ of a triangle ABC is obtuse. Let $D$ denote an arbitrary point on side $AB$, and let $E$ be an arbitrary point on side $AC$. Show that $CD$+$BE$>$BD$+$DE$+$EC$.

The inscribed circle of the right-angled triangle ABC touches the leg $AC$ at $P$, the leg $BC$ at $Q$ and the hypotenuse $AB$ at $R$. Let $M$ denote the orthocentre of the triangle PQR . Prove that $RM$=$PQ$.

Define the sequence $a_n$ ($n$ is a natural number) as follows: $a_0=2 \text{and} a_n=a_{n-1}- \frac{n}{(n+1)!}, \text{if} n>0.$ Express $a_n$ in terms of $n$.

There were 31 people at a party. For any 15 of them, there is a further member of the group who knows all of them. (Acquaintances are mutual.) Prove that there is a guest of the party who knows all the others.

Find the distance between the (non-intersecting) diagonals of two adjacent faces of a unit cube.

The centroid of the triangle ABC is $S$, and the midpoint of $AB$ is $F$. For an interior point $P$ of the line segment $AF$, consider that point $Q$ of the line $PS$ for which $QC$and $AB$ are parallel. Let $R$ be the intersection of the lines $QA$ and $BC$. Prove that the line segment $PR$ halves the area of the triangle ABC .

Prove that the simultaneous equations $x$+$y$+$z$=0, $x^2$+$y^2$+$z^2$=100 have no rational solution.

An escribed circle of a convex polygon is defined as a circle that touches one side of the polygon externally, and also touches the extensions of the two adjacent sides. Let $k$ denote the sum of the areas of the escribed circles of a polygon, and let $t$ denote the total area of the circles drawn over the sides of the polygon as diameters. Assume that the polygon has an inscribed circle, and let $K$ and $T$ denote the perimeter and area of the inscribed circle, respectively. Show that $\frac{k}{t}\le \frac{K}{T}$ .

Prove that every positive integer $n$ has a (positive) multiple smaller than $n^2$, such that in decimal notation it does not contain all the ten different digits.

BAD and ACE are the regular triangles drawn over two sides of a triangle ABC on the outside. Show that the reflection of the line $BE$ about the line $CD$ passes through the point $A$.

Prove that if the difference of the integers $a$, $b$ is divisible by 100, then $a^{100}$-$b^{100}$ is divisible by 10 000.

Three unit circles all pass through the point $P$. Their other intersections are $A$, $B$and $C$. Find the radius of the circumscribed circle of the triangle ABC .

The radius of the iscribed circle of a triangle is $r$, the radius of its circumcircle is $R$, and one of the angles is $\alpha$. Given that $r=4R\cos\alpha \sin^2 \frac{\alpha}{2},$ show that the triangle is isosceles.

The diagonals of a cyclic quadrilateral ABCD are perpendicular. A perpendicular is dropped from the intersection of the diagonals onto the line $AB$. In what ratio does the perpendicular divide the side $CD$?

We built a large solid cube out of more than ten wooden cubes of unit edge. Then we painted all faces of the large cube. Is it possible that the number of unit cubes that have some faces is a factor of the number of unpainted ones?

For what integers $m$>1 is there an ordering $a_1$,$a_2$,...,$a_m$ of the numbers 1,2,...,$m$, such that the sums $a_1$,$a_1$+$a_2$,...,$a_1$+$a_2$+...+$a_m$ leave different remainders when divided by $m$?

Here is a magician's trick that I have seen: He asked one of the spectators to choose 5 cards at random from a deck of 52 French cards and hand them to his assistant. (The assistant was a permanent partner of the magician, so they could have talked before.) The assistant looked at the 5 cards, and handed four of them over to the magician one by one. Then the magician, without receiving any further information, named the fifth card. After the trick, the magician tried to convince mathematically minded spectators as follows: ``My assistant showed me four cards, and those four cards may be any four of the deck, therefore the only way he could convey information to me was through the order of the four cards he showed to me. Since, as you know, there are 4!=24 possible orders of four cards, and the fifth card could be any of 52-4=48, exactly 1 bit of information was missing. It is the guessing of this bit that needs a miracle.'' Does the trick really need supernatural powers?

There are 200 participants at an international conference. Each of them speaks at least 4 languages, and in any three of them there are two who speak a language in common. Prove that there is a language that at least 26 participants speak.

Given that $x$ and $y$ are rational numbers, such that $x^5$+$y^5$=2$x^2 y^2$. Prove that 1-$xy$ is the square of a rational number.

The president of the Pyramid Bank lives in a fancy residential district outside the city. On weekdays, he is driven to the city in a car that leaves the bank in the city at the same time every morning, picks up the president and they arrive at the bank exactly when it opens. One morning, the driver called the president to say that something was wrong with the car and he would be late. So the president left home one hour earlier than usual on foot. Meanwhile, the driver fixed the car and still managed to leave the city at the usual time. He met the bank president on the way, picked him up, and they arrived at the bank 20 minutes before opening. How long did the president walk? (Assume that the speed of the car is constant, and picking up a passenger does not take any time.)

Solve the equation ( $x^2$+$y^2$ ) $^3$= ( $x^3$-$y^3$ ) $^2$ on the set of real numbers.

The lines of the legs $AB$ and $CD$ of a trapezium ABCD intersect at the point $M$. A line $e$ passing through $M$ intersects the line $AD$ at a point $E$ different from $A$ and $D$, and it intersects the line $BC$ at $F$. The intersection of the diagonals is constructed in each of the trapezia ABFE and CDEF , denote them by $P$ and $Q$. Show that $PQ$ is parallel to the bases of the trapezia.

The edges of a square based right prism are $a$=$b$=8, $c$=27 units long. Cut the prism into four parts, such that it is possible to put them together to form a cube.

Consider the points where the lines of the altitudes of an acute scalene triangle intersect the perpendicular bisectors of the sides. Prove that these points determine two congruent triangles that are similar to the original triangle.

For which values of $n$ is it possible to assign values to the variables $x_1,x_2,\ldots,x_n$ from the set {+1,-1} to make the equality $x_1x_2+x_2x_3+x_3x_4+\ldots+x_{n-1}x_n+x_nx_1=0$ true?

It is known that the line segments connecting the midpoints of the opposite edges of a tetrahedron are concurrent. Is it possible to select a point on each edge that divides the edge 1:2, such that the line segments connecting the selected points of opposite edges are concurrent?

Is it possible for the product of four consecutive positive integers to be a perfect cube?

Find all pairs ($a_1$,$a_2$) of positive integers, such that the sequence defined by the recurrence $a_{n+2}=\frac{a_n+a_{n+1}}{(a_n,a_{n+1})} (n\ge 1)$ is periodic.

What is the locus of those points in the plane where the tangents drawn to a given parabola enclose a $30^o$ angle?

$P$ is an interior point of the triangle ABC . The perimeter of the triangle is 2$s$. Show that $s$<$PA$+$PB$+$PC$<2$s$.

Solve the following equation: $x-\sqrt{\frac{x}{x+3}}=\frac{2}{x+3}.$

ABC is an isosceles triangle. Drop a perpendicular from the midpoint $D$ of the base $BC$onto the leg $AC$ and denote the foot of the perpendicular by $E$. The midpoint of the line segment $DE$ is $F$. Show that the lines $BE$ and $AF$ are perpendicular.

How many solutions does the equation $y^2$=$x^2$-$x$+1 have in the set of $a$) integers; $b$) rational numbers?

The skew lines $e$ and $f$ are perpendicular. The unit line segment $AB$ has its endpoint $A$on $e$ and endpoint $B$ on $f$. What is the locus of the midpoint of $AB$ as its endpoints are moving along the lines?

Solve the following equation: sin $x$+cos $x$+sin $x$cos $x$=1.

Prove that $2^{3n}$ +1 is divisible by $3^{n+1}$ for all real numbers $n$.

A fair coin is tossed 12 times in a row and the outcomes are listed. What is the probability that there are no three consecutive heads?

$AB$ is a diameter in the circle $k$. Choose an arbitrary point $E$ in the interior of the circle. The other intersections of the lines $AE$ and $BE$ with the circle are $C$ and $D$, respectively. Prove that the value of the expression $AC^. AE$+$BD^. BE$ is independent of the position of $E$.

Prove that it is possible to inscribe an ellipse in any acute triangle, such that the foci are the orthocentre and the circumcentre of the triangle, respectively.

Prove that every convex quadrilateral has a vertex, such that its reflection about the midpoint of the line segment connecting the two adjacent vertices does not lie outside the quadrilateral.

$P$ is an arbitrary point on the side $AB$ of the parallelogram ABCD , and $Q$ is an arbitrary point of side $CD$. Let $M$ denote the intersection of the lines $DP$ and $AQ$, and let $N$ denote the intersection of the lines $CP$ and $BQ$. Show that the line $MN$ is passing through the centre of the parallelogram.

Each side of the triangle ABC is divided into $p$ equal parts, where $p$ is a prime number. Then one point of division on each side is connected to the opposite vertex so that the three connecting lines are concurrent. Determine the possible values of $p$.

Prove that the inequality $\frac{x^2+xy+y^2}{3}\le\frac{x+y}{2}\cdot \sqrt{\frac{x^2+y^2}{2}}$ holds for all positive numbers $x$, $y$.

Prove that an open interval of length $\frac{1}{n}$ may contain at most $\left[\frac{n+1}{2}\right]$ rational numbers whose denominator is at most $n$.

Find the solutions of the equation cos 4$x$-tan 5$x$=cot 5$x$-sin 6$x$ in the interval (-$\pi$;$\pi$).

Construct a point $P$ in the interior of an acute-angled triangle ABC , such that $PA^. BC$=$PB^. CA$=$PC^. AB$.

Solve the equation $3^x +4^y =5^z$ in the set of positive integers.

A convex solid is built out of four regular hexagons of unit side and four regular triangles. Prove that a sphere can be circumscribed about the solid, and find the radius of the sphere.

I have a six-digit number in mind. By transferring the first digit to the end, I would obtain three times the original number. What is my number?

A diagonal is drawn in a few fields of a chessboard, such that no two of them have a common point. What is the maximum possible number of diagonals that can be drawn in this way?

27 regular dice are glued together to form a 3×3×3 large cube. On every pair of faces of the dice glued together, there are the same number of spots. How many spots are there on the surface of the large cube?

Ann and Bob are playing the following game: There are 10 heaps of pebbles in front of them. The first heap contains 1 pebble, the second one contains 2, the third one contains 3, and so on, the tenth heap contains 10 pebbles. They take turns in carrying out one of the following two moves: they either divide a heap into two smaller ones or remove one pebble from one heap. A player who cannot do so any more will lose the game. Determine which player has a winning strategy, given that Ann starts the game.

Vertex $A$ of a tetrahedron ABCD is reflected about $B$, $B$ is reflected about $C$, $C$about $D$ and $D$ about $A$. Let the respective reflections be $A$', $B$', $C$' and $D$'. By what factor is the volume of the tetrahedron $A$'$B$'$C$'$D$' larger than that of the original tetrahedron ABCD ?

$P$ is an interior point of a circle of radius $r$. Let the arms of a right angle with vertex $P$ intersect the circle at $A$ and $B$. Complete the triangle BPA to form a rectangle PAQB . What curve will the point $Q$ describe while the right angle turns about $P$?

Let {$r$} denote the number obtained by taking away from $r$ the greatest integer not greater than $r$. Prove that the equation {$x^3$}+{$y^3$}={$z^3$} has infinitely many solutions, such that $x$, $y$ and $z$ are rational numbers, but none of them are integers.

Prove that by drawing some noncrossing diagonals it can be divided into triangles such that each vertex of the polygon belongs to an odd number of them.

Find a polynomial $p$ of rational coefficients, such that $p\big(\sqrt{2} + \sqrt{3}\,\big)=\sqrt{2}$.

Find all positive prime numbers that satisfy equation $x^2$+$y^2$=$z$-16.

A sequence $a_n$ is defined as $a_1$=1, $a_{2n}$=$a_n$, $a_{2n+1}$+$a_n$=1. Determine the value of $a_{2006}$.

Some characters in a Hamlet play are paired, for example, both members of a pair can play the role of Gertrude and the role of Player Queen. Before every performance it is decided by a lot who will play Gertrude and who Player Queen. Other pairs are also chosen by a similar lot. Sarah has already seen Hamlet, but she wants to see it again with a different actor playing the other member of the pair Gertrude/Player Queen, Claudius/Player King and Ophelia/Fortinbras, although not necessarily in one performance. How many tickets should she still buy to see all three roles in the other casting with 90% probability?

Determine the conditions on the real numbers $a$, $b$ and $c$ such that for every positive integer $n$ there exists a triangle with side lengths $a^n$, $b^n$ and $c^n$.

A triangle is made by soldering three thin metal rods of length 8, 15 and 17 cm together. A solid ball of radius 5 cm is put on this triangle frame held horizontally. In what ratio does the plane of the triangle divide the volume of the ball?

The sum of integers $a_1,a_2,\ldots, a_{10}$ all greater than 1 is 2006. What is the minimal value of $\binom{a_1}{2}+\cdots+ \binom{a_{10}}{2}$?

Find the angle encompassed by the hypotenuse of a right-angled triangle with circumference of 2 units, as seen from a point lying on the inner angle bisector of the right angle at a distance of $\sqrt{2}$ from that vertex.

Let $a$, $o$ and $c$ be three lines in the plane. Consider all squares ABCD such that vertex $A$ lies on line $a$, the opposite vertex $C$ on line $c$, and the centre of the square $O$lies on line $o$. Find the locus of vertices $B$ and $D$.

Find all positive rational triples ($p$;$q$;$r$) such that $p$+$q$+$r$, $\frac{1}{p} +\frac{1}{q} +\frac{1}{r}$ and pqr are all integers.

Find all two-digit even numbers $\overline{ab}$ , whose fifth powers end in $\overline{ab}$ . (Mathematics competition for teacher training colleges, 1973).

The diagonals of the trapezium ABCD intersect at the point $M$. The areas of the triangles ABM and CDM are 18 and 50 units, respectively. Find the area of the trapezium.

Sketch in the cartesian plane the region consisting of the points ($x$,$y$) such that $\frac{x}{y}+\frac{1}{x}+y \ge \frac{y}{x}+\frac{1}{y}+x$ .

Solve the following simultaneous equations: $x^3$+$y^3$+$z^3$=8, $x^2$+$y^2$+$z^2$=22, $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{z}{xy} =0$.

The angle bisector drawn from vertex $C$ of an acute angled triangle ABC intersects the opposite side at the point $F$. The feet of the perpendiculars drawn from the point $F$ to the sides $BC$ and $CA$ are $P$ and $Q$, respectively. Let $M$ denote the intersection of the lines $AP$ and $BQ$. Prove that $AB$ and $CM$ are perpendicular to each other.

Let $h$ denote the length of the tangents drawn to a circle from an exterior point $P$, and let the midpoint of the line segment connecting the points of contact be $F$. Prove that a chord $AB$ of the circle satisfies the equality $AP^. PB$=$h^2$ if and only if the line $AB$passes through the point $P$ or the point $F$.

$a$ and $b$ are real numbers, such that $a^2$+4$b^2$=4. What is the largest possible value of 3$a^5 b$-40$a^3 b^3$+48$ab^5$? (Suggested by Z. Horváth, Veresegyház)

For what values of the positive integer $n$ is there a simple graph in which the degree of each vertex is at most $n$, and there are $i$ vertices of degree $i$ for all 1$\le i \le n$? (Suggested by G. Mészáros, Kemence)

Let $H$ be the set of integers not greater than 2006: $H$={1,2,...,2006}. Let $D$denote the number of those subsets of the set $H$ in which the number of elements divided by 32 leaves a remainder of 7, and let $S$ denote the number of those subsets of the set $H$in which the number of elements divided by 16 leaves a remainder of 14. Prove that $S$=2$D$. (Based on an OKTV (National Competition) problem)

Let $a$, $b$, $n$, $k$ be positive integers, such that $n$ is odd, $p$ is an odd prime number, and $a^n$+$b^n$=$p^k$. Prove that $n$ is a power of $p$ with a non-negative integer exponent.

Santa Claus is putting Christmas candies in small bags. If he puts 35 candies in each bag, there will be 7 candies left. What can be the overleft if he puts only 15 candies in each bag?

Given are 100 real numbers whose sum is equal to zero. At least how many pairs can be chosen from them such that the sum of the numbers in each pair is not negative?

Solve the following simultaneous equations: $x^2 - 4\sqrt{3x-2} + 6 = y,$ $y^2 - 4\sqrt{3y-2}+6 = x.$

In a theatre performance of Hamlet, Acts 2 to 5 are played in a part of the auditorium chosen at random by means of a draw. The stalls are divided into four sectors. The part of the audience with seats in the sector selected will move to another sector, taking also their chairs with them. Provided that each of the four sectors are large enough (that is, if one chooses a sector, then there will always be enough space for one's chair there) and assuming that each sector is equally likely to be chosen, what is the proportion between the probabilities that one needs to move twice during the play or only once?

2006 points are given on the plane. At most how many of them forms acute-angled triangles, respectively, with every pair chosen from the remaining 2005 points?

$D$ is an interior point of the side $AB$ of a triangle ABC . The inscribed circle of the triangle ABC touches the side $AB$ at the point $P$. The inscribed circles of the triangles ADC and DBC are touching the side $DC$ at $Q$ and $R$, respectively. Show that $DP$=$QR$.

The triangle ABC is right angled at $C$ and its incentre is $O$. The perpendiculars drawn at the point $O$ to the line segments $OA$ and $OB$ meet the side $AB$ at $P$ and $Q$, respectively. The foot of the perpendicular from $P$ to the side $BC$ is $P$', and that from $Q$ to the side $AC$ is $Q$'. Prove that the points $P$', $Q$' and $O$ are collinear.

Let $K$ be an arbitrary point inside the triangle $A_1 A_2 A_3$ and denote the centroid of the triangle $KA_j A_k A_s$ by $S_i$ ($i$=1,2,3,). Prove that the line segments $A_i S_i$ are concurrent.

Draw perpendiculars to the real axis of a hyperbola at its endpoints. Let $P$ and $Q$ denote the intersections of these perpendiculars with an arbitrary tangent to the curve, respectively. Prove that the line segment $PQ$ subtends right angles at the foci.

Let $a$ and $b$ denote positive integers, such that $a^n$+$n$ divides $b^n$+$n$ for all positive integers $n$. Prove that $a$=$b$.

How many solutions does the equation $x$+$y$+$z$=100 have on the set of positive integers?

In a Fun Shop selling party novelties, a packet of streamers costs $p$ percent more than a packet of confetti. In other words, a packet of confetti is $q$ percent cheaper than a packet of streamers. The difference of $p$ and $q$ is 90. How many packets of streamers can one buy for the price of 10 packets of confetti?

At most how many concave interior angles may a 2006-sided polygon have?

Mihály Tímár (the hero of a famous Hungarian novel) is in a trouble. The red crescent that marked the sack containing the treasure has come off. He knows, however, that the heaviest of the four sacks hides the treasure. From three measurements, he has found out that the first sack together with the second one are lighter than the other two, the first and third together are equal in weight to the other two, and the first and fourth sacks together are heavier than the other two. Which sack hides the treasure?

Three sides of a convex polygon are 1 cm, 4 cm and 8 cm long, and its diagonals are perpendicular to each other. How long may the fourth side be?

The values of 10 banknotes add up to 5000 forints (HUF). What banknotes may this sum consist of? (The Hungarian bank notes are: 200 HUF, 500 HUF, 1000 HUF, 2000 HUF, 5000 HUF, 10 000 HUF, 20 000 HUF.)

Paul the postman was delivering letters in a long street. First he walked down the street on one side (where the odd-numbered blocks were) and then he walked back on the other side (delivering the letters to the even-numbered blocks). On the odd side he spent one third as much time standing in front of the letter boxes as he spent by walking back on the even side. On his way back on the even side his total standing time in front of the letter boxes was one third as much as his time in motion down the street on the odd side. Finally, it turned out that delivering the letters took the same time on each side. Find the ratio of his average walking speed (not counting the stops) on the odd side to his average walking speed back on the even side.

We built a large solid cube out of more than ten wooden cubes of unit edge, each. Then we painted all faces of the large cube. Is it possible that the number of unit cubes that have some faces painted is a multiple of the number of the unpainted ones?

In a triangle ABC , $\angle$ BAC $=45^o$. Let $P$ denote the point of side $AC$ for which $AP$:$PC$=1:2. Given that $\angle$ ABP $=15^o$ find $\angle$ ACB .

The perimeter of the axial section of a cylinder is 90 cm. What is the maximum possible volume of the cylinder?

In an algebra class on the first of April, the students were practising. The task was to simplify the fraction $\frac{{(x+2)}^3+{(y+x)}^3}{{(x+2)}^3-{(y-2)}^3}.$ Agnes, who was the best at mathematics in the class, suggested that if the denominator is not zero, they should just cross out the threes in all the indices, that is, write $\frac{x+2+y+x}{x+2-(y-2)}= \frac{2x+y+2}{x-y+4}.$ Check the result.

In how many different orders may the digits 0, 1, 2, 3, 4, 5, 6 form a seven-digit number divisible by four? (A number cannot start with 0.)

Find a line that halves both the area and the perimeter of the ``Egyptian'' triangle (the sides of which are 3, 4, 5).

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

Evaluate cot $x$, given that cot $x$=sin $x$?

An arbitrary interior point of a regular hexagon of unit side is reflected about the midpoint of each side, respectively. Calculate the area of the resulting hexagon.

A fair coin is tossed 12 times in a row and the outcomes are listed. How many sequences of outcomes are possible in which there are no two consecutive heads?

Solve the following inequality on the set of real numbers: $x^2-3\sqrt{x^2+3}\le 1$.

Consider four rays in the space starting from a common point and enclose pairwise equal non-zero angles. What is that angle?

Prove that the following equality is true for every positive integer $n$: $\frac{1^3+3^3+5^3+\ldots+ {(2n-1)}^3}{1+3+5+\ldots+(2n-1)}= 2n^2-1.$

Find all natural numbers $n$, such that $5^n$+12$n^2$+12$n$+3 is divisible by 100.

Find the pairs ($c$,$d$) for which the equation $x^3$-8$x^2$+$cx$+$d$=0 has three (not necessarily) different positive integer roots.

The perimeter and the area of a rectangle are equal to the perimeter and the area of a rhombus, respectively, that has an angle of $30^o$. Find the ratio of the sides of the rectangle.

Prove that the equality $c^2+2ab\sin (\gamma+30^{\circ}) =b^2+2ac\sin (\beta+30^{\circ}) =a^2+2bc\sin (\alpha+30^{\circ}).$ holds in every triangle.

A survey carried out on a sample of 500 revealed that 46% of those asked liked strawberry ice cream, 71% liked vanilla and 85% liked chocolate ice cream. Is there anyone in the sample who likes all three flavours?

In a partial Solar Eclipse, the apparent diameters of the Sun and the Moon were equal. In the time instant when the eclipse was at its maximum, the edge of the Moon's shadow passed through the centre of the Sun's face. What percentage of the Sun's face was eclipsed?

Find the number pairs $x$, $y$ for which the following inequality is true: 2|$x$+$y$|$\le$|$x$|+|$y$|.

Find the integer solutions $x$, $y$ of the equation $x^6$-$y^2$=648.

A triangle is drawn on squared paper. The lengths of the sides are $2\sqrt{10}$, $3\sqrt5$ and 5 units. Prove that the measure of the smallest angle is $45^o$.

Which number has the form 503 in base $n$ and 305 in base ($n$+2)?

Find the value of parameter $a$ such that the distance between two roots of the equation $x^2$-4$ax$+5$a^2$-6$a$=0 is the greatest.

Four distinct points are given in the plane. The pairwise distance between 4 points is 1, and the distance between a certain pair is 1.2. What can be said about the distance between the remaining 2 points?

A cylinder of height $\frac{4}{3} R$ is inscribed into a sphere of radius $R$. Determine the ratio of their volumes.

A car dealer sold 7 cars per day on the average in a certain period. If the day of the smallest number of cars sold is not counted, the average number of cars sold per day will be 8. If the day of the largest number of cars sold is not counted, the average will be 5. Finally, if both of these days are ignored them, the average will be 5.75 cars per day. What was the total number of cars sold by the dealer in the given period?

Prove that if the expression $\frac{x^2}{(x-y)(x-z)} +\frac{y^2}{(y-x)(y-z)} +\frac{z^2}{(z-x)(z-y)}$ is well defined, then its value is independent of the values of the variables $x$, $y$and $z$.

Consider a quadrant of a disc of radius 12 cm. A semicircle is drawn over one of the radii bounding the quadrant, and cut out of it. What is the radius of the largest possible circle that can be inscribed in the remaining figure?

For what real values of $x$ will the value of the expression $\sqrt{2\sin x}-\sin x$ be a maximum?

The roof of a newsstand with a square base 3 m on a side consists of two regular triangular prisms that intersect each other. One lateral face of each prism coincides with the ceiling of the newsstand. (The prisms are rotated through $90^o$ relative to each other.) Calculate the surface area of the roof.

There are two kind of rolls sold in Hungary, their prices are 15 forints and 12 forints, respectively. (Forint is the Hungarian currency, HUF.) We buy some of each. $a$) Is it possible that we pay 500 forints? $b$) Is it possible that we pay 600 forints?

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

When a rectangular sheet of paper is folded along a diagonal, the vertices form a cyclic trapezium with three equal sides, and the length of the fourth side is $10\sqrt3$. Find the dimensions of the rectangle?

A cube is inscribed in a regular four-sided pyramid whose height is the double of the length of the sides of its base. What proportion of the volume of the pyramid is occupied by this cube. (The inscribed cube has four vertices on the lateral edges and the other four are lying on the base of the pyramid.)

I go shopping with 5000 forints. (Forint is the Hungarian currency, HUF.) I drop in three shops, and I find some gift in each of them that I like and buy it if I have enough money. The price of each gift is one of 1000, 1500 or 2000 forints with probability 1/3, independently in each shop. What is the probability that I will end up with three presents and even with some money left?

In a television quiz, 10 people are asked questions. They all answer simultaneously by pressing the button assigned to the answer they think correct. Those who answer correctly get as many points each as the number of incorrect answers. After the fifth question answered, the players had 116 points altogether. 32 of these were collected by Géza. Show that Géza answered all the five questions correctly.

Connect an interior point of a parallelogram to each vertex. Show that it is possible to construct a quadrilateral with the four segments hence obtained as sides such that its vertices are lying on the sides of the parallelogram.

Out of the digits of a three-digit number of different digits, all the possible two-digit numbers of different digits are formed, and these two-digit numbers are added. Given that the sum is equal to the original three-digit number, find all such three-digit numbers.

A merchant paid 6000 Ft for a product. What price should he write on it so that he having reduced this price by 10 percent he still make 20% profit.

The diameter of the wheels on Jack's bike is 28'' (inches), and the diameter of the wheels of Jill's bike is 14'' (inches). Jack's wheel turns around twice while he makes 3 turns on the pedals. Jill's wheels turns around 3 times while she makes two turns on the pedals. Jack turns the pedals twice as many times a minute as Jill does. Cycling at a uniform speed, Jill covers 2 kilometers in 20 minutes. How long does Jack take to cover the same distance?

The lengths of the edges in cm of a square-based prism are whole numbers. With a plane perpendicular to the base and parallel to one lateral face, a 4-cm-thick part is cut off the prism. The volume of the remaining solid is 126 $cm^3$. How long are the edges of the original prism?

ABCD is a square of 5-cm sides. An equilateral triangle CDE is drawn outwards to the side $CD$. $a$) Find the angles of the triangle ABE . $b$) What is the radius of the circle that passes through all three vertices of triangle ABE ?

The point $K$ lies on each of the circles of radius $r$ centred at the points $K_1$, $K_2$, $K_3$. The other common points of two of the three circles are $P$, $Q$, and $R$. Show that the triangles $K_1 K_2 K_3$ and PQR are congruent.

2006 small circles are placed in a large circle, so that each touches the next one in a row on the outside, the centres of the small circles all lie on the same diameter of the large circle, and the sum of the diameters of the small circles equals the diameter of the large circle. Compare the total perimeter of the small circles and the perimeter of the large circle.

The Silly family bought a tape measure that can be hung on the wall for measuring heights of people. One end of the tape reads 80 cm'' and the other end reads 180 cm'' (If he person to be measured stands on the floor, the tape can measure heights between 80 and 180 cm if properly used.) The Sillies hung the tape in a vertical position, but its lower end was not fixed at the appropriate height.To make it even worse, it also turned out that even the centimetre scale on the tape was wrong: the divisions of were not actually 1 cm apart. Thus the head of the 130-cm-tall Pete reached up to the 120-cm mark while the tape read 145 cm for his 150-cm-tall sister. What is the maximum height of a person that can be measured with this measuring tape with improper scale hung improperly on the wall?

If the measure of the surface area of a cuboid (in unit squares) is subtracted from the sum of the measures of its edges (in length units), and the measure of the volume of the cuboid (in unit cubes) is added to the number obtained, the result will be 8. Prove that the measure of one of the edges is 2.

Polyminoes are like dominoes: they are made up of small squares that are joined together along whole sides. Define the ``boundary number'' of a polymino $P$ as the number of different ways to place congruent copies of $P$ around its boundary. (There must be a polymino touching it along each of its edges, and touching edges touch along their full length.) For example the boundary number of the cross is 2, since it can be bounded by its copies in the following two ways: (Arrangements that are mirror images of each other are considered different.) What is the boundary number of a 3×3 square (i.e. one consisting of 9 small squares)?

How many six-digit numbers of the form ABAABA are there can be expressed as a product of exactly four primes?

We have two cubes of not necessarily the same size with integer edges. They are placed on top of each other, so that the upper cube has a whole face touching a face of the lower cube. The volume of the solid obtained in this way expressed incube units is the same as the measure of the surface area in square units. How long are the edges of the original cubes?

Consider a convex hexagon in the plane with four distinct points given in its interior, such that no three points out of the vertices of the hexagon and the four given points lare collinear. The hexagon is dissected into triangles, such that the vertices are those of the hexagon and the given points. (All the 10 points must be used.) Show that the number of triangles obtained is always 12.

Is there a square number, such that the sum of its digits is 2006?

Annie and Bennie were playing with a chair. When Annie was standing on the chair and Bennie on the floor, the top of Annie's head was 30 cm higher than Bennie's. When they exchanged places, Bennie standing on the chair and Annie on the floor, the top of Bennie's head was half a metre higher than Annie's. How tall is the chair?

A large cube is put together out of small unit cubes. There are 80 small cubes altogether that lie along the edges or at the corners of the large one. How many small cubes does the large cube consist of?

There are four ninth-grade classes at a school. The four classes went on a four-day camp. Each class went hiking on one of the four days, while the other classes took part in presentations and other activities. On Monday, class $A$ went hiking and thus 81 students remained in the camp. On Tuesday, class $B$ were hiking and thus 79 students remained in the camp. On Wednesday, class $C$ were hiking and 75 students stayed in the camp. Finally, on Thursday, with class $D$ hiking, 80 students stayed in the camp. How many students are there in each class?

2163 is a four-digit number in which the digit in the units' place is three times the digit in the hundreds' place, and the digit in the tens' place is two times the sum of the digits in the hundreds' place and the thousands' place. Is it true that such four-digit numbers are all divisible by 3?

Pilots use the face of a clock to express bearings. The actual flight direction is identified with 12 o'clock, so, for example, instead of 90 degrees to the right'' they say 3 o'clock'', or behind me'' equals to 6 o'clock''. Follow the flight path of a reconnaissance plane that takes off from the base and flies 3 minutes in a given direction, then turns to 2 o'clock and flies 4 minutes, then turns again to 2 o'clock and flies 3 minutes, finally the plane turns to 4 o'clock and flies 9 minutes. In what clock direction should the plane be turned to in order to fly straight back to the base? How many minutes does this take?

Sherlock Holmes was investigating a crime in the Musgrave property famous for a beautiful elm tree. Looking for the hidden treasure, he was to ``walk ten and ten steps to the north, five and five steps to the east, two and two steps to the south, and one and one step to the west.'' Holmes found out, however, that one of the directions was wrong, one should walk in the opposite direction instead (for example, south instead of north). Since he did not know which was wrong, he tried all possibilities. Each walk ended within the park of the Musgraves. Given that Holmes' steps are 80 cm long and the park is a rectangle with sides in the north-south and east-west directions, find the minimum possible area of the park.

When the five-digit number $\overline{ABCDE}$ is multiplied by 4, the five-digit number $\overline{EDCBA}$ is obtained. Find the value of $\overline{ABCDE}$ . ($A$, $B$, $C$, $D$, $E$denote different digits.)

A one-digit number is written in each field of a 3×3 chessboard. A chessman is moved in an L-shaped path from one field to another (the way that a knight moves). Such an L consists of four fields. In how many different ways is it possible to fill out the 3×3 table with (not necessarily different) one-digit numbers so that the sum of the four numbers in every such L-shaped move is the same?

The sides of an equilateral triangle ABC are 16 cm long. The side $AC$ is extended beyond $A$ by one quarter of the length $AC$ and the point $P$ is obtained. The point $P$ is connected by a line to the point dividing the side $AB$ in a 1:3 ratio (so that the shorter segment of $AB$ lies next to $A$.) This line divides the side $BC$ into two pieces. Find their lengths.

There are 7 houses in Dragon Street, numbered 1 to 7. Last week, Mr. D. E. Livery the postman delivered letters to two houses every day. There are two houses where he did not brought anything during the whole week, but he visited all the other houses twice. The sum of the number of the houses visited by Mr. Livery was 5 on Monday, 8 on Tuesday, 9 on Wednesday, 13 on Thursday and 7 on Friday. (There is no mail on Saturday and Sunday.) Which are the two houses that did not receive any letter last week?

In a certain year, the 1st of January was not a weekend day but the 31st of December was. The school year started on the 1st of September. Which day of the week was that?

Solve the following equation on the set of real numbers: $x^2+6+\frac{1}{x^2}=4x+\frac{4}{x}$ .

George is 17 years older than Anna. If George's age in years is written after Anna's age, a four-digit square number is obtained. The same will be true in 13 years' time. How old are they now?

Four cardboard discs, two of 32 $dm^2$ and two of 21 $dm^2$ are placed on a tabletop of 1 $m^2$. None of them overhang the edge of the table. Is it true that there are two discs that overlap on an area of at least 4 $dm^2$?

Andrew, Bill, Charlie and Dennis are playing cards. In each game, only three of the four boys are playing. They get points from one another, depending on the outcome of the game: either one player gets points from the other two or two players get points from the third one. (If there are two losers or two winners then they do not necessarily lose or gain the same number of points.) At the beginning, everyone had 100 points. The boys have played four games, each time leaving out a different person. The points that each participant had during the card play are tabulated on the diagram. Which player was left out from the respective games? Andrew Bill Charlie Dennis 100 100 100 100 110 110 90 80 125 105 80 110 85 85 100 130

We have a large supply of boxes: each of them is either small, or medium or large. 11 large boxes were put on the table. Some were left empty and 8 medium boxes were put in each of the rest of them. Some of the medium boxes were also left empty, and 8 small boxes (all empty) were put in each of the remaining ones. Thus there were 102 empty boxes on the table. How many boxes are there on the table altogether? (Boxes of the same size cannot be put into each other.)

A book contains 700 psalms, numbered from 1 to 700. The congregation sings one of them every Sunday. The number of the chosen psalm is displayed as a 3-digit number on a special counter, fabricated from three small wooden cubes. (One or two-digit numbers are displayed with leading zeros, e.g. 3 appears as 003 and 28 as 028.) The is one digit on faces of the cubes, respectively, and 6 can also be interpreted as a 9. $a$) How many cubes are needed to display every possible number? How the digits are arranged on the faces? $b$) How many more psalms can be included in the book without adding further cubes?

The sum of the squares of three consecutive odd natural numbers is a four-digit number of identical digits. Find all possible sets of three such numbers. Are there five consecutive odd natural numbers, such that the sum of their squares is a six-digit number of equal digits?

If two digits of the four-digit number 1234 are cancelled in every possible way and the remaining digits are read as a two-digit number, the numbers 12, 13, 14, 23, 24, 34 are obtained. Their sum is 120. Find a four-digit number for which this sum is $a$) 540; $b$) 220.

A square patch of grass, 100 m on a side is surrounded by a paved road along its boundary. Most people want to get from corner $A$ of the square to the midpoint $F$ of an opposite side. They walk along the path ABF , but some of them have started a path through the grassy area. They walk along the side $AB$ for a while, then they are heading towards $F$ along a straight line through the grass, as shown in the diagram. Thus they walk 25% less compared to the path along the border. How far is the turning point $K$ of the path from the point $A$?