KöMaL 2005

The interior bisector of the angle $A$ of the triangle ABC cuts the incircle $c$ at two points; the one closer to $A$ is denoted by $O_A$. Define the points $O_B$ and $O_C$ similarly on the internal bisectors of the angles $B$ and $C$, respectively. The circle $c_A$ is drawn about $O_A$ and it is touching the sides $AB$ and $CA$. Similarly, $c_B$ is about $O_B$ and it is touching the sides $BC$ and $AB$, finally the circle $c_C$ is about $O_C$ and it is touching the sides $CA$ and $BC$. Taken any two of the circles $c_A$, $c_B$ and $c_C$ consider their common external tangents different from the corresponding sides of the triangle, respectively. Prove that these three tangents are concurrent.

Find those positive integers $n$ for which $\sigma(n!)=\frac{(n+1)!}{2}$ holds.

For any two guests on a party of $kn$ participants there is some guest who knows both of them. Prove that if $k

Given that $a$+$b \le c$+1, $b$+$c$ $\le a$+1, $c$+$a \le b$+1 for the numbers $a$,$b$,$c$ $\ge$0, show that $a^2$+$b^2$+$c^2 \le$2 abc +1.

An equilateral triangle of side $n$ is divided into equilateral triangles of unit side. How many castles can be placed on the lattice points of the triangular lattice obtained so that no two of them attack each other? Castles can move parallel to the sides of the triangle, in six directions altogether. (Suggested by A. Egri, Hajdúszoboszló)

$P$ is a point in the interior of the quadrilateral $A_1 A_2 A_3 A_4$ that does not lie on either diagonal. The points $B_i$ lie in the interior of each line segment $A_i P$. Let $C_{ij}$ be the intersection of the lines $A_i B_j$ and $A_j B_i$ (1$\le i$<$j$ $\le$4). Prove that the line segments $C_{12} C_{34}$, $C_{13} C_{24}$, $C_{14} C_{23}$ are all concurrent.

The real numbers $x_1 \ge x_2 \ge$...$\ge x_n$>0 and $y_1 \ge y_2 \ge$...$\ge y_n$>0 satisfy $\prod_{i=1}^kx_i\ge\prod_{i=1}^ky_i.$ for all 0<$k \le n$. Prove that $\sum_{i=1}^nx_i\ge\sum_{i=1}^ny_i.$

Find all continuous functions $f\colon\mathbb{R}\to\mathbb{R}$, such that $f$ ( $x$+$y$+$f$($xy$) ) =$xy$+$f$($x$+$y$). for all $x$,$y$ $\in R$.

Is it possible to obtain the array (5,13,42) starting from (1,21,42) by repeating the following steps: It is allowed to swap any two numbers in an array of three numbers ($a$,$b$,$c$), or replace the array by ($a$,$b$,2$a$+2$b$-$c$).

The inscribed circle of the triangle ABC touches the side $AB$ at $C_1$, the side $BC$at $A_1$, and the side $CA$ at $B_1$. It is known that the line segments $AA_1$, $BB_1$ and $CC_1$ pass through a common point. Let $N$ denote that point. Draw the three circles that pass through $N$ and touches two of the sides. Prove that the six points of tangency are concyclic.

Does there exist a function $f:\mathbb{Q}\to\{-1,1\}$ such that $f$($x$)=-$f$($y$) whenever $x$ and $y$ are different rationals and $xy$=1 or $x$+$y \in${0,1}?

Find all real numbers $\lambda$ for which there exists a non-zero polynomial $P$, such that $\frac{P(1)+P(3)+P(5)+\dots+P(2n-1)}{n} = \lambda P(n)$ for all $n$. List all such polynomials for $\lambda$=2.

The convex $n$-sided polygon $K$ lies in the interior of a unit square. Show that it is possible to select three vertices of the polygon that form a triangle of smaller area than $\frac{80}{n^3}$ units.

A die of $n$ faces is rolled until each of the $n$ possible outcomes is obtained at least once. What is the expected value of the number of rolls needed?

The diameter $AB$ of the circle $k$ is perpendicular to the line $\ell$, which does not pass through either $A$ or $B$. Let $C$ be a point of $\ell$ outside $k$, and let $D$ denote the other intersection of the line $AC$ with the circle. Draw one of the tangents from $C$ to $k$and denote the point of contact by $E$. Let $F$ be the intersection of the lines $BE$ and $\ell$, and finally let $G$ denote the intersection of the line segment $AF$ with $k$. Show that the line $DF$ passes through the reflection of the point $G$ about $AB$.

$a_0$,$a_1$,...,$a_n$ and $b_0$,$b_1$,...,$b_k$ are non-negative real numbers, such that $a_0$=$b_0$=1 and ($a_0$+$a_1 x$+...+$a_n x^n$)($b_0$+$b_1 x$+...+$b_k x^k$)=1+$x$+...+$x^{n+k}$. Prove that each of the numbers $a_i$ and $b_i$ is either 0 or 1.

Find all sequences $a_1$,$a_2$,... of non-negative numbers, such that the sum $S=\sum_{k=1}^\infty a_k$ is finite and $\sum_{k=1}^\infty a_{kn} = \frac{S}{n^\alpha}$ for all positive integers $n$.

Consider one of the two arcs of a rectangular hyperbola centred at $O$, and choose an arbitrary point $P$ on it. Let $Q$ and $R$ denote the intersections of the hyperbola and the circle of radius 2$OP$ drawn about $P$. Prove that $\angle$ QPR $=120^o$.

We have $n$ marbles that look all alike, but we know that one of them is radioactive. We want to find that marble. We also have a detector that can determine whether any chosen subset of the marbles contains the radioactive marble or not. We want to choose a measurement strategy for which the expected number of measurements is a minimum. Find that smallest expected value possible.

In the hexagon ABCDEF we have $AB$=$BC$, $CD$=$DE$ and $EF$=$FA$. Prove that $\frac{AB}{BE}+\frac{CD}{DA}+\frac{EF}{FC}\ge\frac32.$

Given are seven real numbers in such a way that the sum of any three of them is less than the sum of the remaining four. Show that each number is positive.

Are there three cubes whose edges are integers and the sum of their volumes is equal to $10^{20}$+30?

I have some coins in my pocket. One of them is a 1 penny coin and a few more (there are at least four of them) are 2 penny coins. I start drawing them randomly, one by one without replacement, and stop if the sum is a positive multiple of 3. How many drawings are necessary for that on the average?

The lengths of the sides of a cyclic quadrilateral are $a$, $b$, $c$, $d$, respectively, its perimeter is 2$s$ and the measure of the angle of the sides $a$ and $b$ is $\alpha$. Prove that $\tan^2\frac{\alpha}{2}=\frac{(s-a)(s-b)}{(s-c)(s-d)}.$

Given are two disjoint circular discs in the plane. There are also given two diameters, respectively, which, when extended are touching the other circle. Prove that the endpoints of these diameters form a trapezium.

The degree of each vertex of a graph is equal to 3 and it is also given that the graph does not contain cycles whose length is less than 6. Find the lowest possible number of vertices in such a graph.

The real numbers $x$ and $y$ are both in the interval [0;12] and satisfy $xy$=(12-$x$)$^2$(12-$y$)$^2$. Find the maximum value of the product $xy$.

The triangle ABC is isosceles, namely $AB$ = $BC$. The points $C_1$, $A_1$, $B_1$ lie on the sides $AB$, $BC$, $CA$, respectively, and such that $\angle BC_1 A_1$=$\angle CA_1 B_1$=$\angle$ CAB . The lines $BB_1$ and $CC_1$ meet at $P$. Prove that the quadrilateral $AB_1 PC_1$ is cyclic.

Denote the greatest odd divisor of the positive integer $n$ by $k$($n$). Let $A$($n$)=$k$(1)+$k$(2)+...+$k$($n$), $B$($n$)=1+2+...+$n$. Prove that there are infinitely many values of $n$ such that 3 $A$($n$) = 2$B$($n$).

Find the radius of the sphere inscribed into a four sided pyramid whose edges are 1 unit long each.

Find all positive integers $n$, such that $a$) $7^{399}$|$n$! , but $7^{400}\nmid n!\,$, and $b$) $7^{400}$|$n$! , but $7^{401}\nmid n!\,$.

Prove that there is exactly one single point $M$ inside an arbitrary triangle ABC , such that $MA$+$BC$=$MB$+$AC$ =$MC$+$AB$.

$n$ and $k$ are positive integers, such that 2$kn$|$k^2$+$n^2$-$k$. Prove that $k$ is a square number.

A plane intersects each line segment of a closed polygon $P_1 P_2$...$P_n P_1$ of the space. The intersection with the line segment $P_i P_{i+1}$ is the interior point $Q_i$. Prove that $\frac{P_1Q_1}{Q_1P_2}\cdot\frac{P_2Q_2}{Q_2P_3}\cdot\ldots\cdot \frac{P_{n-1}Q_{n-1}}{Q_{n-1}P_n}\cdot\frac{P_nQ_n}{Q_nP_1}=1.$

The centre of the inscribed circle of the triangle ABC is $O$. The extensions of the line segments $AO$, $BO$, $CO$ beyond $O$ intersect the circumscribed circle at $A_1$, $B_1$, $C_1$, respectively. Prove that the area of the triangle $A_1 B_1 C_1$ is $\frac{R^2}{2}(\sin\alpha+\sin\beta+\sin\gamma),$ where $R$ is the radius of the circumscribed circle, and $\alpha$, $\beta$, $\gamma$ are the angles of the original triangle ABC .

The sequence $a_n$ is defined as follows: $a_1 \ne$2, $a_2=\frac{a_1}{a_1-2}$, $a_{n+1}=\frac{na_n^2}{na_n^2-(n+1)a_n+(n+1)}$, if $n \ge$2. Prove that the inequality $a_1$+2$a_2$+3$a_3$+...+$na_n$=$a_1 ^. a_2 ^. a_3 ^. ...^. a_n$ is true for all $n$.

The length of all three edges from a certain vertex of a tetrahedron is unity, and they pairwise enclose 45 angles. Find the volume of the tetrahedron. (based on the idea of

Show that if $A_1 B_1$, $A_2 B_2$ and $A_3 B_3$ are three parallel chords of a circle, then the perpendiculars dropped from the point $A_1$, $A_2$ and $A_3$ onto the line $B_2 B_3$, $B_3 B_1$ and $B_1 B_2$, respectively, are concurrent.

Solve the following equation: ($5^x -2^{x-2}$)$^2$+ 2lg ($5^x +2^{x-2}$)=$x$.

$a$, $b$, $c$ are positive numbers, such that $a^2$+$b^2$+$c^2$=1. Find the smallest possible value of the sum $S=\frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b}.$

Is it possible to obtain the array (1,3,8) starting from (2,5,13) by repeating the following steps: It is allowed to swap any two numbers in an array of three numbers ($a$,$b$,$c$), or replace the array by ($a$,$b$,2$a$+2$b$-$c$).

Let $x$ and $y$ denote integers, such that 4$x$ + 5$y$ =7. Find the smallest value of 5|$x$|-3|$y$|.

Each vertex of a tetrahedron is reflected in the centroid of the opposite face. Show that the volume of the tetrahedron formed by the reflections is at least four times that of the original tetrahedron.

$n$ is a positive integer, such that 2$n$+1 and 3$n$+1 are both square numbers. Prove that $n$ is divisible by 40.

What kind of quadrilateral is the base of the truncated pyramid if every pair of space diagonals of the solid intersect each other?

Is it true that if the product of the sides of a quadrilateral equals the square of its area then the quadrilateral has at least two right angles?

Let $a_1$, $a_2$, ..., $a_n$ denote positive numbers. Prove that $\frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+\ldots+\frac{a_{n-1}^2}{a_{n-1}+a_n}+ \frac{a_n^2}{a_n+a_1}\ge\frac{1}{2}(a_1+a_2+\ldots+a_n).$

Consider the centres of the squares ABDE and BCGH drawn inside above the sides $AB$ and $BC$ of a triangle ABC , and consider the midpoints of the line segments $AC$ and $DH$. What is the shape formed by these four points?

There is a square of side 2005 units long drawn on squared paper. The sides of the square are lattice lines. Draw a closed non self-crossing polygon inside the square, such that each side of the polygon is drawn along a lattice line, and it passes through every lattice point inside or on the boundary of the square exactly once. Show that the area bounded by the polygon is more than half the area of the square.

$P$ is an arbitrary point of the hypotenuse $AB$ of a right-angled triangle ABC . The foot of the altitude drawn from vertex $C$ is $C_1$. The projection of $P$ onto the leg $AC$ is $A_1$, and its projection onto the leg $BC$ is $B_1$. $a$) Prove that the points $P$, $A_1$, $C$, $B_1$, $C_1$ lie on a circle. $b$) Prove that the triangles $A_1 B_1 C_1$ and ABC are similar.

Given the points $A$, $B$, $C$ and $D$ on the plane, construct a circle passing through $A$ and $B$, such that the tangents drawn to it from $C$ and $D$ are equal.

What is the largest value of the whole number $n$, such that with the help of two appropriately chosen weights and an equal-arm balance it is possible to determine the mass of every object that weighs a whole number of kilograms from 1 to $n$? (It is allowed to carry out as many measurements as needed, only using the two weights and the object to be weighed, and neither the weights nor the objects to be weighted are allowed to be cut in parts.)

This spring there were three Hungarian teams among the best eight in the EHF women's handball championship. When the eight teams were paired randomly, each Hungarian team found that they were playing an opponent from a foreign country. What was the probability of this?

Represent the values of the number pair $p$, $q$ on the coordinate plane, such that the equation $x^2$-2$px$+$q$=0 $a$) has two roots; $b$) is satisfied by the number 2; $c$) is satisfied by the single number 2.

Let $P$ and $Q$, respectively, denote the centres of the squares ABDE and BCGH drawn on the sides $AB$ and $BC$ of the triangle ABC outwards. The midpoints of the sides $AC$ and $DH$ are $R$ and $S$, respectively. Show that the points $P$, $Q$, $R$ and $S$are the vertices of a square. ( 4 points )

The binary form of the positive integer $A$ consists of $n$ digits of 1. Prove that the sum of the digits of the number $nA$ in binary notation is $n$.

The bisector of the angle $A$ of triangle ABC intersects the side $BC$ at $D$. The lines $b$ and $c$ drawn through the points $B$ and $C$, respectively, are parallel to each other and equidistant from $A$. Let $M$ and $N$ denote the points of the lines $b$and $c$, such that $AB$ bisects the line segment $DM$ and $AC$ bisects the line segment $DN$. Prove that $DM$=$DN$.

It is known that the planes of the faces of a tetrahedron divide the space into 15 parts. What is the largest possible number of these parts that a line may pass through?

The problem below appears in the article ``Szalonpóker'' (Hungarian title) of the current issue. Suppose that a deck of 52 cards is suffled in the same way. How many times does it need to be shuffled in order to get back the initial order. Solve the problem for that case, too, when the shuffling starts with the bottom card of the deck on the right, that is, when the card in the 26th place originally is put at the bottom of the new deck. (Little John and Old Firehand drop in the famous casino of Black Jacky to play cards. They play with a deck of 32 cards numbered 1 to 32. Before they agree on the rules, Black Jacky shuffles the cards as follows: He places the deck on the table, removes the top 16 cards and puts them on the table, to the right of the remaining deck without turning over. Then he forms one deck of them, with the bottom card of the deck on the left lying at the bottom, followed by alternating cards from the two decks. Then he repeats the procedure several times with the deck obtained in this way. Little John is sure that this way of shuffling the cards is not fair. Show that repeating the procedure several times will lead to a surprising result.)

Given are five 15-litre vessels, containing 1, 2, 3, 4, and 5 litres of water, respectively. We are allowed to double the amount of water in any chosen vessel from another one. What is the largest possible amount of water that can be collected in a vessel by repeating this step?

Is it possible to select four distinct points on the plane, such that through every pair of points there is a circle that has equal tangents from the other two points?

The legs of a right-angled triangle are $a$ and $b$, its hypotenuse is $c$. The radius of the escribed circle drawn to the leg $b$ (i.e. the one touching that leg on the outside and also touching the extensions of the other two sides) is $\varrho_b$. Prove that $b$+$c$=$a$+2$\varrho_b$.

Construct the triangle with ruler and compass ABC , given the lengths of the sides $AB$ and $AC$, and the distance of the vertex $A$ from the point that divides the side $BC$ in the ratio 2:1.

Given an angle and a point in its interior, construct the line through the point that encloses a triangle of minimum area with the arms of the angle.

A convex 100-sided polygon $K$ lies in the interior of a square whose side is 1 metre. Show that it is possible to select three vertices of $K$, such that they form a triangle of smaller area than 8 $cm^2$.

A fair coin is tossed until both outcomes, i.e. heads and tails are obtained at least once. What is the expected value of the number of tosses needed?

Consider the areas and perimeters of the rectangular sections of a regular tetrahedron of unit edge. In what intervals do they vary?

Consider the functions obtained as compositions of the functions $f\colon x \mapsto \frac{1}{x}$ , $g\colon x \mapsto 1-\frac{1}{x}$ , each used a finite number of times; for example $f\circ g\circ f\colon x \overset{f}\mapsto \frac{1}{x} \overset{g}\mapsto 1- \dfrac{1}{\frac{1}{x}} = 1- x \overset{f}\mapsto \frac{1}{1-x}$ . Is it possible to obtain the function $x \mapsto x$+1 in this way? (It is allowed to cancel finitely many numbers from its domain.)

$a$, $b$ are integers, such that $2005 \mid a^3+b^3$ and $2005 \mid a^4+b^4$. Prove that $2005 \mid a^5+b^5$.

Construct by ruler and compass a line of a given direction that halves the area of a given triangle.

Is there a triangle, such that the triangle formed by its medians is similar to the triangle?

What is the smallest possible number of acute-angled triangles that can be put together without gaps or overlaps to form an isosceles triangle with an apex angle of $120^o$?

Show that if $a^2$+$b^2$=$a^2 b^2$, and |$a$|$\ne$1, |$b$|$\ne$1, then $\frac{a^7}{{(1-a)}^2}- \frac{a^7}{{(1+a)}^2}= \frac{b^7}{{(1-b)}^2}- \frac{b^7}{{(1+b)}^2}.$

One base and the two legs of a trapezium each have unit length. What should be the length of the other base so that the trapezium has the largest possible area?

There are traffic lights at eight junctions along a main road. The probability that a certain light is red when one arrives there is 0.4. What is the probability that a car travelling down the road is not stopped by red lights at two consecutive junctions?

Find all natural numbers $n$ greater than 1, for which there is an order $a_1$,$a_2$,...,$a_n$ of the numbers 1,2,3,...,$n$, such that the products $a_1$, $a_1 a_2$, $a_1 a_2 a_3$, ..., $a_1 a_2 ^{...} a_n$ divided by $n$all leave different remainders.

The length'' of an elliptical tabletop is 160 cm and its width'' is 1 m. Is it possible to cover the table completely with a 140 cm×130 cm tablecloth?

In how many different ways is it possible to place 32 knights on a 8×8 chessboard, such that no pair of them should attack each other?

Show that the number $1^{2006}-2^{2006}+3^{2006}-4^{2006}+\ldots-2004^{2006}+2005^{2006}$ is divisible by 1003. Is it also divisible by 2006?

ABCD is a cyclic quadrilateral, and $AC$=$BC$. The perpendicular projections of the point $C$ onto the lines $AD$ and $BD$ are $E$ and $F$, respectively. Show that $AE$=$BF$.

The triangles $AB_1 C_1$ and $AB_2 C_2$ are similar, and their vertices are labelled in the same direction around the triangle. The lines $B_1 B_2$ and $C_1 C_2$ intersect at the point $D$. ($a$) Show that the triangles $AB_1 B_2$ and $AC_1 C_2$ are similar. ($b$) Show that the circumscribed circles of $AB_1 C_1$ and $AB_2 C_2$ pass through the point $D$.

For which point $P$ of an arc $AB$ of a circle does the triangle ABP have the longest perimeter?

Let $n$ denote a positive integer, such that $4^n +2^n$+1 is a prime number. Prove that $n$ is an integer power of 3.

It is well known that cos 2$x$ can be expressed as a polynomial of cos $x$: cos $2$x$=2cos^2 x$-1. Is it possible to express sin 2$x$ as a polynomial of sin $x$?

Point $M$ lies on the angle bisector drawn from vertex $A$, inside an acute-angled triangle ABC . The other intersections of the lines $AM$, $BM$, $CM$ with the circumscribed circle are $A_1$, $B_1$ and $C_1$, respectively. The lines $AB$ and $C_1 A_1$ intersect at $L$, and the lines $AC$ and $B_1 A_1$ intersect at $N$. Prove that $LN$ is parallel to $BC$.

$P$ is a point lying on an ellipse of foci $F_1$ and $F_2$, different from the endpoints of the major axis. Show that the value of $\tan \frac{\angle PF_1F_2}{2}\cdot \tan \frac{\angle PF_2F_1}{2}$ is independent of $P$.

Solve the equation $x^2 -$x$-1=2^x -log_2$($x^2 +2^x$) on the set of positive numbers.

Find those positive integers that are 14 times larger than the sum of their decimal digits.

There is a rectangle inscribed in the pythagorean triangle whose sides are 3, 4, 5 units long, respectively. The vertices of the rectangle are incident to the sides of the triangle and the ratio of its perpendicular sides is 1:3. Determine the dimensions of the rectangle.

Solve the simultaneous equations $\sqrt{7x+y}+\sqrt{x+y}=6$ $\sqrt{x+y}-y+x=2.$

The first term of a geometric progression is 6, the sum of the first $n$ terms is $\frac{45}{4}$ and the sum of the reciprocals of the same terms is equal to $\frac{5}{2}$. Find this geometric progression.

Find all sets of three integers, such that their product is four times their sum, and one number is twice the sum of the other two.

Find all positive integers divisible by 7 that end in 5 in the decimal notation and the rest of their digits are 1.

The lengths of two adjacent sides of a quadrilateral are 2 units and 1 unit, and they enclose an angle of $60^o$. The quadrilateral is cyclic and it is also a tangent quadrilateral. What are the lengths of the other two sides?

Solve the equation {3$x$}$^2$+{$x$}$^2$=1.

Find those three-digit numbers divisible by 45 in which the digits (in that order) form an arithmetic progression.

A steam locomotive is going to Kanizsa from Zalakomár that is 21 km away. The journey takes 16 minutes. The locomotive accelerates uniformly to 90 km/h, then continues at constant speed, and then decelerates uniformly to rest. How long is it travelling at constant speed?

One side of a rectangle is 10 cm long. How long is the other side if a 10 cm x1 cm rectangle can be just placed in it in a diagonal position?

Solve the following simultaneous system with a $t$ real parameter: $x$+$y$+$z$=$t$, $x$+($t$+1)$y$+$z$=0, $x$+$y$-($t$+1)$z$=2$t$.

The product of the real numbers $a$ and $b$ is 1, and $\frac{a+b+2}{4}=\frac{1}{a+1}+\frac{1}{b+1}.$ Find the values of $a$ and $b$.

The price of Aunt Margaret's favourite chocolate has gone up by 30%, while her pension is only raised by 15%. By what percentage will Aunt Margaret's chocolate consumption decrease, if she can only spend 15% more on chocolate.

Having calculated that $6^2$+8=44, Clare noticed that $66^2$+88=4444 was also true. Is it true for every $n$ that $(\underbrace{6\dots6}_{n\text{ digits}}{)}^2+ \underbrace{8\dots8}_{n\text{ digits}}=\underbrace{4\dots4}_{2n\text {digits}}$?

A square tablecloth is layed on a round table so that their centres coincide. The areas of the circle and the square are equal. What percentage of the area of the tabletop is covered by the cloth?

A fly is sitting at the centre $K$ of a regular hexagon ABCDEF , there is another fly at vertex $B$ and a spider sitting at vertex $A$. The flies start clawling from $B$ towards $C$ and from $K$ towards $E$ simultaneously at the same speed. (The spider stays in placed.) Show that the three of them form a regular triangle at every time instant.

Let 0$\le a$,$b$,$c$,$d \le$2005 be whole numbers. What is the probability that the number $ab$+$cd$is even?

A unit square ABCD is rotated about its vertex $C$ through $90^o$. Determine the area swept out by the side $AB$.

Let $x$ and $y$ denote non-negative real numbers. Prove that $\sqrt{\frac{x}2}+\sqrt{\frac y2}\le \sqrt{x+y}$ .

On its second birthday, a department store introduced sales prices. Those buying merchandise for more than 50 000 forints (HUF) enjoyed two successive discounts. Both percentage reductions were whole numbers less than 10. A television set that originally cost 69 000 forints was sold for 60 306 forints. Calculate the percentages.

Consider the truncated cone determined by the circumscribed circle of the base of a cube, and the inscribed circle of the top face of the cube. Find the ratio of the volume of the truncated cone to that of the cube.

Prove that the product of four consecutive integers can always be expressed as the product of two consecutive even numbers.

Solve the following equation: $\frac{3x^3}{x^3-1}- \frac{x}{x-1}=2.$

You are sitting in an aeroplane. Looking through a window 20 cm from your eye, you can see two boats in the direction of each of the lower corners of the 25 cm×40 cm window, respectively. Given that the plane is flying at an altitude of 10.3 km, and your eye is level with the horizontal bimedian of the window, determine the distance between the two boats.

Find the area of the triangle bounded by the lines $x+y=2005, \frac{x}{2005}+ \frac{y}{2006}=1, \frac{x}{2006}+ \frac{y}{2005}=1$

The five numbers drawn in the lottery of 10 September, 2005 were as follows: 4, 16, 22, 48, 88. All the five numbers are even, exactly four of them are divisible by four, three are divisible by 8 and two by 16. In how many different ways is it possible to select five numbers out of the whole numbers 1 to 90?

Lord Moneybag said to his grandson, ``Listen, Bill. Christmas is coming up. I have got some money between 300 and 500 pounds on me. It is an integer multiple of 6 pounds. I will give you 5 pounds out of it in one-pound coins. As I hand you the coins one by one, the amount that remains on me will first be divisible by 5, then by 4, then by 3, then by 2 and finally only by 1. If you can tell me how many pounds I have got on me, you'll get an extra ten-pound note.'' How much money did the lord have on him?

Find the sum of all three-digit numbers whose digits are all odd.

Given are the points $A$(2,1), $B$(3,4), $C$(2,11) on the cartesian plane, show that the ray $OB$ bisects the angle AOC .

The base edges and the height of a square-based right pyramid are all 40 cm long. We want to connect one vertex of the base to the opposite vertex of the base with a line drawn on the lateral faces. What is the length of the shortest path?

Solve the following equation: $\frac{1}{\sin x}-\frac{1}{\sin2x}=\frac{2}{\sin4x}.$

At the confectioner's, we bought 5 pieces of candy in a paper bag, and 10 pieces in another, identical bag. (Each piece of candy has the same mass.) The first bag was weighed to be 85 grams and the second one 165 grams. How much did we pay for a paper bag if the candy cost 1200 forints (HUF) a kilo.

An octagon is inscribed in a circle. Determine the sum of four interior angles, no pair of which are adjacent.

In a jeep there are 3 seats in front and 3 in the back. In how many different ways can six persons of different heights be seated in the jeep, so that everyone in front is shorter than the person directly behind. (Each of them is allowed to sit in the driver's seat.)

The half of the square of an integer and one sixth of the cube of the same integer are added to one third of the same integer. Prove that the result is an integer.

There are two tribes living in Binumeria. The two tribes use numbers expressed in different notations, but the base of each notation is smaller than ten. We asked one man from each tribe how many people there were in the tribe. They both answered 10011, and one of them added that the other tribe had 10 more members and that 1+2=3. How many people live in Binumeria?

Two students are talking after class. ``What is the average of your computer science marks in September? -- 4.6 exactly. -- That is impossible. The school year has just started, you cannot have so many marks yet.'' What may the student in doubt have thought of? (The grades in Hungary are varying from 1 to 5.) (Based on the idea of

In how many different ways is it possible to place a king and a castle on the chessboard so that neither attacks the other? (The fields of the chessboard are labelled by combinations of letters and numbers in the conventional way. Two configurations count as different if at least one of the two chessmen is placed on different fields in the two cases.)

The values of a gold artefact in Aurum is proportional to the square of its mass. A 100-dollar piece is stolen by thieves. The thieves cut it up into smaller pieces of equal mass to prepare pendants of them, whose total value is 10 dollars. The pendants are bought up by a jeweller who assembles them into bracelets (of not necessarily the same mass). How much are the individual bracelets worth, given that each bracelet is made of a whole number of pendants, and their total value is 46 dollars?

There are two kinds of people living on an island: the good and the bad. The good always tell the truth and the bad always lie. Naturally, every inhabitant of the island is either a boy or a girl. Here is a conversation of two young people on the island: $A$: If I am good, $B$ is bad.'' $B$: If I am a boy, $A$ is a girl.'' Find out if each of them is good or bad and what sex they belong to.

Bob has three stencils for drawing circles. The areas of the circles are 6, 15 and 83 $cm^2$. Bob wants to draw a few circles that have a total area of 220 $cm^2$. How many of each should he draw? (Suggested by D. Szilágyi, Budapest)

15 teams participated in a football tournament. Every team played each of the other teams once. 3 points were awarded for winning, 2 for a draw and 1 point for losing the game. At the end of the tournament, every team had a different number of points, 21 being the lowest score. Prove that the team with the highest score has played at least one draw.

In the quadrilateral ABCD , $AB$=$AC$=$DB$, and the diagonals are perpendicular. Find the sum of the angles ACB and ADB .

How many number sets of at least two elements are there, such that the elements are consecutive positive integers and their sum is 100?

How many six-digit numbers of the form ababab can be obtained by multiplying six different primes?

A printing press is working on a board game that uses two kinds of figures moving on the board: dogs and cats. The dog and cat figures are to be cut out of cardboard. The cat to dog ratio in the set of figures packed in the box with each game is 5:3. To reduce wastes, the figures are printed on large sheets of cardboard in two different arrangements. One type of sheet contains 28 dogs and 28 cats, and the other one contains 42 cats. In what proportion should the two types of sheets printed in order to obtain them in the right proportion after cutting out?

There are two milk bars in Cowton, and both of them have hot chocolate with frothed chocolate topping on their menus. Each milk bar serves in a cylindrical glass of the same height. (On serving, one half of the volume of the drink is liquid chocolate, and the other half is frothed chocolate.) In a short time, the froth turns into liquid chocolate of one quarter as much volume. In the Jolly Cowboy, chocolate is served in glasses of radius 6 cm, and sold for 12 Cowton cents a glass. In the Happy Cowgirl, the radius of the glasses is 5 cm, but they fill up the glass again when the froth of the first filling has settled. They charge 11 Cowton cents for a glass. In which milk bar is chocolate cheaper? [To obtain the volume of a cylinder, the area of its base is multiplied by its height.] (Based on a problem of the 24th József Öveges Memorial Competition )

Eskimos greet each other by rubbing their noses together, and they greet the members of an arctic expedition by a tap on the hand. The members of the expedition say hello to each other as well as to the Eskimos. At a certain event, ``Hello'' was said 415 times. How many nose rubbings and how many taps on the hand were there? (Everyone greeted everyone else in the appropriate way.)

How many factors of 857 304 000 are perfect squares?

In the triangle ABC , $X$ is the midpoint of side $AB$ and $Y$ is the midpoint of side $AC$. $BY$ and $CX$ intersect at $S$. Show that $a$) the areas of the triangles SBX and SCY are equal; $b$) the areas of triangle SBC and quadrilateral AXSY are equal.

Express the following sum as a ratio of two integers: $\frac{4}{1989\cdot 1993}+ \frac{4}{1993\cdot 1997}+ \frac{4}{1997\cdot 2001}+ \frac{4}{2001\cdot 2005}.$

$x$ is a real number, such that $x+\frac{1}{x}=5$. Determine the exact values of $x^2+\frac{1}{x^2}$ and $x^3+\frac{1}{x^3}$.

There are only 5 candies left on the Christmas tree: two of them filled with jelly and the other three filled with almond paste. The wrapping of the candies does not show the flavour. Hanga decided to eat one of them every day. Which is more probable: that she will eat the second jelly-filled candy on the third day, or that she will eat the third almond paste candy on the fourth day?