KöMaL 2009

In the triangle ABC , denote by $A_1$, $B_1$, and $C_1$ the feet of altitudes. Let $P$ be the perpendicular foot $C_1$ on line $A_1 B_1$, and let $Q$ be that point of line $A_1 B_1$ for which $AQ$=$BQ$. Show that $\angle$ PAQ =$\angle$ PBQ =$\angle PC_1 C$.

Prove that if each vertex of a simple graph is of degree at least 3 then the graph contains a cycle with length not divisible by 3.

Call a finite sequence ( $p_1$($x$),...,$p_k$($x$) ) of polynomials with integer coefficients Euclidean if there exist polynomials $q_1(x),\ldots,q_k(x)$ with integer coefficients such that $d$($x$)=$q_1$($x$)$p_1$($x$)+...+$q_k$($x$)$p_k$($x$) is a common divisor of $p_1$($x$),...,$p_k$($x$), i.e. there are polynomials $r_1(x),\ldots,r_k(x)$ with integer coefficients for which $p_i$($x$)=$r_i$($x$)$d$($x$) for every 1$\le i \le k$. Prove that whenever $p_1(x),\ldots,p_n(x)$ are polynomials with integer coefficients and every two of them form a Euclidean pair, then the sequence $\big(p_1(x),\ldots,p_n(x)\big)$ is Euclidean as well.

Show that the spherical triangles with equal area can be dissected into each other; in other words, any two spherical triangles, having the same area, can be decomposed into spherical polygons such that the two decompositions consist of the same number of pieces and the respective pieces are congruent.

The point $Q$ is located in the convex $n$-gon $P_1P_2\ldots P_n$. Show that $\sum_{k=1}^n \big(\cot QP_{k-1}P_k\sphericalangle + \cot QP_{k+1}P_k\sphericalangle\big) \cdot \overrightarrow{QP_k} = 0.$ (The indices are considered modulo $n$, e.g. $P_0$=$P_n$ and $P_{n+1}$=$P_1$.)

To each vertex of a regular $n$-gon a real number is assigned in such a way that the sum of all $n$ numbers is positive. If three consecutive vertices are assigned the numbers $x$, $y$, $z$ respectively and $y$<0 then the following operation is allowed: the numbers $x$, $y$, $z$are replaced by $x$+$y$, -$y$, $z$+$y$ respectively. Such an operation is performed repeatedly as long as at least one of the $n$ numbers is negative. Determine whether this procedure necessarily terminates after a finite number of steps.

Let $n \ge$3 be an odd integer, and let $A$={0,1,...,$n$-1} denote the set of residual classes modulo $n$. Call a non-empty subset $B \subset A$ a Dutch set, if for every $a \in A$ and for every $b \in B$ at least one of $b$+$a$ and $b$-$a$ lies in $B$. Determine the smallest possible cardinality of a Dutch set in terms of $n$.

Let $G$ be the complete graf with 2$n$ vertices, and suppose that $S_1$,...,$S_k$ are subgraphs of $G$ with the following properties: ($a$) Every $S_i$ is a complete bipartite graph; ($b$) Every edge of $G$ is contained by an odd number of subgraphs $S_i$. Show that $k \ge n$.

If $a_1$,$a_2$,...,$a_n$ are positive numbers with $a_1$+$a_2$+...+$a_n$=1, prove $a_1\cdot a_2^{2/3}+a_2\cdot a_3^{2/3}+\dots+a_{n-1}\cdot a_n^{2/3}<\frac37$.

Decide whether there exists a positive integer $n$ that is divisible by 103 and satisfies $2^{2n+1}\equiv2\pmod{n}$ .

Let $p$($z$) be a complex polynomial of degree $n$, and suppose that all (complex) roots of $p$($z$) are of unit modulus. For every real number $c \ge$0, show that the roots of the polynomial 2$z$($z$-1)$p$'($z$)+ ( ($c$-$n$)$z$+($c$+$n$) ) $p$($z$) are of unit modulus as well.

Prove that there are infinitely many $n$, for which there exist simple graphs $S_1$,...,$S_n$ with the following properties: ($a$) each $S_i$ is a complete bipartite graph; ($b$) the union of the graphs $S_1$,...,$S_n$ is a complete graph on 2$n$ vertices; ($c$) each edge of this complete graph is contained in an odd number of the graphs $S_i$.

Let $n$ be a positive integer. Prove that $\sum_{k=-n}^n {(-1)}^k \left(\cos\frac{k\pi}{2n+1}\right)^{2n+1} > 0$.

For abritrary integers 0<$k \le n$ and $a$>1, define ${\binom{n}{k}}_{a} = \frac{(a^n-1)(a^{n-1}-1)\cdot \ldots \cdot(a^{n-k+1}-1)}{(a^k-1)(a^{k-1}-1) \cdot \ldots \cdot(a-1)}$. ($a$) Show that ${\binom{n}{k}}_{a}$ is an integer. ($b$) Do there exist integers 0<$k$<$n$<$m$and $a$>1, for which ${\binom{m}{1}}_{a}$ divides ${\binom{n}{k}}_{a}$?

In the Cartesian plane, a convex polygonal disk $K$ is given. Apply the perpendicular affinity ($x$,$y$)$\mapsto$($x$,$py$) to the points of $K$, where 0<$p$<1. Denote the resulting polygonal disk by $K$'. Prove that $K$' can be covered by a polygonal disk which is congruent to $K$.

Let ABCD be a tetrahedron with circumcenter $O$. Suppose that the points $P$, $Q$ and $R$ are interior points of the edges $AB$, $AC$ and $AD$, respectively. Let $K$, $L$, $M$ and $N$ be the centroids of the triangles PQD , PRC , QRB and PQR , respectively. Prove that if the plane PQR is tangent to the sphere KLMN then $OP$=$OQ$=$OR$.

Denote by $\nu$($n$) the exponent of 2 in the prime factorization of $n$!. Show that for arbitrary positive integers $a$ and $m$ there exists an integer $n$>1 for which $\nu(n) \equiv a \pmod{m}$ .

Let $x$, $y$, $z$ be positive numbers satisfying xyz $\ge$1. Prove that $\frac{x}{x^3+y^2+z}+ \frac{y}{y^3+z^2+x}+ \frac{z}{z^3+x^2+y}\le 1$.

Let $P_1 P_2 P_3$ be a triangle with circumcenter $O$, the point $Q$ is in the triangle. Denote $t_i$ and $O_i$ the area and the circumcenter of the triangle $QP_{i+1} P_{i+2}$, respectively, where $i$=1,2,3 (the vertices are counted cyclically: $P_4$=$P_1$ and $P_5$=$P_2$). Prove that $t_1\cdot \overrightarrow{OO_1} + t_2\cdot \overrightarrow{OO_2} + t_3\cdot \overrightarrow{OO_3} = 0$.

Does there exist a non-constant polynomial which always takes non-square integers at integer values of the variable?

The two base faces of a prism are equilateral triangles and the other three faces are squares. At the beginning it stands on its triangle face. Then it is rolled around one of its edges that lays on the table. After some rollings, the prism will stand in the original position. Prove that then all vertices will be in the same position as at the beginning. (Suggested by L. Csirmaz, Budapest)

In the triangle $A_1A_2A_3$, for each $i=1,2,3$, the excircle, which is tangent to the side $A_{i+1}A_{i+2}$, touches the half lines $A_iA_{i+1}$ and $A_iA_{i+2}$ at $P_i$ and $Q_i$, respectively. (The indices are considered modulo 3, e.g. $A_4=A_1$ and $A_5=A_2$.) The lines $P_iP_{i+1}$ and $Q_iQ_{i+2}$ meet at $R_i$; finally, the lines $P_{i+1}P_{i+2}$ and $Q_{i+1}Q_{i+2}$ meet at $S_i$ ($i=1,2,3$). Prove that the lines $R_1S_1$, $R_2S_2$ and $R_3S_3$ are concurrent.

For every finite, nonempty set $H$ of positive integers, denote by $\mathrm{gcd}(H)$ the greatest common divisor of the elements in $H$. Show that if $A$ is a finite, nonempty set of positive integers then the following inequality holds: $\sum_{\emptyset\ne H\subseteq A} {(-2)}^{|H|-1} \mathrm{gcd}(H) > 0.$

Prove that $\sum_{i=1}^n\ \sum_{j=1}^n \frac{a_ia_j}{{(p_i+p_j)}^c}\ge 0$ holds for arbitrary reals $a_1,a_2,\ldots,a_n$, and positive numbers $c,p_1,p_2,\ldots,p_n$.

Let $p_1$,...,$p_k$ be prime numbers, and let $S$ be the set of those integers whose all prime divisors are among $p_1$,...,$p_k$. For a finite subset $A$ of the integers let us denote by $\mathcal{G}(A)$ the graph whose vertices are the elements of $A$, and the edges are those pairs $a$,$b \in A$ for which $a$-$b \in S$. Does there exist for all $m \ge$3 an $m$-element subset $A$ of the integers such that ($a$) $\mathcal{G}(A)$ is complete? ($b$) $\mathcal{G}(A)$ is connected, but all vertices have degree at most 2?

In the acute triangle ABC we have $\angle$ BAC =$\alpha$. The point $D$ lies in the interior of the triangle, on the bisector of $\angle$ BAC , and points $E$ and $F$ lie on the sides $AB$and $BC$, respectively, such that $\angle$ BDC =2$\alpha$, $\angle$ AED $=90^o$+$\frac{\alpha}{2}$, and $\angle$ BEF =$\angle$ EBD . Determine the ratio $BF$:$FC$.

Let $a_1$,$a_2$,...,$a_{2k}$ be distinct integers and let $M$ be a set of $k$ integers not containing 0 and $s$=$a_1$+$a_2$+...+$a_{2k}$. A grasshopper is to jump along the real axis, starting at the point 0 and making 2$k$ jumps with lengths $a_1$,$a_2$,...,$a_{2k}$ in some order. If $a_i$>0 then the grasshopper jumps to the right; while if $a_i$<0 then the grasshopper jumps to the left, to the point in the distance |$a_i$| in the respective steps. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in $M$.

Two adjacent vertices of a square lie on a unit circle. What are the smallest and largest possible distances of the other two vertices from the centre of the circle?

Prove the following inequality: 2($x^4$+$x^2 y^2$+$y^4$)$\ge$3$xy$($x^2$+$y^2$).

The longer base of a symmetrical trapezium ABCD is at most twice as long as the shorter base. $P$ is a point in the interior of the trapezium. Prove that there exists a quadrilateral whose vertices lie on the sides of the trapezium and whose sides are of lengths $AP$, $BP$, $CP$ and $DP$ in some order.

Show that the equation 5$x^2$+3$y^2$=1 has no solutions in the set of rational numbers.

Prove that any right angled triangle can be extended to a rectangle by attaching to its sides three right angled triangles which are similar to each other.

Solve the following system of equations: $x^3 y$+$xy^3$=10, $x^4$+$y^4$=17.

The orthogonal projections of a point onto the lines of the altitudes drawn from vertices $A$, $B$, $C$ of a triangle are denoted by $A_1$, $B_1$, $C_1$, respectively. Prove that there is exactly one point in the plane of the triangle for which the line segments $AA_1$, $BB_1$, $CC_1$ are equal, and then the length of these line segments is equal to the diameter of the inscribed circle of triangle ABC .

Two different digits of a number of at least 100 digits and not divisible by ten are interchanged, the resulting number has the same prime factors as the original number. Show an example for such a number.

Let $\alpha$=$\pi$/14. Evaluate sin $\alpha^.$sin 3$\alpha^.$sin 5$\alpha$.

Given $n$ different subsets of the set of numbers 1,2,3,...,$n$, prove that there exists a number, such that if it is left out of each subset, the remaining subsets will still be different.

The centre of the circumscribed circle of triangle ABC is $O$, and its orthocentre is $M$. Mark the points $E$ and $F$ on the lines of sides $AC$ and $AB$, respectively, at distances of $AO$ and $AM$ from $A$. Prove that $EF$=$AO$.

This problem is a classic: A town is surrounded by a circular wall. There are 12 guards serving on the wall. At twelve noon, each guard leaves his watchpost and starts walking the wall in some direction, at a speed at which it would take exactly one hour to walk around the whole town. If two guards meet, they both turn around immediately and walk at the same speed in the opposite direction. Prove that at twelve midnight each guard will be back at his watchpost.

Solve the following equation: $a^2 b^2$+$b^2 c^2$+$c^2 a^2$+$a^2$+$b^2$+$c^2$+4($a$+$b$+$c$)+12=6 abc +4($ab$+$bc$+$ca$).

Solve the following equation: $\tan x + \cot x + 1 = \cos \left(x +\frac{\pi}{4}\right).$

Solve the following equation: [$x$]=$x^4$-2$x^2$. [$x$] denotes the greatest integer that is not greater than $x$.

The midpoint of the line segment $AB$ on the face ABCD of a cube is $F$. The cube is cut into two parts with a plane passing through the line segment $CF$, such that the ratio of the volumes of the parts containing $B$ and $D$, respectively, is 1:2. Find the angle enclosed between the plane and the face ABCD .

In triangle ABC , $a$=2$b$. Given the vertices $A$, $B$ and a line passing through vertex $C$, construct the triangle.

Is there a bounded set of points in the plane that has infinitely many axes of symmetry but is not centrally symmetric?

Assume that a set $S$ obtained by leaving a finite number of positive elements out of the set of natural numbers is closed under addition. Let $k$ be an element of $S$. If $k$ is subtracted from each element of $S$, in how many cases will the result not belong to $S$?

60 bars of chocolate are distributed among a class of 30 students. Everyone gets at least one but no one gets 31 bars. Prove that it is possible to select a group of students who received a total of exactly 30 bars of chocolate together.

A cuboid is built out of 5×10×20-cm bricks without leaving any gaps between them. Prove that the same cuboid can also be built out of the same bricks with all edges of equal lengths being parallel.

Prove that if a triangle can be constructed from every three line segments selected out of a set of 5, then there exist three line segments in the set that form an acute-angled triangle.

The diagonals of a convex quadrilateral ABCD intersect at $O$. Prove that $AB^2$+$BC^2$+$CD^2$+$DA^2$=2($AO^2$+$BO^2$+$CO^2$+$DO^2$) is true if and only if the diagonals $AC$ and $BD$ are perpendicular or one of them has its midpoint at $O$.

Construct a point $D$ on the side $BC$ of triangle ABC , such that the inscribed circles of triangles ABD and ACD touch each other on the line $AD$.

For every positive integer $n$, let $f$($n$) denote the number obtained by reversing the order of digits in the decimal form of $n$. (For example, $f$(2500)=52, $f$(1456)=6541.) Find all positive integers $k$, such that for any multiple $n$ of $k$, $k$ also divides the number $f$($n$).

The centre of the inscribed circle of triangle ABC is $K$, the midpoint of side $AB$ is $F$, and the point of tangency of the escribed circle on side $AB$ is $G$. Prove that the lines $CG$ and $KF$ are parallel.

Prove that if $a$, $b$, $c$ are pairwise distinct positive integers then $S$=(42$a$+43$b$+43$c$)$^3$+(43$a$+42$b$+43$c$)$^3$+(43$a$+43$b$+42$c$)$^3$ -3(42$a$+43$b$+43$c$)(43$a$+42$b$+43$c$)(43$a$+43$b$+42$c$) is divisible by 128 but is not a perfect power of 2.

A plane intersects the edges $AB$, $BC$, $CD$ and $AD$ of a tetrahedron ABCD at points $K$, $L$, $M$ and $N$, respectively. Prove that $\frac{AK}{AB} \cdot \frac{BL}{BC} \cdot \frac{CM}{CD} \cdot \frac{DN}{AD} \le \frac{1}{16}$.

In every second, the probability that a certain bacterium will die is $p$ and the probability that it will divide into two identical bacteria is 1-$p$. (The offsprings will die or divide independently of each other.) What is the probability of extinction?

Let $n$ be a positive integer and let $k_1, k_2, k_3,\ldots,k_n$ denote an arbitrary order of the integers 1 to $n$. What is the largest possible value of the expression (1-$k_1$)$^2$+(2-$k_2$)$^2$+(3-$k_3$)$^2$+...+($n$-$k_n$)$^2$ of $n$ terms?

Determine those convex quadrilaterals ABCD that have an interior point $P$ for which the triangles ABP , BCP , CDP , DAP have equal areas.

Solve the following simultaneous equations on the set of real numbers: (1) 4$a$+$bc$=32, (2) 2$a$-2$c$-$b^2$=0, (3) $a$+12$b$-$c$-$ab$=6.

Let $A$, $B$, $C$, $D$ be any points in the plane. Prove that if the circles ABC and ABD intersect each other at right angles then the circles ACD and BCD also intersect each other at right angles.

Solve the following equation: (sin $x$+sin 2$x$+sin 3$x$)$^2$+(cos $x$+cos 2$x$+cos 3$x$)$^2$=1.

The tangents drawn to the circumscribed circle of triangle ABC at the points $B$ and $C$intersect each other at point $M$. The line drawn through $M$ parallel to $AB$ intersects line $AC$ at $N$. Prove that $AN$=$BN$.

Prove that for all positive integers $n$ and $k$, the greatest common divisor of the numbers $\binom{n}{k}, \binom{n+1}{k}, \ldots, \binom{n+k}{k}$ is 1.

The vertex $C$ of a parabola is the centre of a circle that passes through the focus $F$ of the parabola. Let the intersections of the parabola and the circle be $A$ and $B$, let $AB$and $CF$ intersect at $E$, and let $D$ denote the point of the circle that lies opposite to $F$. Show that $DE$ is the geometric mean of $FE$ and the diameter of the circle.

Prove that the sequence $a_n=\big[n \sqrt{2}\,\big]$ contains infinitely many integer powers of 3.

The opposite edges of a tetrahedron are equal in length and they pairwise enclose equal angles. Prove that the tetrahedron is regular.

The endpoints of the hypotenuse of a right-angled triangle of area $t$ are the foci of an ellipse, and the third vertex lies on the ellipse. Prove that the area of the ellipse is at least $\sqrt{2}\pi t$ .

Find those arithmetic progressions in which the sum of the first $n$ elements is a perfect square for every $n$.

The vertices of a cyclic quadrilateral divide the circumscribed circle into four arcs. The midpoints of the arcs are, in this order, $F_1$, $F_2$, $F_3$ and $F_4$. Show that the line segment $F_1 F_3$ is perpendicular to $F_2 F_4$.

Show that every non-zero polynomial can be multiplied by an appropriate non-zero polynomial, such that the exponent in each term of the product polynomial is divisible by 3.

In an acute-angled triangle ABC , circles are drawn around the altitudes as diameters. In each circle, a chord is drawn that is perpendicular to the appropriate altitude and passes through the orthocentre of the triangle. Prove that the three chords obtained in this way are equal in length.

Let $a$, $k$ and $n$ be positive integers. Find the greatest common divisor of the numbers $a^n$+1 and $a^k$+1.

Sides $AB$ and $CD$ of a quadrilateral ABCD are not parallel. Let $E$ and $F$, respectively, be interior points of these sides. Prove that the midpoints of the line segments $AF$, $CE$, $BF$ and $DE$ form a convex quadrilateral whose area is independent of the choice of points $E$ and $F$.

Determine all monotonic functions $f$ defined on the set of real numbers, such that $f$($x$)+2$x$=$f$ ( $f$($x$) ) for all real $x$.

Find all positive integers $a$, $b$, $c$ for which $2^a$-1 is divisible by $b$, $2^b$-1 is divisible by $c$ and $2^c$-1 is divisible by $a$.

The numbers 1 to 2009 are written on a sheet of paper. Then the double of each number is also written on it. Finally, all those numbers are erased that occur twice. This procedure is repeated in the following way: In the $i$th step, $i$ times the numbers $1,2,\ldots,2009$ are written on the sheet, and all those numbers are erased that occur twice. How many numbers will be on the sheet after 2009 steps?

Out of a set of one million lottery tickets issued with numbers 000000 to 999999, a group of friends bought all tickets that they considered ``lucky'', that is the ones with numbers $\overline{abcdef}$ , such that $af$+$be$+$cd$=100. Prove that the sum of the numbers of the remaining tickets is divisible by 1001.

The angle at vertex $C$ of a triangle is a right angle. The angle bisector and altitude drawn from $C$ intersect the circumscribed circle at $D$ and $E$, respectively. The not shorter leg of the triangle is $b$. Show that the length of the broken line CDE is $b\sqrt{2}$.

The length of the altitudes of a triangle are 10, 12 and 15. How long are the sides?

Let $n$ be a positive integer. What is the first digit after the decimal point in the number $\sum_{k=1}^{n} \frac{k(k+1)}{n}$ ?

Prove that if the sides of a triangle satisfy 2$b^2$=$a^2$+$c^2$, then the opposite angles satisfy 2cot $\beta$=cot $\alpha$+cot $\gamma$.

The midlines are drawn on the base of a regular tetrahedron. The midpoint of each midline is connected to the vertices of the face whose plane is parallel to the midline. Consider the volume of the intersection of the three tetrahedra obtained. What fraction of the volume of the original tetrahedron is it?

A triangulation of a finite point set $A$ in the plane is defined as a partition of its convex hull into triangles such that the vertices of each triangle are elements of $A$, and no triangle contains any element of $A$ apart from its vertices. Prove that all triangulations of a set $A$ consist of the same number of triangles.

For a finite point set $A$ in the plane, let $v(A)$ denote the number of triangles in a triangulation of set $A$ , and let $A+A=\{x+y\mid x,y\in A\}$, where the sum of two points is defined as the point whose position vector is the sum of the position vectors of the terms. Prove that $v(A+A)\ge 4v(A)$.

Prove that the inequality $\frac{ab}{a^2+3b^2} +\frac{bc}{b^2+3c^2} + \frac{ca}{c^2+3a^2} \le \frac{3}{4}$ is true for all positive numbers $a$, $b$, $c$.

The numbers 1 to 2009 are written on a sheet of paper. In the second step, the double of each number is also written on the sheet and then all those numbers are erased that occur twice on it. This step is repeated as follows: in step $i$, every number on the sheet is multiplied by $i$, the results are written down and then all those numbers are erased that occur twice. Prove that there will be at least 2009 numbers on the sheet after every step.

A common tangent of two intersecting circles touches them at the points $A$ and $B$, and the line segment connecting their centres intersects them at $C$ and $D$, respectively. Prove that ABCD is a cyclic quadrilateral.

$a$, $b$, $c$, $d$ are four positive numbers. Five of the products $ab$, $ac$, $ad$, $bc$, $bd$, $cd$ are known: they are 2, 3, 4, 5 and 6. What is the value of the sixth product?

$A$, $B$, $C$, $D$ are moving points in the plane, such that $AD$=$BC$=2 and $AC$=$BD$=4 remain valid, and the line segments $AC$ and $BD$ intersect each other for all positions of the points. How does the distance $CD$ depend on the distance $AB$?

Let $p$>3 be a prime number and let $k$ and $m$ be non-negative integers. Prove that $p^k$+$p^m$ cannot be a perfect square. (Suggested by P. Kutas )

Is it true that every polygon has a vertex from which it is possible to draw a diagonal in the inside of the polygon to a non-adjacent vertex that is closest to it? (Suggested by P. Maga )

Let $n$ be a positive integer. Determine the first digit following the decimal point in the number $\sum_{k=1}^{n}{\frac{\sqrt{k(k+1)}}{n}}.$ (Suggested by M. Bencze, Brasov)

In an acute triangle ABC , the feet of the altitudes drawn from $A$ and $B$ are $A_1$ and $B_1$, respectively, and the orthocentre of the triangle is $M$. The median drawn from $B$intersects line $A_1 B_1$ at point $P$. Prove that $BPM\sphericalangle$ is a right angle if and only if $B_1 C$=3$AB_1$.

An acute-angled triangle has sides $a$, $b$, $c$ and area $t$. The sides satisfy the equality abc =$a$+$b$+$c$. Prove that $1<t\le \frac{3\sqrt{3}}{4}\,.$

Show that there is no polynomial of rational coefficients that takes a non-integer value at exactly one integer. Is there a polynomial of this property with real coefficients? (Suggested by P. Maga )

In a cinema, 80% of the grown-up spectators are men, 40% of the male spectators are children, and 20% of the children are boys. What is the minimum possible number of spectators?

What is the largest possible number of sides of a convex polygon which can be dissected into right-angled triangles with acute angles of 30 and 60 degrees?

The points $A$ and $B$ lie on one side of line $e$ in the plane. Let $M$ be the point on the line for which the sum $AM +MB$ is a minimum, and let $N$ be the point of the line with $AN = BN$ . Prove that the points $A$ , $B$ , $M$ , $N$ lie on a circle.

Find all perfect squares of the form $\underbrace{aaa\dots a}_{n\ digits} \underbrace{bbb\dots b}_{n\ digits}$.

Let $a$, $b$, $c$ denote the sides of an acute triangle and let $\gamma$, $\beta$, $\alpha$, respectively, denote the opposite angles (in radians). Show that $\frac{\pi}{3}\le \frac{\alpha a^2+\beta b^2+\gamma c^2}{a^2+b^2+c^2}<\frac{\pi}{2}$.

In a graph, no vertex is connected to all of the others. For any pair of vertices not connected there is a vertex adjacent to both. The sum of the squares of the degrees of vertices is $n^2-n$ where $n$ is the number of vertices. What is the length of the shortest possible circuit in the graph?

Let $f$ , $g$ , $h$ be different lines in the space, such that if any line $e$ in the space that has common points with both $f$ and $g$ than also has a common point with $h$ . What can be stated about the mutual position of the lines $f$ , $g$ , $h$ ?

Solve the following simultaneous equations: $1-\frac{12}{3x+y} =\frac{2}{\sqrt{x}},$ $1+\frac{12}{3x+y} =\frac{6}{\sqrt{y}}.$

Let $a$denote the side of the regular polygon of 18 sides inscribed in a circle of radius $r$. Show that $a^3$+$r^3$=3$ar^2$.

The students in a class of 30 organized 16 trips during the school year. Eight students went on the trip each time in a van. Show that there are two students in the class who went on at least two trips together.

Consider the number pairs (1,36), (2,35), ..., (12,25). Is it possible to select one number from each given pair, such that the sum of the numbers selected equals the sum of the numbers not selected? Will the answer change if the last two pairs are left out? (Suggested by J. Pataki, Budapest)

The lengths of the diagonals of a rhombus with side 2 units are added. What integer values may the sum have? (Suggested by G. Nyul )

$a$<$b$<$c$ are the sides of a triangle $H$. Consider the three rhombuses, such that one vertex coincides with a vertex of $H$ and the other three vertices lie on the sides of $H$. Given that two of these rhombuses have the same area, show that $b^2$=$ac$.

Is it true that if there exists a quadrilateral with sides $a$, $b$, $c$ and $d$ then there exists such a cyclic quadrilateral, too?

The sequence $p_n$ is defined recursively: Let $p_1$=2 and let $p_{n+1}$ be the largest prime factor of $p_1p_2\ldots p_n+1$. Is 11 a term of the sequence?

In the parallelogram ABCD , 2$BD^2$=$BA^2$+$BC^2$. Show that the circumscribed circle of triangle BCD goes through one of the points that trisect the diagonal $AC$. (Suggested by L. Koncz, Budapest)

Each edge of a regular pyramid of square base has unit length. Determine the distance between the lines of two skew edges.

Show that the number of terms with coefficients not divisible by 3 in the binomial expansion of ($x$+$y$)$^n$ is not divisible by 5.

Rose, Ivy and Violet decided to solve all problems in their book of mathematics problems. Rose solves $a$ problems, Ivy solves $b$ problems and Violet solves $c$ problems a day. (Each problem is only solved by one of the girls.) If Rose solved 11 times as many problems a day, Ivy solved 7 times as many and Violet solved 9 times as many, they would finish the job in 5 days. If Rose solved 4 times as many, Ivy solved 2 times as many and Violet solved 3 times as many problems a day, they would take 16 days. How many days does it take them to get done with the solutions?

Is there a plane figure different from the unit square that has an area of 1 and a perimeter of 4?

There are 10 mail boxes next to the gate of a building. A man distributing flyers walks by and puts flyers in 5 boxes. Later another one walks by and also puts flyers in 5 boxes. What is the probability that there are fliers in at least 8 boxes?

Solve the equation: 1+cos 3$x$=2cos 2$x$.

ABCD is a cyclic trapezium. Diagonal $BD$ is extended beyond vertex $D$ by the length of the leg and the point $L$ is obtained. $K$ is the point on diagonal $AC$ whose distance from $C$is equal to the leg. Prove that the areas of the rectangles formed by the line segments $AB$and $CD$ and the line segments $AK$ and $BL$ are equal.

In triangle ABC , the altitude drawn from vertex $C$ intersects side $AB$ at $T$. Right-angled triangles CAD and CBE are drawn on sides $AC$ and $BC$ on the outside, such that the right angles are at $A$ and $B$. Given that $AD$=$TB$ and $BE$=$TA$, prove that $\angle$ CDE =$\angle$ CED .

The sum of three positive numbers is multiplied by the sum of their reciprocals. Find the smallest possible interval where the product may lie.

An entrepreneur has taken out a subsidized loan for 12 million forints (HUF, Hungarian currency) at a fixed annual interest rate of 8%. How much will he owe in 10 years if he pays back the loan in annual instalments of 1.2 million forints?

A café has tables for two. At the moment, there are six customers sitting at three tables. Three of them are having coffee and three are having tea. What is the probability that there is a table where both customers are having tea?

We have a red bouncy ball with thirty round white spots on it. (The spots form spherical caps.) The perimeter of the great circle of the ball is 54 cm, and the perimeter of the spots is 11 cm. What percentage of the surface area of the ball is covered by the spots?

Assume that there are relatively more teenagers among allergic people than in the whole population, and a higher proportion of teenagers do sports than the population average. Does it follow that there are relatively more allergic people among those doing sports?

The captain of a ship wrote down the formula $d=p\sqrt h$ in his log book for the distance to the horizon. Unfortunately, the number $p$ got blurred. $d$ denotes the distance to the horizon in kilometers, and $h$ stands for the height of the eyes of the observer above sea level in metres. Determine a value of $p$ that provides a practicable formula. (Use 6370 km for the radius of the Earth.)

Prove that $5^{2008}$+4 is a composite number.

Given the line $BD$ and the feet of the perpendiculars dropped from point $A$ onto line $CD$, from point $B$ onto line $DA$, from point $C$ onto line $AB$ and from point $D$ onto line $BC$, construct the quadrilateral ABCD . (It is not required to discuss the number of possible solutions.)

$a$, $b$ and $c$ are three not necessarily consecutive terms of an arithmetic progression of positive numbers. Given that $\frac{c-b}{a}+\frac{a-c}{b}+ \frac{b-a}{c}=0$, find the common difference of the arithmetic progression.

A two-digit number is multiplied by 4, and the original two-digit number is written behind the number obtained. The resulting number has exactly six divisors. What may be the original two-digit number?

Find all integers that may be the measure, in degrees, of the angles of a regular polygon.

The lengths of the sides of a triangle cut out of paper are 8 cm, 10 cm and 12 cm. The triangle is folded along a line through the common vertex so that the shortest side overlaps with the longest side. A double-layer part and a single-layer part are obtained. Prove that the single-layer part is an isosceles triangle.

Given that 4 passengers in a metro train of 6 carriages have colds, what is the probability that there are at most two carriages in which there is a passenger who has a cold?

A cubical playing die is made out of a spherical body by cutting off six identical spherical caps. Each of the circular sections is tangent to the four adjacent ones. What percentage is the total area of the six circles of the whole surface area of the resulting die?

Find all integers for which the value of the expression 6$x^2$-167$x$-4823 is $a$) a prime number; $b$) the smallest possible integer.

A right-angled triangle of sides 3, 4, 5 is cut into two parts with a line perpendicular to the hypotenuse. One part is a quadrilateral that has an inscribed circle, and the other part is a right-angled triangle. Find the lengths of the sides of the quadrilateral.

To boost the sales of a certain kind of sweets, the manufacturing company encloses gift vouchers in some of the boxes. The managers consider that the campaign will be efficient and costs will be bearable if the probability for a customer buying 10 boxes of sweets to find at least 1 voucher is about 50%. One in how many boxes should contain a voucher to achieve that?

The midpoint of the base $AB$ of an isosceles triangle ABC is $F$. Let $D$ denote the orthogonal projection of point $F$ onto side $BC$. Let $E$ be the midpoint of the line segment $DF$. Prove that $CE$ is perpendicular to $AD$.

Let $x$<$y$<$z$. Solve the following equation on the set of natural numbers: $3^x+3^y+3^z=179 415$.

Show that the simultaneous equations $x$-$y$+2$z$=0, -2$x$+$y$-2$z$=-2, 2$x$+$cy$+3$z$=1 have a solution that is independent of the value of the parameter $c$.

Given six different points in the plane such that at least three of any four are collinear. Prove that at least five of the points lie on the same line.

Prove that every fourth term of the Fibonacci sequence is divisible by three. (In the Fibonacci sequence, $a_1$=1, $a_2$=1 and $a_n$=$a_{n-1}$+$a_{n-2}$ for all $n\in \mathbb{N}$ , $n \ge$3.)

The points $A$, $B$, $C$ and $D$ lie on a line in this order, and $AB$=$BC$. Draw perpendiculars to $AD$ at $B$and $C$. The perpendicular drawn at $B$ intersects the circle of diameter $AD$ at $P$ and $Q$. The perpendicular drawn at $C$intersects the circle of diameter $BD$ at $K$ and $L$. Show that the centre of the circle passing through points $P$, $K$, $L$ and $Q$ is point $B$.

A coin is tossed four times. Then it is tossed again, the number of times a head was obtained in the first four tosses. What is the probability of getting at least 5 heads altogether?

There are 30 people sitting at a round table. Some of them are liars, the others tell the truth. We know that out of the two neighbours of every liar, exactly one is a liar. Each of the 30 people is asked how many liars are sitting next to them. 12 say exactly one and the others say that both of their neighbours are liars. How many liars are there around the table? (Based on a problem from Subcarpathia)

A certain integer has two prime factors. The number of its divisors is 6, and the sum of the divisors is 28. Which number is it?

Find those right-angled triangles of integer sides for which the measures of the perimeter and area are equal.

A wholesale merchant sells drugstore products and stationery. He has a container of volume 12 $m^3$ that has a capacity of 5 tonnes of goods. One tonne of drugstore products fill up a volume of 1 $m^3$, and a tonne of stationery products take up 3 $m^3$. He makes a profit of 100,000 forints (HUF, Hungarian currency) per tonne on drugstore products and 200,000 forints per tonne on stationery. What is the maximum profit that the merchant can make by selling a container of goods?

An arbitrary line is drawn through vertex $A$ of a square ABCD , and perpendiculars are dropped onto the line from the vertices $B$ and $D$. The feet of the perpendiculars are $B_1$ and $D_1$, respectively. Prove that $AB_1 ^2$+$AD_1 ^2$=$BB_1 ^2$+$DD_1 ^2$.

An intercity train overtakes a freight train travelling on a parallel track. On its way back, the intercity also passes the freight train travelling in the opposite direction. The ratio of the speed of the IC to that of the freight train equals the ratio of the time of overtaking to the time of passing each other in opposite directions. By what factor is the IC faster than the freight train if each train travels at a constant speed?

Prove that six-digit numbers of the form $\overline{ababab}$ cannot have prime factors of more than two digits.

Prove that the diameter of the inscribed circle of a right-angled triangle is the geometric mean of the difference between the hypotenuse and one leg, and the double of the difference between the hypotenuse and the other leg.

The digits 0, 1, 2, 3, 4, 5, 6 are written in a random order. What is the probability that a seven-digit number divisible by four is obtained? (The number is not allowed to start with zero.)

Given that two perpendicular tangents can be drawn from the origin to the circle $x^2$-6$x$+$y^2$-2$py$+17=0, find the possible values of $p$.

Santa divides 53 Christmas candies in three bags, so that there is a different number of candies in each bag but the total number of candies in any two is greater than the number of candies in the third one. In how many different ways is that possible?

Prove that the value of the expression $a^3$-3$ab^2$+2$b^3$ is non-negative if $a$ and $b$ are non-negative real numbers.

A circle passing through the centre of a square is drawn about each vertex. The circles intersect the sides of the square at 8 points altogether. Prove that the intersections form a regular octagon.

Represent the set of points ($x$,$y$) in the coordinate plane the coordinates of which satisfy the two conditions below: $x^2$+$y^2 \le$25, $-1\le \frac{x}{x+y}\le 1$.

The number of persons who booked ticket for the New Year's concert is a perfect square. If 100 more persons booked ticket then the number of spectators would be a perfect square plus 1. If still 100 more persons booked ticket then the number of spectators would be again a perfect square. How many persons booked ticket for the concert?

Four girls and four boys went dancing together. The first dance was a waltz. Each boy asked a girl to dance (each girl was only asked by one boy). The next dance was a tango. This time, it was the girls who asked the boys to dance. Each girl asked a boy (and each boy was asked by only one girl). There was no couple that danced both dances together. Given the following information, find out who danced each dance with whom. $a$) Charles waltzed with the girl who danced tango with Daniel. $b$) Andrew asked that girl to waltz who danced tango with the boy who waltzed with Emily. $c$) Bernard waltzed with the girl who danced tango with Mary's waltz partner. $d$) Gillian did not dance tango with Bernard. $e$) Helen danced tango with a boy who did not waltz with Gillian. After the two dances, when the dance instructor asked everyone to stand behind the back of the partner whom he or she asked to dance, the result was one big circle of people.