KöMaL 2015

Let $ABCD$ be a convex quadrilateral. In the triangle $ABC$ let $I$ and $J$ be the incenter and the excenter opposite to vertex $A$ , respectively. In the triangle $ACD$ let $K$ and $L$ be the incenter and the excenter opposite to vertex $A$ , respectively. Show that the lines $IL$ and $JK$ , and the bisector of the angle $BCD$ are concurrent.

Prove that if $n$ is a sufficiently large positive integer then among any $n$ distinct positive integers there are four whose least common multiple is greater than $n^{3.99}$.

Let $n\ge2$ be a in integer and let $f\colon \mathbb{R}\to[-1,1]$ be an $n$ times differentiable function. Show that the equation $f^{(n)}(x)=0$ has at least $n-1$ distinct solutions.

Show that for every positive real number $c>0$ there is a positive integer $n$ such that $\varphi\big(\sigma(n)\big)>cn$ . (For an arbitrary postive integer $k$ , $\varphi(k)$ denotes the number of positive integers not exceeding $k$ that are co-prime with $k$ . $\sigma(k)$ is the sum of positive divisors of $k$ .)

There is given a convex quadrilateral $ABCD$ and a point $P$ in the interior of the triangle $BCD$ in such a way that the quadrilateral $ABPD$ has an inscribed circle, and the three inscribed circles of the quadrilateral $ABPD$ , the triangle $BCP$ and the triangle $CDP$ , respectively, are pairwise tangent to each other. Denote by $Q$ and $R$ the points tangency on the line segments $BP$ and $DP$ , respectively. Let the lines $BP$ and $AR$ meet at $S$ , let the lines $DP$ and $AQ$ meet at $T$ , and let the lines $BT$ and $DS$ meet at $U$ . Show that the line $CU$ bisects the angle $BCD$ .

Let $n$ be a positive integer. Let $\mathcal{F}$ be a family of sets that contains more than half of all subsets of an $n$ -element set $X$ . Prove that from $\mathcal{F}$ we can select $\lceil\log_2n\rceil+1$ sets that form a separating family on $X$ , i.e., for any two distinct elements of $X$ there is a selected set containing exactly one of the two elements.

Does there exist a simple, closed polyline in the 3-dimensional Cartesian co-ordinate system whose perpendicular projection onto every co-ordinate plane is a cycleless graph?

The angle trisectors of a triangle bound a convex hexagon in the interior of the triangle. Prove that the diagonals of this hexagon, that connect opposite vertices, are concurrent.

Find all primes $p$ and positive integers $n$ for which the numbers ${(k+1)}^n-2k^n$ ($k=1,2,\ldots,p$ ) form a complete residue system modulo $p$ .

Determine whether there is a finite, nonempty subset $S$ of the square grid in the plane such that every element of $S$ has at least two neighbours in $S$ and $S$ does not contain four points that are the vertices of a square (with sides not necessary parallel to the coordinate axes)?

Let $n\ge3$, let $x_1,\ldots,x_n$ be nonnegative numbers, and let $A=\sum_{i=1}^n x_i$, $B= \sum_{i=1}^n x_i^2$ and $C= \sum_{i=1}^n x_i^3$. Prove that $(n+1)A^2B + (n-2)B^2 \ge A^4 + (2n-2)AC$ .

For every positive integer $n$ , let $P(n)$ be the greatest prime divisor of $n^2+1$. Show that there are infinitely many quadruples $(a,b,c,d)$ of positive integers that satisfy $a<b<c<d$ and $P(a)=P(b)=P(c)=P(d)$.

Let $f(x,y)$ be a polynomial with two variables and integer coefficients such that $f$ is constant neither in $x$ - nor in $y$ -direction. Prove that $\max_{a,b\in[-2,2]}\big|f(a,b)\big|\ge4$.

Do there exist infinitely many (not necessarily convex) 2015-gons in the plane such that every three of them have a common interior point, but no four have a common point?

Ginger and Rocky play the following game. First Ginger hides two bones in the corners of a rectangular garden. She may dig 45 cm deep altogether, that is, she may either hide the two bones in two different corners, where the sum of their depths may be at most 45 cm, or she may hide them in the same place, both bones at a maximum depth of 45 cm. She levels the ground carefully so that it is impossible to see where she has dug. Then Rocky may dig holes with a total depth of 1 m. Rocky's goal is to maximize the probability of finding both bones, while Ginger's goal is to maximize the probability of keeping at least one for herself. $(a)$ Show that if Ginger plays well she can achieve a probability of more than $1/2$ for at least one bone remaining hidden, independently of Rocky's search strategy. $(b)$ What are the chances of Ginger's success if both dogs play optimally?

Let $k$ be a nonnegative integer. Prove that there are only finitely many positive integers $n$ for which there exist two disjoint sets $A$ and $B$ satisfying $A \cup B = \{1; 2; \ldots; n\}$ and $\left|\prod \limits_{a \in A} {a} - \prod\limits _{b \in B} {b}\right|=k$ .

In the acute angled triangle $ABC$ , the midpoints of the sides $BC$ , $CA$ and $AB$ are $D$ , $E$ and $F$ , respectively. The foot of the altitude of the triangle starting from $C$ is $T_1$. On some line, passing through point $C$ but not containing $T_1$, the feet of the perpendiculars starting from $A$ and $B$ are $T_2$ and $T_3$, respectively. Prove that the circle $DEF$ passes through the center of the circle $T_1T_2T_3$.

A convex polyhedron has only quadrilateral faces. Show that it is possible to split every face into two triangles by drawing one of its diagonals in such a way that at each vertex of the polyhedron, an even number of triangles meet.

There is given an acute-angled triangle $ABC$ with a point $X$ marked on its altitude starting from $C$ . Let $D$ and $E$ be the points on the line $AB$ that satisfy $\sphericalangle DCB=\sphericalangle ACE=90^\circ$. Let $K$ and $L$ be the points on line segments $DX$ and $EX$ , respectively, such that $BK=BC$ and $AL=AC$ . Let the line $AL$ meet $BK$ and $BC$ at $Q$ and $R$ , respectively; finally let the line $BK$ meet $AC$ at $P$ . Show that the quadrilateral $CPQR$ has an inscribed circle.

Determine all real polynomials $P(x)$ that satisfy $P\big(x^3-2\big)=P{(x)}^3-2.$

Prove that there exists a real number $C>1$ with the following property: whenever $n>1$ and $a_0<a_1<\cdots<a_n$ are positive integers such that $\frac1{a_0},\frac1{a_1},\ldots,\frac1{a_n}$ form an arithmetic progression, then $a_0> C^n$.

Let $n\ge2$ be an integer. Prove that there exist integers $a_1,\dots,a_{n-1}$ such that $a_1 \arctg 1 + a_2 \arctg 2 +\ldots+ a_{n-1}\arctg(n-1) = \arctg n$ if and only if $n^2+1$ divides $(1^2+1)(2^2+1)\ldots\big((n-1)^2+1\big)$.

Let $p(x)$ be a polynomial of degree at most $n$ such that $\big|p(x)\big|\le\frac{1}{\sqrt{x}}$ for $0<x\le 1$. Prove that $\big|p(0)\big|\le 2n+1$.

Two circles, $k_1$ and $k_2$ meet at points $A$ and $B$ . Points $C$ and $D$ lie on $k_1$, while points $E$ and $F$ lie on circle $k_2$ in such a way that $A$ , $C$ , $E$ are collinear and $B$ , $D$ , $F$ are collinear, too. Points $G$ and $H$ are other two points on lines $ACE$ and $BDF$ , respectively. The line $CH$ meets $FG$ and $k_1$ the second time at $I$ and $J$ , respectively. The line $DG$ meets $EH$ and $k_1$ the second time at $K$ and $L$ , respectively. Circle $k_2$ meets the lines $EHK$ and $FGI$ the second time at $M$ and $N$ , respectively. The points $A,B,C,\ldots,N$ are distinct. Show that $I$ , $J$ , $K$ , $L$ , $M$ and $N$ are either concyclic or collinear.

Let $p(x)=a_0+a_1x+\dots+a_nx^n$ be a polynomial with real coefficients such that $p(x)\ge0$ for $x\ge0$. Prove that for every pair of positive numbers $c$ and $d$ , $a_0 + a_1(c+d) + a_2(c+d)(c+2d) + \dots + a_n(c+d)(c+2d)\dots(c+nd) \ge0$.

Let $\{x_n\}$ be the van der Korput sequence, that is, if the binary representation of the positive integer $n$ is $n = \sum_i a_i2^i$ ($a_i\in\{0,1\}$), then $x_n = \sum_i a_i2^{-i-1}$. Let $V$ be the set of points $(n,x_n)$ in the plane where $n$ runs over the positive integers. Let $G$ be the graph with vertex set $V$ that is connecting any two distinct points $p$ and $q$ if and only if there is a rectangle $R$ which lies in a parallel position to the axes and $R\cap V = \{p,q\}$. Prove that the chromatic number of $G$ is finite.

We call a bar of width $w$ on the surface $S^2$ of the unit sphere in $3$-dimension, centered at the origin a spherical zone which has width $w$ and is symmetric with respect to the origin. Prove that there exists a constant $c>0$ such that for every positive integer $n$ the surface $S^2$ can be covered with $n$ bars of the same width so that every point is contained in no more than $c\sqrt{n}$ bars.

Ann and Bill take turns writing digits on a sheet of paper, left to right. Ann starts with a nonzero digit, and they continue until a 100-digit number is formed. Bill wins if the resulting number divided by 11 leaves a remainder of 5, otherwise Ann wins. Both players are good at mathematics. Who will win the game?

Prove that among any 39 consecutive natural numbers there is a number in which the sum of the digits is divisible by 11.

Find the integer solutions of the equation $3^{n}=2n^2+1$

What is the area of the pentagon in exercise C. 1240. ?

For a given positive integer $k$ , find the largest positive integer $m$ such that the following statement should be true: If at most $m$ of $3k$ different points in the plane are collinear, then the points can be divided into $k$ groups of three such that the points in each group form a triangle.

Is there a plane that intersects a right pyramid of regular pentagonal base in a hexagon with $a)$ line symmetry, $b)$ central symmetry?

The diagonals of a quadrilateral $ABCD$ are perpendicular, they intersect at $E$ . From point $E$ , drop a perpendicular onto the line of each side. Consider the intersection of each perpendicular with the opposite side. Prove that the four points all lie on a circle centred at a point of the line segment connecting the midpoints of the diagonals.

Determine the smallest possible value of $x^{2}+y^{2}+z^{4}$, where $x$ , $y$ and $z$ are positive numbers that add up to 34.

A flea is jumping on the points of the plane with integer coordinates. Is it possible for the flea to follow a route touching every lattice point exactly once, in which each jump has an integer length, and each integer jump length occurs exactly once?

Samson wrote the number 123456789 on a sheet of paper. Then he may or may not have placed a multiplication sign between any two consecutive digits. (He may have used several of them, or none at all.) By reading the digits between multiplication signs as a single number, he obtained a product of numbers. For example, $1234 \cdot 56 \cdot 789$. What is the maximum possible value of the number obtained in this way?

Which of the numbers below is greater if each expression contains 50 twos and 50 threes alternating: $2^{3^{\scriptstyle 2^{\scriptstyle 3^{\scriptstyle .^{\scriptstyle .^{\scriptstyle .^{\scriptstyle 2^{\scriptstyle 3}}}}}}}}$ or $3^{2^{\scriptstyle 3^{\scriptstyle 2^{\scriptstyle .^{\scriptstyle .^{\scriptstyle .^{\scriptstyle 3^{\scriptstyle 2}}}}}}}}$?

Is there a pentagonal pyramid that can be intersected by a plane in a regular hexagon?

9 mathematicians of a university went to a conference together. Since the presentations were boring, they fell asleep several times, each of them at most 4 times. For any pair of mathematicians, there was a time when they were both sleeping. Show that there was a time instant when at least three of them were sleeping.

$a$ , $b$ , $c$ and $d$ are four parallel lines in the plane, in this order. The distance of $a$ and $b$ is 1, the distance of $b$ and $c$ is 3, and the distance of $c$ and $d$ is also 1. Consider the rectangles that have exactly one vertex on each line. How can we obtain the rectangle of minimal area, and what is this area?

The sides of an acute-angled triangle are $a$ , $b$ , $c$ , the opposite angles are $\alpha$, $\beta$, $\gamma$, and the lengths of the corresponding altitudes are $m_a$, $m_b$, $m_c$, respectively. Prove that $\frac{m_a}{a} + \frac{m_b}{b} + \frac{m_c}{c} \ge 2\cos \alpha \cos\beta \cos \gamma \left(\frac{1}{\sin 2\alpha} + \frac{1}{\sin 2\beta} + \frac{1}{\sin 2\gamma}\right) + \sqrt{3}\,$.

$K$ is a point on side $AC$ of triangle $ABC$ such that $AK=2KC$ and $\angle ABK = 2 \angle KBC$ . Let $F$ denote the midpoint of side $AC$ , and let $L$ be the orthogonal projection of point $A$ onto the line segment $BK$ . Prove that the lines $FL$ and $BC$ are perpendicular.

Find all real numbers $p_1$, $q_1$, $p_2$, $q_2$ such that $p_2$ and $q_2$ are roots of the equation $x^3+p_1x+q_1=0$, and $p_1$ and $q_1$ are roots of the equation $x^3+p_2x+q_2=0$.

Given that a permutation of the points of the plane maps any three concyclic points to three concyclic points, show that three points are collinear if and only if their images are collinear.

How many positive integers $n$ are there such that the geometric mean and the harmonic mean of $n$ and 2015 are both integers?

Prove that if a right-angled trapezium has an inscribed circle, then the shorter leg equals the line segment cut out by the legs from the line parallel to the bases, passing through the intersection of the diagonals.

Give an example for sets $H_1,H_2,\ldots\subset\mathbb{N}$ for which the following conditions hold: $a)$ $|H_n|=n$ for all positive integers $n$ . $b)$ For all positive integers $n$ and $k$ , $H_n \cap H_k = H_{(n,k)}$, where $(n,k)$ is the greatest common divisor of $n$ and $k$ .

Construct a bicentric kite if given the radius of its circumscribed circle and the distance between the centres of its circumscribed and inscribed circles.

Solve the equation $\big(\sqrt{1+\sin^2 x}-\sin x\big) \big(\sqrt{1+\cos^2 x}-\cos x\big) =1.$

Let $A_1 B_1 C_1 D_1$ be a quadrilateral. For any set of four points $A_n B_n C_n D_n$ already defined for a positive integer $n$ , let $A_{n+1}$ be the centroid of the triangle $B_{n}C_{n}D_{n}$. $B_{n+1}$, $C_{n+1}$ and $D_{n+1}$ are defined analogously, with a cyclic permutation of the points. Show that for any starting quadrilateral, the point sequence $A_n$ only has a finite number of points lying outside the unit circle drawn about the centre of mass of the quadrilateral $A_1 B_1 C_1 D_1$.

Show that there are infinitely many points equidistant from three pairwise skew edges of a given cube.

Given that the absolute values of the numbers $x_1$, $x_2$, $x_3$, $x_4$, $x_5$, $x_6$ are at most 1, and their sum is 0, prove that $3\sum_{i=1}^{5} {\sqrt{1-x_i^2}} \le \sum_{i=1}^{5} {\sqrt{9-{(x_i+x_{i+1})}^2}}\,.$

The circles $k_2$ and $k_3$ have different radii. A circle $k_1$ touches both of them from the inside. The circles $k_2$ and $k_3$ are tangent to a circle $k_4$ from the inside. Show that the radical axis of $k_1$ and $k_4$ passes through the external point of similitude of $k_2$ és $k_3$.

Let $p$ denote an odd prime number. Show that the equation $x^2 + px = y^2$ has exactly one solution on the set of pairs of positive integers.

The sides of rectangle $ABCD$ are $AB= \frac{\sqrt{5}+1}2$ and $BC=1$. Let $E$ be the point in the interior of line segment $AB$ with $AE=1$. Show that $\angle ACE= 2\cdot \angle EDB$ .

Let $t>1$ be an odd integer. Prove that there exist only a finite number of pairs of integers $n$ and $k$ , not smaller than $t$ such that $S=\binom{n}{t} + \binom{k}{t}$ is a prime.

$O$ is the centre of the circumscribed circle of triangle $ABC$ , and $M$ is the orthocentre. Point $A$ is reflected in the perpendicular bisector of side $BC$ , $B$ is reflected in the perpendicular bisector of side $CA$ , and finally $C$ is reflected in the perpendicular bisector of side $AB$ . The reflections are denoted by $A_1$, $B_1$, $C_1$, respectively. Let $K$ be the centre of the inscribed circle of triangle $A_1B_1C_1$. Prove that point $O$ bisects line segment $MK$ .

Solve the simultaneous equations $x^{2}+y^{2}=13,$ $x^{3}+y^{3}=35.$

$\mathcal P$ is a set of points in the plane such that every disc of unit radius has at least one point of $\mathcal P$ in its interior. Is it true that there exists a closed disc of unit radius that contains at least three points of $\mathcal P$ ?

Let $f(x)=\frac{4^x}{4^x+2}$. Calculate the value of the sum $f(0/2015)+ f(1/2015)+f(2/2015)+\dots +f(2014/2015)+f(2015/2015)$.

What percentage of a pencil gets wasted? Assume that a pencil is a cylinder, infinitely long, and the graphite rod inside is also cylindrical. The axes of the two cylinders coincide. When the pencil is sharpened, its point is a perfect cone with an apex angle of 12 degrees. When we write with the pencil, its axis always encloses a 42-degree angle with the plane of the paper. We keep using the pencil until we can no longer write with it since no matter how we rotate it about its axis, the wood will scratch the paper. Then the pencil is sharpened again to the shape of a 12-degree cone, but never longer, that is, the tip of the pencil never changes during sharpening, it only wears in writing. What percentage of the graphite is wasted by scraping it off with the sharpener? Will someone holding the pencil at a 45-degree angle waste more than that or less? If so, by how much?

A circle passing through vertices $B$ and $C$ of triangle $ABC$ intersects side $AB$ at $D$ , and side $AC$ at $E$ . The intersection of lines $CD$ and $BE$ is $O$ . Let $M$ denote the centre of the inscribed circle of triangle $ADE$ , and let $N$ denote the centre of the inscribed circle of triangle $ODE$ . Prove that line $MN$ bisects the smaller arc $DE$ .

Given 2015 points in the plane, show that if every four of them form a convex quadrilateral then the points are the vertices of a convex 2015-sided polygon.

Find all pairs of positive integers $(a,b)$ such that $a^{(b^2)}=b^a$.

The rhombus $ABFE$ determined by sides $AB$ and $AE$ is cut out of a regular pentagon $ABCDE$ . Determine the centre of mass of the remaining concave pentagonal plate $BCDEF$ .

Solve the equation $|1-x| = \left|2x-57-2\sqrt{x-55} +\frac{1}{x-54-2\sqrt{x-55}}\right|$.

The midpoint of edge $B'C'$ of a cube $ABCDA'B'C'D'$ is $E$ , and the midpoint of edge $C'D'$ is $F$ . The plane $AEF$ divides the cube into two parts. Find the ratio of the volumes of the two parts.

Show that $\sum_{j=0}^{b} \sum_{i=j}^{a-b+j} \binom{i}{j} \binom{a-i}{b-j} =(a+1)\binom{a}{b}$ for all positive integers $a \ge b$ .

Let $a$ and $n$ denote positive integers such that $a^n-1$ is divisible by $n$ . Prove that the numbers $a+1$, $a^2+2$, ..., $a^n+n$ all leave different remainders when divided by $n$ .

A circle $k$ touches the legs $AB$ and $AC$ of an isosceles triangle $ABC$ , and intersects the base $BC$ at $K$ and $L$ . Line segment $AK$ intersects the circle $k$ again at point $M$ . The reflections of point $K$ in $B$ and in $C$ are $P$ and $Q$ , respectively. Prove that $k$ is tangent to the circumscribed circle of triangle $PMQ$ .

Each permutation of an $n$ -element set is coloured in either red, white or green. Let $N_{\rm RWG}$ denote the number of ways to perform a red permutation followed by a white permutation and then a green permutation, such that each element is restored to its initial position at the end. Analogously, let $N_{\rm GWR}$ denote the number of ways to perform a green permutation followed by a white permutation and then finally a red permutation, such that each element is restored to its initial position. Show that $N_{\rm RWG}=N_{\rm GWR}$.

Are there infinitely many difference prime numbers, such that the sum of any 2015 of them is a composite number?

Solve the inequality $x+y-xy+\frac{1}{x} +\frac{1}{y} -\frac{1}{xy}\le 2$ and represent it in the coordinate plane.

Show that if a simple graph has 7 vertices and no cycle of length 4 then it has a vertex whose degree is at most 2.

In a square $ABCD$ , points $P$ and $Q$ lie on sides $AB$ and $BC$ , respectively, and $BP = BQ$ . Let $T$ denote the foot of the perpendicular dropped from vertex $B$ to the line segment $PC$ . Prove that $\angle DTQ$ is a right angle.

Determine all functions $f:\mathbb R \to \mathbb R$ such that $f(x+y)+f(x)f(y)=x^2 y^2+2xy$ for all $x$ and $y$ .

Let $e$ be the line of one of the edges of a cube. How many lines of the space intersect exactly those out of the 12 lines containing the edges of the cube which are skew relative to $e$ ?

A given quadrilateral $ABCD$ has right angles at $C$ and at $D$ . Construct that point $P$ of line segment $CD$ for which $\angle APD = 2\angle BPC$ .

The circles $k_1$ and $k_2$ touch at point $E$ . Points $X_i$ and $Y_i$ are marked on each circle $k_i$ ($i = 1,2$) such that the two lines $X_iY_i$ intersect each other on the common interior tangent of the circles. Prove that the line connecting the centres of circles $X_1X_2E$ and $Y_1Y_2E$ , and the other line connecting the centres of circles $X_1Y_2E$ and $X_2Y_1E$ also intersect each other on the common interior tangent of the circles.

Let $0\le a,b,c \le 2$, and $a+b+c=3$. Determine the largest and smallest values of $\sqrt{a(b+1)} + \sqrt{b(c+1)} + \sqrt{c(a+1)}.$

The teacher of a class of 36 students enters the averages of the mathematics test grades in a $6 \times 6$ table. Each student has a different mean grade. The teacher marks the largest entry in each column of the table. He finds that all of the 6 marked numbers lie in different rows. Then he marks the largest entry in each row, and finds that these all lie in different columns. Prove that the two sets of six numbers are equal.

Every edge of a simple connected graph of $n\ge 2$ vertices is labelled with either a 1 or with a 2. Then each vertex is assigned with the product of the numbers on the edges product of the numbers on the related edges on it. Show that there will be a pair of two vertices assigned with the same number.

Some fields (unit cubes) constituting a cubical lattice of edge 2015 units are infected by an unknown disease. The disease will spread if at least $t$ fields in some row parallel to any edge of the cube are infected $(1 \le t \le 2015)$. In that case, every field of that row will become infected in one minute. How many fields need to be infected initially in order to $a)$ make it possible $b)$ be certain that the infection reaches all fields of the cube?

Construct a cyclic quadrilateral, given one vertex, the line of the diagonal through the vertex, and the intersections of the lines of the two pairs of opposite sides.

Let $n$ be a positive integer. Solve the simultaneous equations $\sum_{i=1}^{n} {|x_i|} = \sum_{i=1}^{n} \big|x_i^3\big| = \sum_{i=1}^{n} {\frac{2 {|x_i|}^3}{x_i^2+1}}.$

$D$ is the foot of the altitude drawn to the hypotenuse $AB$ of a right-angled triangle $ABC$ . The angles bisectors of $\angle ACD$ and $\angle BCD$ intersect hypotenuse $AB$ at $E$ and $F$ , respectively. Determine the ratio of the inradius of triangle $ABC$ to the circmradius of triangle $CEF$ .

$C$ is an arbitrary point of a circle $k$ of diameter $AB$ , different from $A$ and $B$ . Drop a perpendicular from $C$ onto diameter $AB$ . The foot of the perpendicular on line segment $AB$ is $D$ , and the other intersection with the circle $k$ is $E$ . The circle of radius $CD$ centred at $C$ intersects circle $k$ at points $P$ and $Q$ . Let $M$ denote the intersection of line segments $CE$ and $PQ$ . Dertermine the value of $\frac{PM}{PE} + \frac{QM}{QE}$ .

Consider all real numbers $x$ for which $\tan x + \cot x$ is a positive integer. Find those of them for which $\tan^3x + \cot^3 x$ is a prime number.

Eight unit cubes, with corresponding edges parallel, are glued together to form a solid. Prove that the surface area of the resulting solid is at least 24 units. Only the outer surface counts, even if the resulting solid contains a cavity.

How many different triangles with an axis of symmetry are there in which one side is twice as long as one of the altitudes? Similar triangles are not considered different.

Show that it is possible to label the edges of a complete graph of $n\ge 3$ vertices with 1, 2 or 3, so that the product of the labels of the edges be different at each vertex.

The inscribed circle of triangle $ABC$ touches sides $BC$ , $AC$ and $AB$ at points $A_1$, $B_1$ and $C_1$, respectively. Let the orthocentres of triangles $AC_1B_1$, $BA_1C_1$ and $CB_1A_1$ be $M_A$, $M_B$ and $M_C$, respectively. Show that triangle $A_1B_1C_1$ is congruent to triangle $M_AM_BM_C$.

Let $n$ be a non-negative integer. Determine the exponent of 7 in the prime factor representation of $3^{7^n} + 4^{7^n}$.

Let $n$ be a positive integer. Solve the equation $\frac{1}{\sin^{2n} x} + \frac{1}{\cos^{2n} x} = 2^{n+1}$.

The inscribed circle of triangle $ABC$ touches sides $BC$ , $AC$ and $AB$ at points $A_1$, $B_1$ and $C_1$, respectively. The other intersection of line segment $AA_1$ with the inscribed circle is $Q$ . The line through point $A$ parallel to $BC$ intersects the lines $A_1 C_1$ and $A_1 B_1$ at points $P$ and $R$ . Prove that $PQR\sphericalangle =B_1 QC_1\sphericalangle$.

In a certain lottery game, 6 numbers are drawn every week, out of the numbers 1 to 45. The draw of the first week of this year produced surprising results, since five consecutive numbers appeared. The numbers drawn were 37, 38, 39, 40, 41, 45. The news spread fast in the press. The question arises whether the excitement was justified: are these numbers so special? Let a number sequence be called perfect if it consists of six consecutive numbers, and nearly perfect if exactly five numbers out of the six are consecutive. How many perfect and nearly perfect combinations are there? Considering that the lottery game has been played for 26 years, and there have been 1227 weekly draws so far, what is the probability that during a time interval of this length at least one perfect or nearly perfect sequence of numbers is drawn?

A triangle $\mathcal H$ is rotated about a line lying in its plane but not intersecting it. Show that the volume of the resulting solid equals the product of the area of $\mathcal H$ and the perimeter of the circle described by the centroid of $\mathcal H$ during the rotation.

The feet of the altitudes drawn from vertices $B$ and $C$ of an acute-angled triangle $ABC$ on the sides $AC$ and $AB$ are $D$ and $E$ , respectively. The midpoint of side $BC$ is $F$ . The intersection of line segments $AF$ and $DE$ is $M$ , and the orthogonal projection of point $M$ onto the line segment $BC$ is $N$ . Prove that line segment $AN$ bisects line segment $DE$ .

Ann claims that a random three-digit number is more likely to contain a digit of 6 than a random five-digit number. Bill says it is the other way around. Who is right?

Prove that $3^{n}+5^{n}$ is not a perfect square for any positive integer $n$ .

Consider circles $k_1$ with center $O_1$ and radius $r_1$, and $k_2$ with center $O_2$ and radius $r_2$. Line $PA$ is tangent to $k_1$ at $A$ and line $PD$ is tangent to $k_2$ at $D$ . Segment $AD$ intersects $k_1$ and $k_2$ at $B$ and $C$ , respectively. Find $PA/PD$ in terms of $r_1$ and $r_2$ if $AB=CD$ .

Prove that $\sqrt{2x \sqrt{(2x+1) \sqrt{(2x+2) \sqrt{2x+3}}}} < \frac{15x+6}{8}$ for all $x>0$.

Lines $AD$ , $BD$ and $CD$ passing through an interior point $D$ of a triangle $ABC$ intersect the opposite sides at $A_{1}$, $B_{1}$ and $C_{1}$, respectively. The midpoints of the segments $A_1B_1$, $B_1C_1$ and $C_1A_1$ are $C_2$, $A_2$ and $B_2$, respectively. Show that the lines $AA_{2}$, $BB_{2}$ and $CC_{2}$ are concurrent.

In a triangle $ABC$ , the escribed circles $k_A$ and $k_B$ drawn to sides $CB$ and $CA$ touch the appropriate sides at $D$ and $E$ , respectively. Show that line $DE$ cuts out equal chords from the circles $k_A$ and $k_B$.

In the interior of a unit cube, there are some spheres with a total surface area of 2015. Show that $a)$ there exists a line that intersects at least 500 spheres, $b)$ there exists a plane that intersects at least 600 spheres.

Let $A_k$ denote the number that consists of $k$ ones in decimal notation. How many positive integers are there that cannot be obtained as the sum of the digits of any multiple of $A_k$?

What is the minimum number of different lines determined by the sides of a (not necessarily convex) 2015-sided polygon?

Solve the equation $5(2n+1)(2n+3)(2n+5) =\overline{ababab}$ , where $n$ denotes a positive integer, $a$ and $b$ stand for different digits, and $\overline{ababab}$ is a six-digit number.

Given a convex angle in the plane and a point $S$ in its interior, determine the line (e.g. by providing a method of construction) that forms a triangle with the vertex of the given angle such that the centroid of the triangle is the given point.

Prove that for all real numbers $a$ and $b$ , $a^4+b^4+2\ge 4ab$ .

What is the minimum number of sides of a regular polygon in which the radius of the circumscribed circle is at most 1.1 times the radius of the inscribed circle?

There are a few lines and a few circles drawn on a sheet of paper. Every two of them intersect each other, but no three pass through the same point. What are the numbers of lines and of circles if the number of intersection points is 75?

Consider circumscribed circle of a right-angled triangle. Draw the semicircle containing the triangle, and draw tangents to it, parallel to the legs. The parallels together with the line of the hypotenuse form a similar right-angled triangle. Find the angles of the triangle if the area of the outer triangle is 6 times the area of the inner triangle.

The sum of an arithmetic progression of 100 terms is 838 without the 68th term, and 849 without the 13th term. What are the values of the terms left out?

Prove that $3^{4n}+4\cdot 7^{4k}$ is divisible by 5 for all $n,k\in \mathbb{N}$ .

The 482 employees of a company are taking a team building trip. They can use vehicles seating 4, 19 or 21 people. How many of each kind of vehicle should be used, provided that each vehicle needs to be full? Find all possible solutions.

Determine the largest five-digit positive integer $\overline{abcde}$ that is divisible by each of the numbers $\overline{bcde}$ , $\overline{cde}$ , $\overline{de}$ and $e$ .

$X$ , $Y$ , $Z$ , $V$ are interior points of sides $AB$ , $BC$ , $CD$ , $DA$ of a parallelogram $ABCD$ , respectively, such that $\frac{AX}{XB} =\frac{BY}{YC} =\frac{CZ}{ZD} =\frac{DV}{VA}=k$ , where $k$ is a positive constant less than $\frac 12$. Find the value of $k$ , given that the area of quadrilateral $XYZV$ is 68% of the area of parallelogram $ABCD$ .

Depending on the value of $r^2$, how many solutions do the simultaneous equations $x^2 + y^2 =r^2$, $|x| + |y| =2$ have?

Determine the value of $n$ , given that $\binom n1$, $\binom n2$ and $\binom n3$ are three consecutive terms of an increasing arithmetic progression.

Determine all squares that have vertex $A$ at the point $(4;-2)$, vertex $B$ on the circle $b\colon {(x-5)}^2+ {(y-2)}^2=25$, and vertex $D$ on the line $d\colon 4x+3y+2=0$.

Given that the natural numbers $m$ and $n$ are relatively primes, prove that the greatest common factor of $m+n$ and $m^2+n^2$ is either 1 or 2.

Let $M$ denote the intersection of the lines of the legs of a trapezium. On a line passing through $M$ and parallel to the bases, let $A$ and $B$ denote the intersections with the extensions of the diagonals. Prove that $|AM|=|BM|$.

How many solutions does the equation $2^{a}+3^{b}+4^{c}+5^{d}+6^{e}=22$ have where $a$ , $b$ , $c$ , $d$ , $e$ are integers?

The longer base, $AB$ , of trapezium $ABCD$ is not greater than three times the base $CD$ . Lines $e$ and $f$ are parallel to the legs $BC$ and $DA$ , respectively, and each of them halves the area of the trapezium. Let $P$ and $Q$ denote the intersections of $AB$ with $e$ and $f$ , respectively, and let $P'$ and $Q'$ denote their intersections with $DC$ . $a)$ Prove that the intersection $M$ of lines $e$ and $f$ lies on the midline of the trapezium. $b)$ Given that the quadrilateral $PQ'P'Q$ is a parallelogram, find the ratio of the area of triangle $MPQ$ to the area of the trapezium $ABCD$ .

Five cards are drawn at random from a deck of German cards. Which of the following events has a higher probability: the event that all five are from the same suit, or the event that there are four cards among them with the same denomination?

The length of the radius of the inscribed circle of an isosceles triangle is divided by the length of the radius of the circumscribed circle. What is the maximum possible value of the ratio?

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

$H$ is the point lying closer to vertex $B$ that divides side $AB$ of parallelogram $ABCD$ in a 1 to 2 ratio. $F$ is the midpoint of side $BC$ . In what ratio does the intersection of line segments $AF$ and $DH$ divide them?

We have 5 five-forint coins, 10 ten-forint coins and 20 twenty-forint coins (HUF, Hungarian currency). In how many different ways can we pay 500 forints with these coins?

Solve the equation $2xy+2x-5y=40$ on the set of pairs of integers $(x;y)$.

Find the point on the $x$ -axis where the line segment formed by points $A(2;4)$ and $B(6;1)$ subtends the greatest angle.

Solve the equation $\big(3\sqrt{3}\,\big)^n- \big(2\sqrt{2}\,\big)^n =2^n+3^n+\sqrt{6}^{ n}$ on the set of positive integers.