KöMaL 2016

For which $n$ are there polynomials $g(x)$ and $h(x)$ with real coefficients and degrees smaller than $n$ such that $g\big(h(x)\big)= x^n+x^{n-1}+x^{n-2}+\dots +x^2+x+1$?

The circle $\omega$ is inscribed in the quadrilateral $ABCD$ . The incenters of the triangles $ABC$ and $ACD$ are $I$ and $J$ , respectively. Let $T$ and $U$ be those two points on the arcs of $\omega$, lying in the triangles $ABC$ and $ACD$ , respectively, for which the circles $ATC$ and $AUC$ are tangent to $\omega$. Show that the line segments $AC$ , $IU$ and $JT$ are concurrent.

Let $K$ be a fixed positive integer. Let $(a_0,a_1,\dots)$ be the sequence of real numbers that satisfies $a_0=-1$ and $\sum_{\substack{i_0,i_1,\dots,i_K\ge0 \\ i_0+i_1+\dots+i_K=n}} \frac{a_{i_1}\cdot\dots\cdot a_{i_K}}{i_0+1} =0$ for every positive integer $n$ . Show that $a_n>0$ for $n\ge1$.

The points $A_1$, $A_2$, $A_3$, $A_4$, $B_1$, $B_2$, $B_3$, $B_4$ lie on a parabola in this order. For every pair $(i,j)$ with $1\le i,j\le4$ and $i\ne j$ , let $r_{ij}$ denote the ratio in which the line $A_jB_j$ divides the segment $A_iB_i$. (That is, if $A_iB_i$ and $A_jB_j$ meet at $X$ then $r_{ij}=\frac{A_iX}{XB_i}$.) Show that if two of the numbers $r_{12} \cdot r_{21} \cdot r_{34} \cdot r_{43}$, $r_{13} \cdot r_{31} \cdot r_{24} \cdot r_{42}$ and $r_{14} \cdot r_{41} \cdot r_{23} \cdot r_{32}$ coincide then the third one is also equal to them.

There are given two positive integers: $k$ and $\ell$. A square with horizontal and vertical sides is divided into finitely many rectangles by line segments such that the following statements are satisfied: $(i)$ every horizontal or vertical line of the plane contains at most one of the segments; $(ii)$ no two segments cross each other in their interiors; $(iii)$ every horizontal line, intersecting the square but not containing any of the segments, intersects exactly $k$ rectangles; $(iv)$ every vertical line, intersecting the square but not containing any of the segments, intersects exactly $\ell$ rectangles. What can be the number of rectangles?

Let $a_1<a_2<\ldots<a_n$ be an arithmetic progression of positive integers. Prove that $[a_1,a_2,\ldots,a_n] \ge [1,2,\ldots,n]$. (The symbol $[\ldots]$ stands for the least common multiple.)

Let $a_1,a_2,\ldots,a_n$ be distinct positive integers. Show that $3\sum_{i=1}^{n}a_i^5+\bigg(\sum_{i=1}^{n}a_i\bigg)^{2} \ge 4\bigg(\sum_{i=1}^{n}a_i^3\bigg)\bigg(\sum_{i=1}^{n}a_i\bigg).$

Let $p$ be a prime, let $k$ be a positive integer, and let $\mathcal{A}$ be a finite set of integers with at least $p^k$ elements. Denote by $N_{\text{even}}$ the number of subsets of $\mathcal{A}$ with even cardinality and sum of elements being divisible by $p^k$. Similarly, denote by $N_{\text{odd}}$ the number of subsets of $\mathcal{A}$ with odd cardinality and sum of elements being divisible by $p^k$. Show that $N_{\text{even}}\equiv N_{\text{odd}} \pmod{p}$ .

On the circumcircle of the scalene triangle $ABC$ , let $A_0$, $B_0$, and $C_0$ be the midpoints of the arcs $BAC$ , $CBA$ and $ACB$ , respectively. Denote by $A_1$, $B_1$ and $C_1$ the Feuerbach points of the triangles $AB_0C_0$, $BC_0A_0$ and $CA_0B_0$, respectively. Show that the triangles $A_0B_0C_0$ and $A_1B_1C_1$ are similar.

There is given a positive integer $k$ , some distinct points $A_1,A_2,\ldots,A_{2k+1}$ and $O$ in the plane, and a line $\ell$ passing through $O$ . For every $i=1,\ldots,2k+1$, let $B_i$ be the reflection of $A_i$ about $\ell$, and let the lines $OB_i$ and $A_{i+k}A_{i+k+1}$ meet $C_i$. (The indices are considered modulo $2k+1$: $A_{2k+2}=A_1$, $A_{2k+3}=A_2$, ..., and it is assumed that these intersections occur.) Show that if the points $C_1,C_2,\ldots,C_{2k}$ lie on a line then that line passes through $C_{2k+1}$ also.

Determine whether the set of rational numbers can be ordered to in a sequence $q_1,q_2,\ldots$ in such a way that there is no sequence of indices $1\le i_1<i_2<\dots<i_6$ such that $q_{i_1},q_{i_2},\ldots,q_{i_6}$ form an arithmetic progression.

Let $a_1,a_2,\ldots$ be a sequence of nonnegative integers such that $\sum_{i=1}^{2n} a_{id} \le n$ holds for every pair $(n,d)$ of positive integers. Prove that for every positive integer $K$ , there are some positive integers $N$ and $D$ such that $\sum_{i=1}^{2N} a_{iD} = N-K.$

Prove that $0< \sum_{i=0}^k {(-1)}^{i}\binom{n+1}{i}{(k+1-i)}^n < n!$ holds for every pair $0<k<n$ of integers.

Point $O$ is the apex of an oblique circular cone. Show that there are some points $F_1$ and $F_2$ in the interior of the base such that $\angle XOF_1 +\angle XOF_2$ is constant when $X$ runs along the perimeter of the base disk.

We have colour pearls placed on an $n\times n$ board; a square may contain more than one pearl. Altogether we used $2n-1$ colours and $n$ pearls from each colour. The pearls are arranged in such a way that no row or column contains more than one pearl of the same colour. Prove that it is possible to select $n$ pearls with distinct colours such that no two of them are in the same row or column.

Let $a_1, \ldots, a_n>1$ be integers satisfying the inequality $\sum_{i=1}^{n} \frac{1}{a_{i}+1} \le \frac{7}{12}.$ Every year, the government of Optimistica publishes its Annual Report with $n$ economic indicators. For each $i=1,\ldots,n$, the possible values of the $i$-th indicator are $1, 2, \ldots, a_i$. The Annual Report is said to be optimistic if at least $n - 1$ indicators have higher values than in the previous report. Prove that the government can publish optimistic Annual Reports in an infinitely long sequence.

Let $r(x)$ be a polynomial with real coefficients whose degree $n$ is odd. Prove that the number of pairs of polynomials $p(x)$ and $q(x)$ with real coefficients satisfying the equation $\big(p(x)\big)^3 + q(x^2) = r(x)$, is smaller than $2^n$.

Let $\mathcal{K}$ be a circle with diameter $OI$. Construct a one-to-one correspondence $f \colon \mathcal{K}\setminus \{O,I\}\to\mathbb{R}\setminus\{0\}$ with the following property: for every four distinct points $A$, $B$, $C$, $D$ of $\mathcal{K}$, other than $O$ and $I$, $f(A)f(B)=f(C)f(D)$ is satisfied if and only if the lines $AB$, $CD$ and $OI$ are concurrent or parallel to each other.

Find all pairs $(p,q)$ of (positive) prime numbers for which $p^3+q^3+1$ is divisible by $pq$.

The convex polyhedron $\mathcal{K}$ has five vertices, $A$, $B$, $C$, $D$ and $E$. The line segment $DE$ intersects the plane of the triangle $ABC$ in the interior of the triangle. Show that $\mathcal{K}$ has an inscribed sphere - with being tangent to all six faces - if and only if the inscribed spheres of the terahedra $ABCD$ and $ABCE$ are tangent to each other.

Let $n = 2^{128}$, $M = \{1, 2, 3, 4\}$, and let $M^n$ denote the set of sequences of length $n$ and consisting of elements of $M$. Decide whether there exist functions $f_1,\ldots,f_n$ and $g_1,\ldots,g_n\colon M^n\to M$ such that for every pair of sequences $(x_1,\ldots,x_n),(y_1,\ldots,y_n)\in M^n,$ at least one of the following statements holds: $\bullet$ $f_i(y_1,\ldots,y_n) = x_i$ for some index $i$ with $1\le i\le n$; $\bullet$ $g_j(x_1,\ldots,x_n) = y_j$ for some index $j$ with $1\le j\le n$.

Let $M(x)$ be a real polynomial having no real root. Prove that for every real polynomial $P(x)$ there exists a real polynomial $Q(x)$ such that $P{(x)}^2+Q{(x)}^2$ is divisible by the polynomial $M(x)$.

The positive numbers $a_1,a_2,\ldots,a_n$ satisfy $\sqrt{a_1}+\sqrt{a_2} +\cdots+\sqrt{a_n}=1$. Show that ${a_1}^{a_1}\cdot {a_2}^{a_2}\cdot \ldots \cdot {a_n}^{a_n} \ge {(a_1+a_2+\ldots+a_n)}^2.$

Let $A_1B_1C_1D_1E_1F_1A_2B_2C_2D_2E_2F_2$ be a cyclic dodecagon such that its diagonals $A_1A_2$, $B_1B_2$ and $C_1C_2$ are concurrent, the diagonals $A_1A_2$, $E_1E_2$ and $F_1F_2$ are concurrent, the diagonals $C_1C_2$, $D_1D_2$ and $E_1E_2$ are concurrent, and finally the diagonals $B_1B_2$, $D_1D_2$ and $F_1F_2$ are concurrent.Let $k_A$, $k_C$ and $k_E$ be three circular arcs in the interior of the dodecagon's circumcircle such that $k_A$ connects $A_1$ and $A_2$, $k_C$ connects $C_1$ and $C_2$, and $k_E$ connects $E_1$ and $E_2$. Let the intersection points of the three arcs be $B=k_A\cap k_C$, $D=k_C\cap k_E$ and $F=k_A\cap k_E$. Prove that the circular arcs $B_1BB_2$, $D_1DD_2$ and $F_1FF_2$ are concurrent. (These arcs degenerate to line segments if their specified points are collinear.)

Let $K=(V,E)$ be a finite, simple, complete graph. Let $\phi\colon E\to\mathbb{R}^2$ be a map from the edge set to the plane, such that the preimage of any point in the range defines a connected graph on the entire vertex set $V$, and the points assigned to the edges of any triangle are collinear. Show that the range of $\phi$ is contained in a line.

Show that there are no rational numbers $x$ and $y$ such that $x - \frac1x + y - \frac1y = 4.$

Let $AB$ and $CD$ be two chords in a circle $\Omega$. Choose a circle $\omega$ tangent to the segments $AB$ and $CD$ at their interior points $M$ and $N$ respectively, and let $\omega$ intersect $\Omega$ at $P$ and $Q$. Suppose that a third circle $\omega'$, distinct from $\Omega$ and $\omega$, and also passing through $P$ and $Q$, meets the line $MN$ at $M'$ and $N'$. Prove that the points $A$, $B$, $C$, $D$, $M'$, and $N'$ lie on a conic section that is tangent to $\omega'$ at $M'$ and $N'$.

Four men and four women are sitting around a round table. Prove that there are four adjacent people such that there are the same number of men and women among them.

Let $n$ be an integer greater than 3. Prove that if each digit occurs exactly once in the base-$n$ representation of an integer then the number cannot be a prime.

In a simple graph, the degree of each vertex is four, and for each edge there exists exactly one vertex connected to both ends of the edge. What is the minimum number of vertices in such a graph?

Let $G$ be a simple undirected graph of $n$ vertices. Prove that it is possible to assign infinite sets $\mathcal{H}_1,\mathcal{H}_2, \dots, \mathcal{H}_n$ of natural numbers to the graph such that the intersection of two sets is infinite if the corresponding vertices are joined by an edge, and empty if there is no edge joining the vertices.

In the rectangular coordinate plane, consider all lines with equations of the form $aX+\frac{Y}{a}=2$, where $a$ is a real number. Determine the set of those points in the plane that do not lie on any of the lines.

In a cyclic quadrilateral $ABCD$ , the angle bisectors of angles $ADB\sphericalangle$ and $ACB\sphericalangle$ intersect side $AB$ at points $E$ and $F$ , and the bisectors of $CBD\sphericalangle$ and $CAD\sphericalangle$ intersect side $CD$ at $G$ and $H$ , respectively. Prove that the points $E$ , $F$ , $G$ , $H$ are concyclic.

The sequence $a_{1}, a_{2}, \dots$ is defined by the following recursion: $a_{1}=1$, $a_{2}=5$, $a_{3}=15$, and in the case of $n\ge 4$, $a_{n}=n^{2}+a_{n-1}+a_{n-2}-a_{n-3}$. $a)$ Calculate the sum $a_{1}+a_{2}+a_{3}+\dots +a_{2015}$. $b)$ Prove that $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}+ \dots +\frac{1}{a_{2015}}<\frac{4}{3}$.

Find those convex polyhedra in which for each vertex $C$ , the angles lying at $C$ on the faces meeting at $C$ add up to $180^\circ$.

Find all fractions in which the numerator and denominator are both two-digit numbers such that the second digit of the numerator equals the first digit of the denominator, and the value of the fraction stays the same if these identical digits are deleted.

In which triangles are the points dividing the sides 2:1 all concyclic?

What may be the last digit of a positive integer $n\ge 3$ if $n+n^{2} +\dots+n^{2n-3} - 4$ is a prime?

In an aeroplane, there are one hundred seats, booked by one hundred passengers, each having their assigned seat. However, the first passenger does not care, and sits down on a random seat. When the other passengers enter one by one, each of them tries to take his or her own seat, or, if that seat is already taken, selects another one at random. What is the probability that the hundredth passenger is able to take his own seat?

Is it true that if the four sides in two convex quadrilaterals are pairwise equal, and the two diagonals are also pairwise equal then the two quadrilaterals are congruent?

The centre of the inscribed circle of triangle $ABC$ is $O$ , and the centre of the circumscribed circle is $K$ . Prove that the vector $\frac{\overrightarrow{AB}}{AB}+ \frac{\overrightarrow{BC}}{BC}+ \frac{\overrightarrow{CA}}{CA}$ is perpendicular to line $OK$ .

The parabolas $p_1$ $\big(y=-x^2+b_1 x+c_1\big)$ and $p_2$ $\big(y=-x^2+b_2 x+c_2\big)$ are tangent to the parabola $p_3$ $\big(y=x^2+b_3x+c_3\big)$. Prove that the line connecting the points of tangency is parallel to the common tangent of $p_1$ and $p_2$.

Find those pairs $(n,k)$ of positive integers for which $\sum_{i=1}^{2k+1} {(-1)}^{i-1} a_{i}^{n}\ge \bigg(\sum_{i=1}^{2k+1} {(-1)}^{i-1} a_{i}\bigg)^{\! n}$ for all real numbers $a_1\ge a_2\ge \dots \ge a_{2k+1}\ge 0$.

Let $\mathcal{O}$ be a regular octahedron. How many different axes of rotation are there such that $\mathcal{O}$ is mapped onto itself by a rotation through an angle of at most $180^{\circ}$?

There are three species of creatures peopling the planet B-4777: the Alpha, the Beta and the Gamma. People of one species (not necessarily the Alpha) have 2 hands, people of another species have 3 hands, and people of the third species have 4 hands. People of one species (not necessarily those with 2 hands) have 4 fingers on each hand, people of another species have 5, and people of the third species have 6 hands on each hand. Every people represent numbers in a notation with a base equal to the total number of fingers on their hands. (For example, if those with 4 hands have 6 fingers on each, then they will use base-24 representation.) The number $64_{\alpha}$ expressed in the notation of the Alpha people coincides with the number $51_{\beta}$ expressed in the notation of the Beta people. How many hands and how many finger per hand do the Alpha, the Beta and the Gamma have?

Let $D$ denote an interior point of an acute-angled triangle $ABC$ . Construct the circles of diameters $AD$ , $BD$ and $CD$ , and draw a tangent from each of the points $A$ , $B$ and $C$ to each of the two circles not passing through it. Prove that the sum of the squares of the six tangents equals the sum of the squares of the sides of the triangle.

The two points dividing side $AB$ of triangle $ABC$ into three equal parts, and the two points dividing side $BC$ in a 1 to 3 ratio form a cyclic quadrilateral. The circumscribed circle of the quadrilateral is tangent to side $CA$ . Determine the angles of triangle $ABC$ .

For what natural numbers $n$ is it true that whenever the midpoints of the sides of a convex $n$ -sided polygon $\mathcal{K}_n$ form a regular $n$ -gon, then $\mathcal{K}_n$ itself is also a regular $n$ -gon?

The rows and the columns of an $n \times n$ chessboard are numbered 1 to $n$ , and a coin is placed on each field. The following game is played: A coin showing tails is selected. If it is in row $k$ and column $m$ , then every coin with row number at least $k$ and column number at least $m$ is turned over. This procedure is repeated. What is the least number $L(n)$ for which it is possible to achieve in at most $L(n)$ steps that all coins on the board show heads, whatever be the initial distribution of heads and tails?

Solve the following equation on the set of real numbers: $8^x+27^x+2\cdot30^x+54^x+60^x= 12^x+18^x+20^x+24^x+45^x+90^x.$

A flea is jumping on the vertices of a square $ABCD$ . It starts at vertex $A$ , and in each jump he moves to an adjacent vertex with a probability of $\frac{1}{2}$ for each. The flea will stop when it has reached all vertices at least once. For each vertex, determine the probability of its being the last vertex.

Prove that the following inequality is true for all real numbers $a$ , $b$ , $c$ : $2\big(a^4+b^4+c^4\big)+\frac{71+17\sqrt{17}}{2}\ge 4abc+ a^2b^2+c^2a^2+3b^2c^2.$

$\mathcal{G}$ is a given sphere in the space. For any line $e$ that has no common point with $\mathcal{G}$ , define the line $f$ as the conjugate of $e$ with respect to $\mathcal{G}$ if $f$ joins the points of tangency on the two planes tangent to $G$ passing through $e$ . Show that two lines of the space passing $\mathcal{G}$ are skew if and only if their conjugates with respect to $\mathcal{G}$ are skew.

Let $p$ and $q$ be positive integers. Prove that at least one of $p^2+q$ and $p+q^2$ is not a perfect square.

The area of a certain right-angled trapezium is equal to the half of the product of the legs. At which point of the perpendicular leg does the other leg subtend a maximum angle?

Solve the following simultaneous equations on the set of real numbers: $x^2+y^3 =x+1,$ $x^3+y^2 =y+1.$

The interior angle bisectors drawn from vertices $A$ and $B$ in a triangle $ABC$ intersect the circumscribed circle again at the points $G$ and $H$ , respectively. The points of tangency of the inscribed circle of triangle $ABC$ on sides $BC$ and $AC$ are $D$ and $E$ , respectively. Let $K$ denote the circumcentre of triangle $DCE$ . Show that the points $G$ , $H$ and $K$ are collinear.

In a scalene triangle $ABC$ , a Thales circle is drawn over each median. The Thales circles of the medians drawn from vertices $A$ , $B$ and $C$ intersect the circumscribed circle of the triangle $ABC$ again at points $A_1$, $B_1$, and $C_1$, respectively. Prove that the perpendiculars drawn to $AA_1$ at $A$ , to $BB_1$ at $B$ and to $CC_1$ at $C$ are concurrent.

Altitudes $AD$ and $CE$ of triangle $ABC$ intersect at point $M$ . Line $DE$ intersects the line of side $AC$ at $P$ . Prove that line $PM$ is perpendicular to the median drawn from vertex $B$ of the triangle.

Is it true that by deleting infinitely many appropriate digits out of the decimal representation of any positive irrational number, we can always get back the original number?

How many permutations do the numbers $1, 2, 3, \ldots, n$ have in which there are exactly $a)$ one, $b)$ two occurrences of a number being greater than the adjacent number on the right of it?

Given that at least three faces of a convex polyhedron are pentagons, what is the minimum number of faces the polyhedron may have?

The hour hand and the minute hand of a clock are identical, and it has no second hand. How many time instants are there after noon and before midnight when it is impossible to tell the time by looking at the clock?

Solve the following equation on the set of real numbers: $x^2-6\{x\}+1=0,$ where $\{x\}$ stands for the fractional part of a number $x$ (that is, the difference obtained when the largest integer not greater than $x$ is subtracted from $x$ ).

In triangle $ABC$ , $D$ , $E$ and $F$ are arbitrary interior points of sides $AB, BC$ and $CA$ , respectively. Let $G$ , $H$ and $I$ denote the centroids of triangles $ADF$ , $BED$ and $CFE$ , respectively. Furthermore, let $S$ , $K$ , $L$ be the centroids of triangles $ABC$ , $DEF$ and $GHI$ , respectively. Prove that the points $K$ , $L$ and $S$ are collinear.

In a cyclic quadrilateral $ABCD$ , diagonals $AC$ and $BD$ are perpendicular, and the centre of the circumscribed circle is $K$ . Prove that the areas of triangles $ABK$ and $CDK$ are equal.

Determine those pairs of non-negative integers $(a,b)$ for which $5^a+3^b$ is a perfect square.

$T$ is a point on line $BC$ , different from the midpoint of line segment $BC$ . Circle $k$ is centred at $T$ , and $A$ is its intersection with the perpendicular drawn to $BC$ at $T$ . The intersections of $k$ with the lines $AB$ and $AC$ are $K$ and $L$ , respectively. Let $k$ intersect the circumscribed circle of $ABC$ again at $M$ . Prove that the lines $KL$ , $AM$ and $BC$ are concurrent.

Define the sequence $f_n$ of functions by the following recurrence relation: $f_0(x) = f_1(x) = 1, \mathrm{~and ~for ~} n\ge 2 f_n(x) = f_{n-1}(x) \cdot 2\cos(2x) - f_{n-2}(x).$ Determine the number of roots of $f_n(x)$ in the interval $[0,\pi]$.

The inscribed sphere of a right circular cone $\mathcal{K}$ is $\mathcal{G}$ . The centres of the spheres $g_1, g_2, \dots, g_n$ of radius $r$ form a regular $n$ -gon of side $2r$ . Furthermore, each sphere $g_i$ is tangent to both the lateral surface and the base of $\mathcal{K}$ , and also tangent to $\mathcal{G}$ . What may be the value of $n$ ?

Is it possible to specify closed intervals of rational endpoints on the number line such that every rational number is the endpoint of exactly one interval, and $a)$ one of any pair of closed intervals contains the other; $b)$ no pair of two intervals are disjoint, but no interval contains another?

With his sword received from the queen of the lakes, Sir Robin wants to free the town of Agnor from a dragon. The dragon has three heads and three tails. With one slash, he may cut off one head, two heads, one tail or two tails. If he cuts off one head, the dragon will immediately grow three heads in its place. If he cuts off two heads, then nothing will grow in its place. If he cuts off one tail, there will be two tails growing immediately. Finally, if he cuts off two tails then the dragon will grow a head. What is the minimum number of cuts needed to kill the dragon (that is, in order that the dragon has no heads or tails remaining at all)?

Solve the simultaneous equations $x+y+z=2$, $xyz=2(xy+yz+zx)$ on the set of real numbers.

Given a circular disc $K$ and two points on it, separated by a distance larger than $2$ units. Show that $K$ contains a disc of unit radius that does not contain either of the two points.

A square is drawn over each side of a right-angled triangle $ABC$ of unit area, on the outside. Denote the centres of the squares by $D$, $E$ and $F$. Prove that the area of triangle $DEF$ is at least two.

A point $(a;b)$ of the coordinate plane, where $a$ and $b$ are positive integers, is coloured red. Then if a point $(x;y)$ is red then point $(x+1;y+1)$, as well as point $\left(\frac x2;\frac y2\right)$ are also coloured red, provided that $x$ and $y$ are both even. If $(x;y)$ and $(y;z)$ have both been coloured, then we will also colour the point $(x;z)$ red. What is the distance from the origin to the closest point coloured red?

Rose colours every point of the number line either blue or red. Violet will perceive a point as purple if however small a neighbourhood of the point is considered, there exist both a red point and a blue point in it. Is it possible that the numbers Violet sees purple $a)$ cover the whole number line, $b)$ are exactly the integers, $c)$ are exactly the rational numbers?

Two angles of triangle $ABC$ are $BAC\sphericalangle=15^{\circ}$ and $ABC\sphericalangle=30^{\circ}$. The perpendicular drawn to side $AC$ at point $C$ intersects line segment $AB$ at $D$, and the perpendicular bisector of $AB$ intersects line $CD$ at $E$. Extend line segment $AB$ beyond $B$ by the length of line segment $BC$ to get $G$. Prove that the points $B$, $G$, $E$, $C$ lie on a circle of diameter $\sqrt{2} \cdot AB$.

Prove that $\frac1{a_1} + \frac1{[a_1,a_2]} + \frac1{[a_1,a_2,a_3]} + \ldots + \frac1{[a_1,a_2,\ldots,a_n]} < 2,$ for all integers $0<a_1<a_2<\ldots<a_n$, where the symbol $[\dots]$ stands for the least common multiple.

Let $C_1$, $A_1$ and $B_1$ denote the feet of the altitudes drawn to sides $AB$, $BC$ and $CA$ of an acute-angled triangle $ABC$, respectively. The perpendicular projections of the orthocentre of triangle $ABC$ on the lines $A_1B_1$, $B_1C_1$, $C_1A_1$ are $C_2$, $A_2$ and $B_2$, respectively. Prove that the lines $AA_2$, $BB_2$ and $CC_2$ are concurrent, and their intersection lies on the Euler line of triangle $ABC$.

Given a prime $p$, find the integer solutions of the equation $\left|\frac{2}{x}-\frac{1}{y}\right|=\frac{1}{p}.$

Let $P$ be a given point in the interior of a sphere. Consider the pairwise perpendicular planes $S_1$, $S_2$ and $S_3$ that intersect at $P$. Show that the sum of the areas of the circles cut out of the planes by the sphere is independent of the choice of the planes $S_1$, $S_2$ and $S_3$.

The binary operation $\circ$ is defined on the set of integers. Given that $a\circ(b+c)=b\circ a+ c\circ a.$ for all integers $a$, $b$ and $c$, show that there exists an integer $k$ such that $a\circ b= k\cdot a \cdot b$ for all $a$ and $b$.

Let $O$ denote the midpoint of line segment $AB$, and let $C$ be the point on line segment $OB$ for which $\frac{AO}{OC}=\frac{OC}{CB}$. Point $D$ is one of the points where the perpendicular drawn at point $C$ to $AB$ intersects the Thales circle of $AB$. Points $E$ and $F$ are the two intersections of line $AD$ with the circle of radius $OC$ centred at $O$. Determine the size of the angle $EOF$.

Solve the following equation in the set of real numbers: $x + y + z = xyz = 8,$ $\frac 1x- \frac 1y- \frac 1z = \frac 18.$

The intersection of diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ is $M$. The angle bisectors of the angles $\angle CAD$ and $\angle ACB$ intersect the circumscribed circle of the cyclic quadrilateral $ABCD$ at points $E$ and $F$, respectively. Prove that line $EF$ is perpendicular to the angle bisector of the angle $AMD$.

Prove that if $0<x<\frac{\pi}{2}$, then ${(\tg x)}^{\sin x}+ {(\ctg x)}^{\cos x}\ge 2.$ For what values of $x$ does the equality hold?

Four lattice points in a lattice of regular triangles of unit side form the vertices of a parallelogram $\mathcal{P}$. The area of $\mathcal{P}$ is $\sqrt{3}$ units. What may be the sum of the lattice segments in the interior of $\mathcal{P}$?

Does there exist an integer $a\ge 1$ such that $x^2+3$ and ${(x+a)}^2+3$ are relatively primes for all positive integers $x$?

Find all natural numbers $p$ for which the equation $x^2+(p+2015)x +2016p+1=0$ has an integer solution.

In an acute-angled triangle $ABC$ ($AB < AC$), the foot of the altitude from vertex $A$ is $D$, and the centre of the circumscribed circle is $O$. Prove that if the exterior angle bisector of $BAC\sphericalangle$ is parallel to $OD$, then the diagonals of the quadrilateral $AODC$ are equal.

Let $x$ and $y$ be real numbers for which $xy=1$. Determine the largest and smallest values of the expression $K=\frac{{(x+y)}^2-(x-y)-2}{{(x+y)}^2+(x-y)-2}.$

The circles $a$ and $b$ intersect at two distinct points $P$ and $Q$. Suppose that $A_0$ is a point that lies on circle $a$. Define the points $B_0$, $A_1$, $B_1$, $A_2$ etc., respectively, as follows: $B_i$ is the intersection of line $A_iP$ with $b$, and then $A_{i+1}$ is the intersection of line $B_iQ$ with $a$. (If $A_i = P$ then line $A_iP$ means the tangent of circle $a$ at $P$, and if $B_i = Q$ then line $B_iQ$ means the tangent of circle $b$ at $Q$.) Show that if there exists an $A_n$ ($n\ge 2$) for which $A_0$ and $A_n$ coincide, then this holds for any point $A_0$ of circle $a$.

Suppose that $p$ is a polynomial of integer coefficients, not identically 0, such that $p(n)$ is divisible by 2016 for all integers $n$. What is the minimum possible value of the sum of the absolute values of the coefficients of $p$?

Let $0<x_1<x_2<\ldots<x_n<2\pi$. Show that $\sum_{i,j=1; i\ne j}^n \frac{1}{|x_i-x_j|}+\frac{1}{2\pi-|x_i-x_j|}\ge \frac{n^2}{\pi}\sum_{k=1}^{n-1} \frac 1k.$ On what condition will the equality hold?

Cover the surface of the unit sphere with great circles, such that every point is contained in at most four great circles.

Solve the following equation in the set of positive integers: $n!=2^a+2^b.$

We have a ,,pocket calculator'' that can only be used for addition, subtraction and taking the reciprocal of a number. Starting with the number $\sqrt{20}+16$, is it possible to get 1 as a result? (During the calculation, we can store the original number, as well as any intermediate result in separate memory caches, and we may access them as many times as we wish.)

Let $a$, $b$, and $c$ denote some positive integers. Prove that there exist two relatively prime positive numbers $r$ and $s$ such that $ar+bs$ is divisible by $c$.

For a sector of a circle of radius $R$, smaller than a semicircle, the radius of the inscribed circle is $r$, and the length of the chord connecting the endpoints of the boundary arc of the sector is $2a$. Prove that $\frac 1r= \frac 1R+ \frac 1a.$

Let $K$ denote the centre of the circumscribed circle of an acute-angled triangle $ABC$. The line drawn through $K$, parallel to $AB$, intersects line $BC$ at $D$, and line $CA$ at $E$. Similarly, let the line drawn through $K$, parallel to $BC$ intersect line $CA$ at $F$, and line $AB$ at $G$. Prove that the circle of radius $EA$ centred at $E$, the circle of radius $FC$ centred at $F$, and the circumscribed circle of the triangle $ABC$ are concurrent.

Solve the following simultaneous equations: $x+y+z =3,$ $x^{2}+y^{2}+z^{2} =7,$ $x^{3}+y^{3}+z^{3} =15.$

In a parallelogram $ABCD$, $BC=\lambda AB$. The intersection of the interior angle bisectors drawn from $A$ and from $B$ intersect at $M$. What fraction of the parallelogram $ABCD$ is covered by the triangle $ABM$?

Find all functions $f\colon \mathbb{R}\to \mathbb{R}$ satisfying $(x+1)\cdot f(x+2)-2(x+2)\cdot f(-x-1)=3x^2+8x+3$.