KöMaL 2018

We say that a sequence $a_1,a_2,\dots$ is expansive if for all positive integers $j$, $i<j$ implies $|a_i-a_j|\ge \frac1{j}$. Find all positive real numbers $C$ for which one can find an expansive sequence in the interval $[0,C]$.

Consider $n\ge 2$ pairwise disjoint disks $D_1,D_2,\dots,D_n$ on the Euclidean plane. For each $k=1,2,\dots,n$, denote by $f_k$ the inversion with respect to the boundary circle of $D_k$. (Here, $f_k$ is defined at every point of the plane, except for the center of $D_k$.) How many fixed points can the transformation $f_n\circ f_{n-1}\circ\cdots\circ f_1$ have, if it is defined on the largest possible subset of the plane?

Let $a$ and $b$ be positive integers. We tile a rectangle with dimensions $a$ and $b$ using squares whose side-length is a power of $2$, i.e. the tiling may include squares of dimensions $1\times 1$, $2\times 2$, $4\times 4$ etc. Denote by $M$ the minimal number of squares in such a tiling. Numbers $a$ and $b$ can be uniquely represented as the sum of distinct powers of $2$: $a=2^{a_1}+\cdots+2^{a_k}$, $b=2^{b_1}+\cdots +2^{b_\ell}$. Show that $M=\sum_{i=1}^k \;\sum_{j=1}^{\ell} 2^{|a_i-b_j|}.$

Let $ABC$ be a triangle and let $D$ be a point in the interior of the triangle which lies on the angle bisector of $\angle BAC$. Suppose that lines $BD$ and $AC$ meet at $E$, and that lines $CD$ and $AB$ meet at $F$. The circumcircle of $ABC$ intersects line $EF$ at points $P$ and $Q$. Show that if $O$ is the circumcenter of $DPQ$, then $OD$ is perpendicular to $BC$.

We say that a positive integer is lazy if it has no prime divisor greater than $3$. Prove that there are at most two lazy numbers strictly between two consecutive square numbers.

Let $\mathbb{R}[x,y]$ denote the set of two-variable polynomials with real coefficients. We say that the pair $(a,b)$ is a zero of the polynomial $f\in\mathbb{R}[x,y]$ if $f(a,b)=0$. If polynomials $p,q\in\mathbb{R}[x,y]$ have infinitely many common zeros, does it follow that there exists a non-constant polynomial $r\in\mathbb{R}[x,y]$ which is a factor of both $p$ and $q$?

Let $ABC$ be a scalene triangle with circumcenter $O$ and incenter $I$. The $A$-excircle, $B$-excircle, and $C$-excircle of triangle $ABC$ touch $BC$, $CA$, and $AB$ at points $A_1$, $B_1$, and $C_1$, respectively. Let $P$ be the orthocenter of $AB_1C_1$ and $H$ be the orthocenter of $ABC$. Show that if $M$ is the midpoint of $PA_1$, then lines $HM$ and $OI$ are parallel.

We call a positive integer lively if it has a prime divisor greater than $10^{10^{100}}$. Prove that if $S$ is an infinite set of lively positive integers, then it has an infinite subset $T$ with the property that the sum of the elements in any finite nonempty subset of $T$ is a lively number.

Let $n\ge 2$ be a positive integer, and suppose $a_1,a_2,\dots,a_n$ are positive real numbers whose sum is $1$ and whose squares add up to $S$. Prove that if $b_i=\frac{a_i^2}{S}$ ($i=1,\dots,n$), then for every $r>0$, we have $\sum_{i=1}^n \frac{a_i}{{(1-a_i)}^r}\le \sum_{i=1}^n \frac{b_i}{{(1-b_i)}^r}.$

The Hawking Space Agency operates $n-1$ space flights between the $n$ habitable planets of the Local Galaxy Cluster. Each flight has a fixed price which is the same in both directions, and we know that using these flights, we can travel from any habitable planet to any habitable planet. In the headquarters of the Agency, there is a clearly visible board on a wall, with a portrait, containing all the pairs of different habitable planets with the total price of the cheapest possible sequence of flights connecting them. Suppose that these prices are precisely $1,2,\dots,\binom n2$ monetary units in some order. Prove that $n$ or $n-2$ is a square number.

Let $f\colon \mathbb{R}\to \mathbb{R}$ be a continuous function such that the limit $g(x)=\lim_{h\to 0}\frac{f(x+h)-2f(x)+f(x-h)}{h^2}$ exists for all real $x$. Prove that $g(x)$ is constant if and only if $f(x)$ is a polynomial function whose degree is at most $2$.

A sphere $\mathcal{G}$ lies within tetrahedron $ABCD$, touching faces $ABD$, $ACD$, and $BCD$, but having no point in common with plane $ABC$. Let $E$ be the point in the interior of the tetrahedron for which $\mathcal{G}$ touches planes $ABE$, $ACE$, and $BCE$ as well. Suppose the line $DE$ meets face $ABC$ at $F$, and let $L$ be the point of $\mathcal{G}$ nearest to plane $ABC$. Show that segment $FL$ passes through the centre of the inscribed sphere of tetrahedron $ABCE$.

Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f\colon \mathbb{R}^+\to \mathbb{R}^+$ satisfying the following equation for all $x,y\in\mathbb{R}^+$: $f\big(xy+f(y)^2\big) = f(x)f(y)+yf(y).$

In triangle $ABC$ with incenter $I$, line $AI$ intersects the circumcircle of $ABC$ at $S\ne A$. Let the reflection of $I$ with respect to $BC$ be $J$, and suppose that line $SJ$ intersects the circumcircle of $ABC$ for the second time at point $P\ne S$. Show that $AI=PI$.

For any finite sequence $(x_1,\ldots,x_n)$, denote by $N(x_1,\ldots,x_n)$ the number of ordered index pairs $(i,j)$ for which $1\le i<j\le n$ and $x_i=x_j$. Let $p$ be an odd prime, $1\le n<p$, and let $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ be arbitrary residue classes modulo $p$. Prove that there exists a permutation $\pi$ of the indices $1,2,\ldots,n$ for which $N\big(a_1+b_{\pi(1)},a_2+b_{\pi(2)},\ldots,a_n+b_{\pi(n)}\big) \le \min \big(N(a_1,a_2,\ldots,a_n), N(b_1,b_2,\ldots,b_n) \big).$

Floyd the flea makes jumps on the positive integers. On the first day he can jump to any positive integer. From then on, every day he jumps to another number that is not more than twice his previous day's place. a) Show that Floyd can make infinitely many jumps in such a way that he never arrives at any number with the same sum of decimal digits as at a previous place. b) Can the flea jump this way if we consider the sum of binary digits instead of decimal digits?

In a cyclic quadrilateral $ABCD$, the diagonals meet at point $E$, the midpoint of side $AB$ is $F$, and the feet of perpendiculars from $E$ to the lines $DA$, $AB$, and $BC$ are $P$, $Q$, and $R$, respectively. Prove that the points $P$, $Q$, $R$, and $F$ are concyclic.

Let $F_n$ be the $n$th Fibonacci number (${F_1=F_2=1}$, ${F_{n+1}=F_n+F_{n-1}}$). Construct infinitely many positive integers $n$ such that $n$ divides $F_{F_n}$ but $n$ does not divide $F_n$.

Let $G=(V,E)$ be a tree graph with $n$ vertices, and let $P$ be a set of $n$ points in the plane with no three points collinear. Is it true that for any choice of the graph $G$ and set $P$, we can embed $G$ in $P$, i.e., we can find a bijection $f\colon V\to P$ such that when we draw the line segment $\big[f(x),f(y)\big]$ for all $(x,y)\in E$, no two such segments intersect each other?

Does there exist an infinite sequence $a_1,a_2,\ldots$ of real numbers which is bounded, not periodic, and satisfies the recursion $a_{n+1}=a_{n-1}a_n+1$?

Circle $\omega$ lies in the interior of circle $\Omega$, on which a point $X$ moves. The tangents from $X$ to $\omega$ intersect $\Omega$ for the second time at points $A\ne X$ and $B\ne X$. Prove that the lines $AB$ are either all tangent to a fixed circle, or they all pass through a point.

For an arbitrary positive integer $m$, not divisible by $3$, consider the permutation $x\mapsto 3x\pmod{m}$ on the set $\{1,2,\ldots,m-1\}$. This permutation can be decomposed into disjoint cycles; for instance, for $m=10$ the cycles are $(1\mapsto3\mapsto9\mapsto7\mapsto1)$, $(2\mapsto6\mapsto8\mapsto4\mapsto2)$ and $(5\mapsto5)$. For which integers $m$ is the number of cycles odd?

For any function $f\colon [0,1]\to[0,1]$, let $P_n(f)$ denote the number of fixed points of the function $\underbrace{f\big(\ldots f}_n(x)\ldots\big),$ i.e., the number of points $x\in[0,1]$ satisfying $\underbrace{f\big(\ldots f}_{n}(x)\ldots\big)=x$. Construct a piecewise linear, continuous, surjective function $f\colon [0,1]\to[0, 1]$ such that for a suitable number ${2<A<3}$, the sequence $\frac{P_n(f)}{A^n}$ converges.

Let $P$ be a point in the plane of triangle $ABC$. Denote the reflections of $A$, $B$, $C$ about $P$ by $A'$, $B'$ and $C'$, respectively. Let $A''$, $B''$, $C''$ be the reflections of $A'$, $B'$, $C'$ over the lines $BC$, $CA$ and $AB$, respectively. Let the line $A''B''$ intersect $AC$ at $A_b$ and let $A''C''$ intersect $AB$ at a point $A_c$. Denote by $\omega_A$ the circle through the points $A$, $A_b$, $A_c$. The circles $\omega_B$, $\omega_C$ are defined similarly. Prove that $\omega_A$, $\omega_B$, $\omega_C$ are coaxial, i.e., they share a common radical axis.

$100$ points are given in space such that no four of them lie in the same plane. Consider those convex polyhedra with five vertices that have all vertices from the given set. Prove that the number of such polyhedra is even.

(Our original problem in December was wrong. Here we post a new one. You can submit it together with the three problems in January.) Consider the following sequence: $a_1=1$, $a_2=2$, $a_3=3$, and $a_{n+3}=\frac{a_{n+1}^2+a_{n+2}^2-2}{a_n}$ for all integers $n\ge 1$. Prove that every term of the sequence is a positive integer.

Let $a_1,a_2,\ldots$ be a sequence of real numbers from the interval $[0,1]$. Prove that there is a sequence $1\le n_1<n_2<\ldots$ of positive integers such that $A=\lim_{\substack{i, j\to \infty\\i\ne j}} a_{n_i+n_j}$ exists, i.e., for every real number $\varepsilon>0$ there is a $N_\varepsilon$ such that $\big|a_{n_i+n_j}-A\big| < \varepsilon$ is satisfied for any pair of distinct indices $i,j>N_\varepsilon$.

Let $n$ and $k$ denote positive integers. Prove that given $n+k$ integers it is always possible to select at least $(k+1)$ numbers out of them such that their sum is divisible by $n$.

Find the integer solutions of the following simultaneous equations: $3x-y^{2} =\frac{z}{2},$ $3y+x^{2} =\frac{3z}{2}.$

The interior angle bisector drawn from vertex $A$ of triangle $ABC$ intersects side $BC$ at $E$, and the interior angle bisector drawn from vertex $B$ intersects side $AC$ at $F$. Let $O$ denote the centre of the inscribed circle of the triangle. What may be the size of the angle at $C$ if the sum of the areas of $\triangle OFA$ and $\triangle OBE$ equals the area of $\triangle AOB$?

Consider the perpendicular lines drawn from the centres of the escribed circles of a triangle to the corresponding sides. Prove that the three lines are concurrent.

Show that if the mean of the non-negative real numbers $a_1,a_2,\dots,a_{2017}$ is 1, then the following inequality holds: $\frac{a_1}{a_1^{2018} + a_2 + a_3 + \ldots +a_{2017}} + \frac{a_2}{a_2^{2018} + a_3 + a_4 + \ldots +a_{2017} + a_1} + \ldots +$ $+ \frac{a_{2017}}{a_{2017}^{2018} + a_1 + a_2 + \ldots +a_{2016}} \le 1.$

In an acute-angled triangle $ABC$, the feet of the altitudes drawn from $B$ and from $C$ are $D$ and $E$, respectively. The reflections of point $E$ in the lines $AC$ and $BC$ are $S$ and $T$, respectively. The circle $CST$, centred at $O$, intersects line $AC$ again at point $X\ne C$. Show that lines $XO$ and $DE$ are perpendicular.

Let $A$ and $B$ be finite sets of vectors, and let $A+B=\{\mathbf{v}+\mathbf{w} \mid \mathbf{v}\in A$, $\mathbf{w} \in B\}$. Show that $|A+B|\ge|A|+|B|-1$.

The trunk of an ever-growing tree forks in two at a height of one foot. In the following, the term branch will refer to a section between two joints, with no further joint along its length. Every branch of the ever-growing tree is straight, and terminates one foot higher than its lower end. The branches starting from the upper end of the branch are considered the children of the branch, also called the siblings of each other. Every branch of the tree has at least two children. If a branch does not have exactly two children then it has a sibling with exactly two children. Siblings always have different numbers of children. If a branch has more than two children then it has a sibling with one fewer children. How many branches start from joints at a height of $n$ feet?

The planes of an ellipse $\mathcal E$ and a hyperbola $\mathcal H$ in the space are perpendicular. The foci of $\mathcal E$ are the endpoints of the real axis of $\mathcal H$, and the foci of $\mathcal H$ are the endpoints of the major axis of $\mathcal E$. Let $A$ be $B$ two fixed points on different branches of hyperbola $\mathcal H$, and let $P$ be an arbitrary point of the ellipse. Prove that the sum of the distances $PA$ and $PB$ is independent of the choice of $P$.

Every inhabitant of a village belongs to one of three religious faiths: they either worship the Sun God, the Moon God or the Earth God. Regulations of these faiths require that a shrine should have the minimum possible total distance from all the houses of the village (whatever the faith of those living in the houses). Given that the worshippers of the Sun God already have a shrine in the village, and those of the Moon God have one, too, show that the worshippers of the Earth God can also build a shrine for themselves. (The village lies on flat terrain, and the shrines and the houses of the village can be considered pointlike.)

Prove that if $a$, $b$, $c$ are the sides of a triangle then $\frac{a^2(b+c)+b^2(a+c)}{abc}>3.$

The Great Bestiary of Wonderland features a dragon for every week of the year. All dragons have different ages. The youngest dragon, named Aloysius has 13 heads. The second youngest one, Bartholomeus has 14 heads, ... (and so on, each dragon in the order of their ages has one more head than the previous one). The oldest dragon, Zebulon has 64 heads. Wonderland monks are writing the Giant Codex of Dragon Tales. A tale may only be included in the Codex if the total number of heads of all the dragons in the tale is exactly 1001. For every pair of tales, the sets of dragons mentioned in the tales are different. Which of the 13-headed Aloysius and the 14-headed Bartholomeus will appear in more tales when the monks are finished with writing down all possible tales?

Find the area of a regular triangle of maximum perimeter inscribed in a unit square.

For any positive integers $n$ and $k$, let $f(n,k)$ denote the number of unit squares cut in two by a diagonal of an $n\times k$ lattice rectangle. How many number pairs $n$, $k$ are there such that $n\ge k$, and $f(n,k)=2018$?

Let $AB$ be a fixed chord that is not a diameter in a circle $k$. The midpoint of $AB$ is $F$. Let $P$ be a point on the circle $k$, different from $A$ and $B$. Let the line $PF$ intersect circle $k$ again at $X$, and let $Y$ be the reflection of $X$ in the perpendicular bisector of $AB$. Prove that there exists a point in the plane that lies on the line $PY$ for all $P$.

In the plane, a set of lattice quadrilaterals with the following property is selected: however the lattice points are coloured with a finite number of colours, there will always be a selected quadrilateral whose vertices all have the same colour. Prove that there are infinitely many selected lattice quadrilaterals, no two of which have a vertex in common.

It is known that it is possible to draw the complete graph with $7$ vertices on the surface of a torus (see the Császár polyhedron, for example). 7 points are marked on the side of a mug. We want to connect each pair of points with a curve, so that the curves have no interior points in common. What minimum number of these curves need to lead across the handle of the mug?

Show that a convex 2018-sided polygon cannot be dissected into triangles in which the angles in degrees are all integers.

What may be the value of the sum $x+y+z$ if $x^4 + 4y^4 + 16 z^4 + 64 = 32xyz?$

The centre $O$ of the circumscribed circle of an acute-angled triangle $ABC$ is reflected in the feet of the altitudes. Prove that the circle formed by the three reflections has the same radius as the circumscribed circle of the triangle.

The one hundred mathematicians participating in an international combinatorial conference were all housed in the same hotel. The receptionist was originally planning to place them in the order of their arrival in the rooms numbered 1 to 100. However, he forgot to give that instruction to the guest arriving first, who has thus chosen a room at random. So the receptionist instructed all the other guests to take the room with their number in the order of arrivals, or, if that room has already been taken, to select any other room they like. How many possible arrangements of the guests in the rooms are there?

There is an ant at each corner of a given face of a rectangular brick. Each ant wants to get to the opposite vertex of the cuboid, that is, to the other endpoint of the space diagonal drawn from his vertex of the cuboid. Is it possible for the ants to crawl to the opposite vertices along the surface of the brick, so that they follow the shortest possible paths and their paths do not intersect?

Let $t$ denote the area of (some) triangle of maximum area inscribed in a convex plane figure $\mathcal{S}$, and let $T$ denote the area of (some) triangle of minimum area circumscribed about $\mathcal{S}$. What is the maximum of the ratio $\frac{T}{t}$?

Find all positive integers $n$ for which $1\cdot 2^0+2\cdot 2^1+ 3\cdot 2^2+ \ldots + n\cdot 2^{n-1}$ is a perfect square.

Let $f(x)$ be a polynomial of real coefficients such that $f(k)$ is an integer for every positive integer $k$ that ends in $5$ or $8$ in decimal notation. $a)$ Prove that $f(0)$ is an integer. $b)$ Give an example of a polynomial $f(x)$ that meets the above conditions, but $f(1)$ is not an integer.

Prove that there is exactly one way of dissecting a cube into five tetrahedra. (Two dissections are not considered different if the resulting pieces are congruent.)

The positive integer $n$ is said to be chunky if it has a prime factor greater than $\sqrt{n}$. For example, $2017$ (a prime number), $2018=2\cdot 1009$ and $2022=2\cdot3\cdot 337$ are chunky, while $2023=7\cdot 17^2$ is not. How many chunky numbers are there which only have prime factors less than 30?

The feet of the altitudes drawn from vertices $B$ and $C$ of an acute-angled triangle $ABC$ are $D$ and $E$, respectively. Let $P$ be an interior point of $AD$, and let $Q$ be an interior point of $AE$ such that $EDPQ$ is a cyclic quadrilateral. Show that the line segments $BP$ and $CQ$ intersect on the median drawn from $A$.

Let $F_n$ denote the $n$th Fibonacci number ($F_1=F_2=1$, $F_{n+2}= F_{n+1}+F_n$), and define the sequence $a_0,a_1,a_2,\dots$ with the following recurrence relation: let $a_0=2018$, and for all $k\ge 0$ let $a_{k+1}=a_k+F_n$, where $F_n$ is the largest Fibonacci number less than $a_k$. Will there be any Fibonacci number in the sequence $(a_k)$?

The elements of a set $V$ are $n$-dimensional vectors (ordered $n$-tuples of numbers) of which each coordinate is $-1$, $0$ or $1$. No three different vectors of $V$ add up to the zero vector. Show that $|V|\le 2\cdot 3^{n-1}$.

Is it possible to dissect a cube with a finite number of straight cuts so that the pieces can be put together to form two smaller congruent cubes?

Prove that $\ln n+\sqrt{\frac12}+\sqrt{\frac23}+\ldots +\sqrt{\frac{n-1}{n}}<\sqrt2+\sqrt{\frac32}+\sqrt{\frac43}+\ldots +\sqrt{\frac{n}{n-1}}\,$ for all integers $n>1$.

Line $\ell$ passes through vertex $A$ of a triangle $ABC$, and it is parallel to $BC$. Let $\ell$ intersect the interior angle bisectors of angles $ABC$ and $ACB$ at $K$ and $L$, respectively. The inscribed circle touches $BC$ at point $D$. Show that the circumscribed circle intersects the Thales circle of line segment $KL$ at two points, and these two points are collinear with $D$.

Let $n$ be a positive integer. What is the largest possible number of ordered triples of non-negative integers $(x_1,y_1,z_1),(x_2,y_2,z_2),\ldots$ such that the following conditions hold: (1) For all $i$, $x_i + y_i + z_i = n$. (2) The numbers $x_1,x_2,\ldots$ are all different, the numbers $y_1,y_2,\ldots$ are all different, and the numbers $z_1,z_2,\ldots$ are also all different. Give an example for such a sequence of maximum length with the required property.

A transformation of central similitude is applied to a tetrahedron $ABCD$ with each vertex as centre. The four diminished tetrahedra obtained are $AA_bA_cA_d$, $B_aBB_cB_d$, $C_aC_bCC_d$ and $D_aD_bD_cD$. Given that these small tetrahedra are pairwise disjoint, prove that the volumes of tetrahedra ${A_b B_c C_d D_a}$, ${A_b B_d D_c C_a}$, ${A_c C_b B_d D_a}$, ${A_c C_d D_b B_a}$, ${A_d D_b B_c C_a}$ and ${A_d D_c C_b B_a}$ are equal.

A set with positive integer elements is said to be jolly good if it does not contain a pair of numbers whose difference is 2. How many jolly good subsets does the set $\{1, 2, 3,\ldots, 10\}$ have?

The sides of a triangle are $a$, $b$, $c$, the radius of the inscribed circle is $r$, and the radius of the circumscribed circle is $R$. Prove that if $a+b+c =\frac{4}{rR} \text{and} \sqrt{ab}+\sqrt{bc}+\sqrt{ca} =6,$ then $R=2r$.

Barnaby has $n$ marbles in his pocket. When he performs a somersault, each marble has a probability of $0<p<1$ to fall out of his pocket, independently of one another. If at least one marble falls out during a somersault then Barnaby will stop performing somersaults. Otherwise he will continue. Given that the probability of having an even number of marbles in his pocket when he stops doing somersaults is 50%, what may be the value of $n$?

The intersection of three unit circles is bounded by the arcs $\widehat{AB}$, $\widehat{AC}$ and $\widehat{BC}$. The perimeter of the intersection is $K$. Calculate the perimeter of the intersection of the unit circles centred at $A$, $B$ and $C$.

Let $n$ be a positive integer. Solve the following simultaneous equations on the set of real numbers: $a_1^2 + a_1 - 1 = a_2$ $a_2^2 + a_2 - 1 = a_3$ $\vdots$ $a_n^2 + a_n - 1 = a_1.$

Let $r_a$ be the radius of the largest escribed circle of a triangle, and let $R$ denote the radius of the circumscribed circle. Prove that $r_a \ge \frac32 R$.

Is it true that if the functions $f,g\colon \mathbb{R}\to[0,1]$ are periodic and the function $f+g$ is also periodic then they have a period in common?

For a vector $\mathbf x\ne \mathbf 0$, let $\mathbf e_{\mathbf x}=\frac{\mathbf x} {|\mathbf x|}$. A given plane $\mathcal{S}$ is parallel, but not identical to the plane of a given (non-degenerate) triangle $ABC$. Show that there exists a unique $P\in \mathcal{S}$, such that the vector $\mathbf e_{\overrightarrow {PA}}+\mathbf e_{\overrightarrow {PB}}+\mathbf e_{\overrightarrow {PC}}$ is perpendicular to $\mathcal{S}$.

Find the nineteenth positive integer in which the sum of the digits is 2018.

Let $P$ be an interior point of $\triangle ABC$, and the midpoint of sides $AB$, $BC$ and $CA$ are $C_1$, $A_1$ and $B_1$, respectively. Through the points $A_1$, $B_1$ and $C_1$, draw parallels to the lines $AP$, $BP$ and $CP$, respectively. Show that these three lines are concurrent.

Solve the following simultaneous equations on the set of positive real numbers: $$\begin{align*} \frac{1}{1+a + ab + abc} + \frac{1}{1+b + bc + bcd} + \frac{1}{1+c + cd + cda} + \frac{1}{1+d + da + dab} & = 1,\\ a+b+c+d & = 4. \end{align*}$$

The sides of a rectangle $T$ are $a\le b$. Given that the union of two appropriate disks of radius $r$ covers $T$ but this is impossible with two disks of radii smaller than $r$, determine the length $r$.

A line $e$ in the plane separates two given points $A$ and $B$ of the plane. On the line $e$, select points $P$ and $Q$ such that $\angle PAQ=90^{\circ}$. Show that there exists a point different from $B$ that lies on all circles drawn through $B$, $P$ and $Q$, independently of the choice of $P$ and $Q$.

For which primes $p$ is there a positive integer $a$ such that $1+a+a^2+\cdots+a^{p-1}$ is divisible by $p^2$?

Let $a_1, a_2, \ldots, a_{2018}$ denote non-negative real numbers that add up to $1$. Find the largest possible value of the sum $S = \sum_{i \ne j, i\mid j} a_ia_j$.

At least how many numbers should be selected out of $1, 2,\ldots, 10$ so that we can be certain that every such selection will contain a set of numbers whose sum is divisible by 11?

Given a point $P$ and four pairwise different lines $e\parallel f$ and $g\parallel h$, construct a line through $P$ that intersects the lines $e$, $f$, $g$, $h$, respectively at points $E$, $F$, $G$, $H$ such that $EF=GH$.

Let $A=\{-4;-3;-2;-1;0;1;2;3;4\}$. First and Second take turns in selecting a number (not selected before) out of the elements of set $A$. The player first collecting three numbers that add up to zero wins the game. Is there a player who has a winning strategy?

Prove that the orthocentre of the triangle formed by the points of tangency of the incircle on the sides of a right-angled triangle lies on the altitude drawn to the hypotenuse.

Let $n\ge 3$ be an integer and let $\alpha$ denote an arbitrary real number. Prove that $\sum_{k=0}^{n-1} \cos^2 \left(\alpha+\frac{2k\pi}{n}\right) = \frac n2.$

In an acute angled triangle $ABC$, $D$ and $E$ are interior points of the sides $AB$ and $AC$, respectively. The line segments $BE$ and $CD$ meet at $F$. Prove that if $BC^2=BD\cdot BA+CE\cdot CA$ then the quadrilateral $ADFE$ is cyclic.

Let $n>3$ be a positive integer, and let $a_1, a_2, \ldots, a_n$ be positive real numbers. Prove that $1 < \frac{a_1}{a_n+a_1+a_2} + \frac{a_2}{a_1+a_2+a_3} + \ldots + \frac{a_n}{a_{n-1}+a_n+a_1} < \left[ \frac{n}{2} \right]$ where the left-hand side of the inequality cannot be replaced by a larger number, and the right-hand side cannot be replaced by a smaller number. ($[x]$ denotes the greatest integer not greater than the number $x$.)

The area of the orthogonal projection of a unit cube onto the plane $xy$ is $A$, and the length of its orthogonal projection onto the $z$-axis is $a$. Prove that $A = a$.

The diagonals $AC$ and $BD$ of a convex kite $ABCD$ intersect at point $E$ such that $AE<CE$. The midpoint of diagonal $AC$ is $F$. The circles $ABE$ and $CDE$ intersect again at $M$. Show that $\angle EMF=90^\circ$.

Find the real solutions of the equation $x^2+2x-3-\sqrt{\frac{x^2+2x-3}{x^2-2x-3}}=\frac{2}{x^2-2x-3}.$

Prove that for any positive integer $x$, there exists a positive integer $y$ such that $x^3+y^3+1$ is divisible by the number $x+y+1$. Is there a positive integer $x$ for which there are infinitely many $y$ with this property?

Given that any three out of four lines determine a triangle, prove that the orthocentres of the four triangles are concurrent.

Consider the 64 points of the space for which each of the three coordinates is 1, 2, 3 or 4. Kate and Peter are playing a three-dimensional tic-tac-toe game on this set of points. Kate starts the game by selecting any point and colouring it blue. In the second step, Peter selects a different point and colours it red. Then they take turns by selecting further points and colouring them in blue or red. Whoever first completes a collinear set of four points of their own colour will win the game. Show that it makes no difference for Kate whether she starts by colouring the point $(1,1,2)$ or the point $(2,2,1)$ blue in the first step.

The circumcentre of an acute-angled scalene triangle $ABC$ is $O$, its orthocentre is $M$, the foot of the altitude drawn from vertex $A$ is $D$, and the midpoint of side $AB$ is $F$. The ray drawn from $F$ through $M$ intersects the circumcircle of triangle $ABC$ at $G$. $a)$ Prove that the points $A$, $F$, $D$ and $G$ are concyclic. $b)$ Let $K$ denote the circle in $a)$, and let $E$ be the midpoint of line segment $CM$. Prove that $EK=OK$.

In an $(m+2)\times(n+2)$ table, we cut out the four $1\times1$ corners''. Arbitrary real numbers are written in each field of the first and last rows, and in the first and last columns of the truncated table obtained in this way. Prove that it is possible to fill in the remaining $m\times n$ interior'' of the table in a unique way with real numbers such that every number is the arithmetic mean of the four adjacent numbers.

The midpoints of sides $BC$, $CA$ and $AB$ of a triangle $ABC$ are $D$, $E$ and $F$, respectively. Let $S$ denote the centroid of the triangle. Assume that the perimeters of triangles $AFS$, $BDS$ are $CES$ equal. Show that triangle $ABC$ is equilateral.

Let $n$ denote a natural number greater than one. Let $d(n)$ denote the number of positive divisors of $n$, and let the sum of the divisors be $\sigma(n)$. Show that $\sigma(n) > d(n)\sqrt n\,$.

Arthur and Belle take turns in colouring the edges of a cube red. In each step, they colour an edge that is skew to the previously coloured edge. Artur starts. The player who cannot do an appropriate coluring will lose the game. Which player has a winning strategy?

To start, each of the numbers $1,2,\ldots, n$ is coloured either red or blue. In every move, three distinct numbers in arithmetic progression are selected, and the colour of each of the three numbers is changed. For what values of $n$ is it true that however the numbers $1,2,\ldots, n$ are coloured initially, it is possible to turn all of them red by applying an appropriate sequence of such steps?

A square is drawn over each of legs $BC$ and $CA$ of a right-angled triangle $ABC$. Let $D$ and $E$ denote the vertices of the squares that lie opposite to $C$. Show that the circumscribed circle of triangle $ABC$ passes through the midpoint of line segment $DE$.

Prove that if the cubic equation $x^3+Ax^2+Bx+C=0$ has three distinct positive roots for some values of the real parameters $A$, $B$ and $C$, then $A^2+B^2+18C>0$.

Let $D$, $E$ and $F$, respectively, denote the midpoints of sides $BC$, $CA$ and $AB$ of a right-angled but not isosceles triangle $ABC$, let $O$ be the centre of the circumscribed circle of the triangle, and let $M$ be the orthocentre. The tangent drawn to the circumscribed circle at $A$ intersects line $EF$ at $P$, and the tangent drawn at point $B$ intersects line $FD$ at $Q$. Show that lines $PQ$ and $OM$ are perpendicular.

Given a line segment and one point that divides it $1:2$, construct the other point dividing the line segment $1:2$ by only using a straight edge.

Consider the following sequence $p_n(x)$ of polynomials with integer coefficients: let $p_0(x)=0$, $p_1(x)=1$, and for all $n\ge 2$ let $p_n(x)=p_{n-1}(x)+x\cdot p_{n-2}(x).$ Prove that if a polynomial $f(x)$ divides the polynomials $p_n(x)$ and $p_m(x)$ for some positive integers $n$, $m$ then it also divides the polynomial $p_{(m,n)}(x)$. ($(n,m)$ denotes the greatest common divisor of $n$ and $m$. A polynomial $P(x)$ divides a polynomial $Q(x)$ if there exists a polynomial $R(x)$ of real coefficients for which $Q(x)=P(x)R(x)$.)

The currency used on a distant island consists of coins of unusual denominations. The basic units are three different one-digit numbers, and there are their multiples, too: ten times, a hundred times, and also a thousand times their value. The price of one kilo of coconut may be paid with two identical coins plus a third coin of different value. In order to pay for a kilo of passion fruit, which costs twice as much, the third coin needs to be replaced by the coin with 10 times its value. Given that no coin has a denomination of 1 and the largest denomination is 7000, what other coins are used on the island?

Prove that no perfect square can be represented in the form $3^a+9^b+1$ ($a$, $b$ are positive integers).

An isosceles right-angled triangle inscribed in a circle is rotated through 45 degrees about the centre of the circle. Find the perimeter and area of the intersection of the two triangles.

Solve the following equation on the set of real numbers: $\sqrt{x+11} + \sqrt{x^2+11x} -\sqrt{x} -x=4.$

Reflect the parabola $y=x^2$ about the point $F\left(0,\frac 14\right)$. At what angle do the two parabolas intersect?

The operation $\circ$ is defined on positive integers. Given that $i)$ $1 \circ 1=3$; $ii)$ $a\circ b=b\circ a$ for all $a$, $b$; $iii)$ $a \circ (b+1) = a\circ b + (a+1) + 2b$ for all $a$, $b$, determine the value of $2017\circ 2018$.

The first term of an arithmetic sequence is $a_1=3$, and its common difference is 9. Prove that for every natural number $k$, the number $3\cdot 4^k$ occurs among the terms.

$M$ is an interior point of a regular triangle $ABC$. The feet of the perpendiculars dropped from $M$ onto the sides $AB$, $BC$ and $CA$ are $H$, $K$ and $P$, respectively. Prove that $(i)$ ${|AH|}^2+{|BK|}^2+{|CP|}^2= {|HB|}^2+{|KC|}^2+{|PA|}^2$; $(ii)$ $|AH|+|BK|+|CP|= |HB|+|KC|+|PA|$.