International Competitions 2018

Let $\Gamma$ be the circumcircle of acute-angled triangle $ABC$. Points $D$ and $E$ lie on segments $AB$ and $AC$, respectively, such that $AD = AE$. The perpendicular bisectors of $BD$ and $CE$ intersect the minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$, respectively. Prove that the lines $DE$ and $FG$ are parallel (or are the same line).

Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \ldots, a_{n+2}$, such that $a_{n+1} = a_1$ and $a_{n+2} = a_2$, and $$a_i a_{i+1} + 1 = a_{i+2}$$ for $i = 1, 2, \ldots, n$.

An anti-Pascal triangle is an equilateral triangular array of numbers such that, except for the numbers in the bottom row, each number is the absolute value of the difference of the two numbers immediately below it. For example, the following array is an anti-Pascal triangle with four rows which contains every integer from 1 to 10.

$$\begin{array}{ccccccc} & & & 4 & & & \\ & & 2 & & 6 & & \\ & 5 & & 7 & & 1 & \\ 8 & & 3 & & 10 & & 9 \end{array}$$

Does there exist an anti-Pascal triangle with 2018 rows which contains every integer from 1 to $1 + 2 + \cdots + 2018$?

A site is any point $(x, y)$ in the plane such that $x$ and $y$ are both positive integers less than or equal to 20.

Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance between any two sites occupied by red stones is not equal to $\sqrt{5}$. On his turn, Ben places a new blue stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from any other occupied site.) They stop as soon as a player cannot place a stone.

Find the greatest $K$ such that Amy can ensure that she places at least $K$ red stones, no matter how Ben places his blue stones.

Let $a_1, a_2, \ldots$ be an infinite sequence of positive integers. Suppose that there is an integer $N > 1$ such that, for each $n \geq N$, the number $$\frac{a_1}{a_2} + \frac{a_2}{a_3} + \cdots + \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}$$ is an integer. Prove that there is a positive integer $M$ such that $a_m = a_{m+1}$ for all $m \geq M$.

A convex quadrilateral $ABCD$ satisfies $AB \cdot CD = BC \cdot DA$. Point $X$ lies inside $ABCD$ so that $$\angle XAB = \angle XCD \quad \text{and} \quad \angle XBC = \angle XDA.$$ Prove that $\angle BXA + \angle DXC = 180°$.

Let $\mathbb{Q}^+$ denote the set of all positive rational numbers and let $\alpha \in \mathbb{Q}^+$. Determine all functions $f\colon \mathbb{Q}^{+}\to (\alpha , + \infty)$ satisfying

$$f \left(\frac {x + y}{\alpha}\right) = \frac {f (x) + f (y)}{\alpha}, \quad \text {for all} \, x, y \in \mathbb {Q} ^ {+}.$$

The two figures depicted below consisting of $6$ and $10$ unit squares, respectively, are called staircases.

Consider a $2018 \times 2018$ board consisting of $2018^2$ cells, each being a unit square. Two arbitrary cells were removed from the same row of the board. Prove that the rest of the board cannot be cut (along the cell borders) into staircases (possibly rotated).

Let $ABC$ be an acute-angled triangle with $AB < AC$, and let $D$ be the foot of its altitude from $A$. Let $R$ and $Q$ be the centroids of the triangles $ABD$ and $ACD$, respectively. Let $P$ be a point on the line segment $BC$ such that $P \neq D$ and the points $P, Q, R$ and $D$ are concyclic. Prove that the lines $AP, BQ$ and $CR$ are concurrent.

(a) Prove that for every positive integer $m$ there exists an integer $n \geq m$ such that

$$\left\lfloor \frac {n}{1} \right\rfloor \cdot \left\lfloor \frac {n}{2} \right\rfloor \cdots \left\lfloor \frac {n}{m} \right\rfloor = \binom {n} {m}. \tag{*}$$

(b) Denote by $p(m)$ the smallest integer $n \geq m$ such that the equation (*) holds. Prove that $p(2018) = p(2019)$.

Remark: For a real number $x$, we denote by $\lfloor x \rfloor$ the largest integer not larger than $x$.

Let $a$, $b$ and $c$ be positive real numbers satisfying $abc = 1$. Prove that $$\frac{a^2 - b^2}{a + bc} + \frac{b^2 - c^2}{b + ca} + \frac{c^2 - a^2}{c + ab} \leq a + b + c - 3.$$

Let $P(x)$ be a polynomial of degree $n \geq 2$ with rational coefficients such that $P(x)$ has $n$ pairwise different real roots forming an arithmetic progression. Prove that among the roots of $P(x)$ there are two that are also the roots of some polynomial of degree $2$ with rational coefficients.

A group of pirates had an argument and now each of them holds some other two at gunpoint. All the pirates are called one by one in some order. If the called pirate is still alive, he shoots both pirates he is aiming at (some of whom might already be dead). All shots are immediately lethal. After all the pirates have been called, it turns out that exactly $28$ pirates got killed.

Prove that if the pirates were called in whatever other order, at least $10$ pirates would have been killed anyway.

Let $n$ be a positive integer and $u_1, u_2, \ldots, u_n$ be positive integers not larger than $2^k$, for some integer $k \geq 3$. A representation of a non-negative integer $t$ is a sequence of non-negative integers $a_1, a_2, \ldots, a_n$ such that $$t = a_1 u_1 + a_2 u_2 + \cdots + a_n u_n.$$

Prove that if a non-negative integer $t$ has a representation, then it also has a representation where less than $2k$ of the numbers $a_1, a_2, \ldots, a_n$ are non-zero.

Let $ABC$ be an acute-angled triangle with $AB < AC$, and let $D$ be the foot of its altitude from $A$. Points $B'$ and $C'$ lie on the rays $AB$ and $AC$, respectively, so that points $B'$, $C'$ and $D$ are collinear and points $B$, $C$, $B'$ and $C'$ lie on one circle with center $O$. Prove that if $M$ is the midpoint of $BC$ and $H$ is the orthocenter of $ABC$, then $DHMO$ is a parallelogram.

Let $ABC$ be a triangle. The internal bisector of $\angle ABC$ intersects the side $AC$ at $L$ and the circumcircle of triangle $ABC$ again at $W \neq B$. Let $K$ be the perpendicular projection of $L$ onto $AW$. The circumcircle of triangle $BLC$ intersects line $CK$ again at $P \neq C$. Lines $BP$ and $AW$ meet at point $T$. Prove that $AW = WT$.

Let $a_1, a_2, a_3, \ldots$ be the sequence of positive integers such that $$a_1 = 1 \quad \text{and} \quad a_{k+1} = a_k^3 + 1, \text{ for all positive integers } k.$$

Prove that for every prime number $p$ of the form $3\ell + 2$, where $\ell$ is a non-negative integer, there exists a positive integer $n$ such that $a_n$ is divisible by $p$.

An integer $n$ is called Silesian if there exist positive integers $a$, $b$ and $c$ such that $$n = \frac{a^2 + b^2 + c^2}{ab + bc + ca}.$$

(a) Prove that there are infinitely many Silesian integers.

(b) Prove that not every positive integer is Silesian.