Srednjoeuropska matematička olimpijada 2018

MEMO 2018 pojedinačno problem 1

Let $\mathbb{Q}^+$ denote the set of all positive rational numbers and let $\alpha \in \mathbb{Q}^+$ . Determine all functions $f : \mathbb{Q}^+ \rightarrow (\alpha, +\infty)$ such that $f\bigg(\frac{x+y}{\alpha}\bigg) = \frac{f(x)+f(y)}{\alpha}$ for all $x, y \in \mathbb{Q}^+$

MEMO 2018 pojedinačno problem 2

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

$Attachment #1$

Consider a $2018 \times 2018$ board consisting of $2018^2$ cells, each being an 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).

MEMO 2018 pojedinačno problem 3

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 triangles $ADB$ and $ADC$ respectively. Let $P$ be a point on the line segment $\overline{BC}$ such that the points $P, Q, R, D$ are concyclic.

Prove that the line $AP, BQ, CR$ are concurrent.

MEMO 2018 pojedinačno problem 4

a) Prove that for very positive integer $m$ there exists an integer $n \geq m$ such that $\bigg\lfloor\frac{n}{1}\bigg\rfloor\cdot\bigg\lfloor\frac{n}{2}\bigg\rfloor\cdot\ldots\cdot\bigg\lfloor\frac{n}{m}\bigg\rfloor = \binom{n}{m}$

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

MEMO 2018 ekipno problem 1

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

MEMO 2018 ekipno problem 2

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

MEMO 2018 ekipno problem 3

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 have been killed.

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

MEMO 2018 ekipno problem 4

Let $n$ be a positive integer and let $u_1, u_2, \ldots u_n$ be positive integers not larger than $2^k$ for some integer $k \geq 3$ . A $\emph{representation}$ of a non-negative integer $t$ is a sequence of non-negative integers $a_1, a_2, \ldots, a_n$ such that $t = a_1u_1 + a_2u_2 + \ldots + a_nu_n$ 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.

MEMO 2018 ekipno problem 5

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 $AB$ , $AC$ respectively, so that points $B'$ , $C'$ and $D$ are collinear and points $B$ , $C$ , $B'$ , $C'$ are concyclic with center $O$ . Prove that if $M$ is the midpoint of $\overline{BC}$ and $H$ is the orthocenter of $ABC$ , then $DHMO$ is a parallelogram.

MEMO 2018 ekipno problem 6

Let $ABC$ be a triangle. The internal bisector of $ABC$ intersects the side $\overline{AC}$ 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 $T$ . Prove $|AW| = |WT|$ .

MEMO 2018 ekipno problem 7

Let $a_1, a_2, \ldots$ be a sequence of positive integers such that $a_1 = 1 \text{ and } a_{k+1} = a_k^3+1, \forall k \in\mathbb{N}$ Prove that for every prime number $p$ of the form $3t+2$ , where $t$ is a non-negative integer, there exists a positive integer $n$ such that $a_n$ is divisible by $p$ .