Srednjoeuropska matematička olimpijada 2017

Zadaci

MEMO 2017 pojedinačno problem 1

Determine all functions $f : \mathbb{R} \to \mathbb{R}$ satisfying $f(x^2 + f(x)f(y)) = xf(x + y)$ for all real numbers $x$ and $y$ .

MEMO 2017 pojedinačno problem 2

Let $n \geqslant 3$ be an integer. A labelling of the $n$ vertices, the $n$ sides and the interior of a regular $n$ -gon by $2n + 1$ distinct integers is called memorable if the following conditions hold:

(a) Each side has a label that is the arithmetic mean of the labels of its endpoints.
(b) The interior of the $n$ -gon has a label that is the arithmetic mean of the labels of all the vertices.

Determine all integers $n \geqslant 3$ for which there exists a memorable labelling of a regular $n$ -gon consisting of $2n + 1$ consecutive integers.

MEMO 2017 pojedinačno problem 3

Let $ABCDE$ be a convex pentagon. Let $P$ be the intersection of the lines $CE$ and $BD$ . Assume that $\angle PAD = \angle ACB$ and $\angle CAP = \angle EDA$ .
Prove that the circumcentres of the triangles $ABC$ and $ADE$ are collinear with $P$ .

MEMO 2017 pojedinačno problem 4

Determine the smallest possible value of $|2^m - 181^n|$ where $m$ and $n$ are positive integers.

MEMO 2017 ekipno problem 1

Determine all pairs of polynomials $(P, Q)$ with real coefficients satisfying $P(x + Q(y)) = Q(x + P(y))$ for all real numbers $x$ and $y$ .

MEMO 2017 ekipno problem 2

Determine the smallest possible real constant $C$ such that the inequality $|x^3 + y^3 + z^3 + 1| \leqslant C|x^5 + y^5 + z^5 + 1|$ holds for all real numbers $x, y, z$ satisfying $x + y + z = -1$ .

MEMO 2017 ekipno problem 3

There is a lamp on each cell of a $2017 \times 2017$ board. Each lamp is either on or off. A lamp is called bad if it has an even number of neighbours that are on. What is the smallest possible number of bad lamps on such a board?
(Two lamps are neighbours if their respective cells share a side.)

MEMO 2017 ekipno problem 4

Let $n \geqslant 3$ be an integer. A sequence $P_1, P_2, \ldots, P_n$ of distinct points in the plane is called good if no three of them are collinear, the polyline $P_1P_2 \ldots P_n$ is non-self-intersecting and the triangle $P_iP_{i + 1}P_{i + 2}$ is oriented counterclockwise for every $i = 1, 2, \ldots, n - 2$ . For every integer $n \geqslant 3$ determine the greatest possible integer $k$ with the following property: there exist $n$ distinct points $A_1, A_2, \ldots, A_n$ in the plane for which there are $k$ distinct permutations $\sigma : \{1, 2, \ldots, n\} \to \{1, 2, \ldots, n\}$ such that $A_{\sigma(1)}, A_{\sigma(2)}, \ldots, A_{\sigma(n)}$ is good.
(A polyline $P_1P_2 \ldots P_n$ consists of the segments $P_1P_2, P_2P_3, \ldots, P_{n - 1}P_n$ .)

MEMO 2017 ekipno problem 5

Let $ABC$ be an acute-angled triangle with $AB > AC$ and circumcircle $\Gamma$ . Let $M$ be the midpoint of the shorter arc $BC$ of $\Gamma$ , and let $D$ be the intersection of the rays $AC$ and $BM$ . Let $E \neq C$ be the intersection of the internal bisector of the angle $ACB$ and the circumcircle of the triangle $BDC$ . Let us assume that $E$ is inside the triangle $ABC$ and there is an intersection $N$ of the line $DE$ and the circle $\Gamma$ such that $E$ is the midpoint of the segment $DN$ .
Show that $N$ is the midpoint of the segment $I_B I_C$ , where $I_B$ and $I_C$ are the excentres of $ABC$ opposite to $B$ and $C$ , respectively.

MEMO 2017 ekipno problem 6

Let $ABC$ be an acute-angled triangle with $AB \neq AC$ , circumcentre $O$ and circumcircle $\Gamma$ . Let the tangents to $\Gamma$ at $B$ and $C$ meet each other at $D$ , and let the line $AO$ intersect $BC$ at $E$ . Denote the midpoint of $BC$ by $M$ and let $AM$ meet $\Gamma$ again at $N \neq A$ . Finally, let $F \neq A$ be a point on $\Gamma$ such that $A, M, E$ and $F$ are concyclic.
Prove that $FN$ bisects the segment $MD$ .

MEMO 2017 ekipno problem 7

Determine all integers $n \geqslant 2$ such that there exists a permutation $x_0, x_1, \ldots, x_{n - 1}$ of the numbers $0, 1, \ldots, n - 1$ with the property that the $n$ numbers $x_0, \hspace{0.3cm} x_0 + x_1, \hspace{0.3cm} \ldots, \hspace{0.3cm} x_0 + x_1 + \ldots + x_{n - 1}$ are pairwise distinct modulo $n$ .

MEMO 2017 ekipno problem 8

For an integer $n \geqslant 3$ we define the sequence $\alpha_1, \alpha_2, \ldots, \alpha_k$ as the sequence of exponents in the prime factorization of $n! = p_1^{\alpha_1}p_2^{\alpha_2} \ldots p_k^{\alpha_k}$ , where $p_1 < p_2 < \ldots < p_k$ are primes. Determine all integers $n \geq 3$ for which $\alpha_1, \alpha_2, \ldots, \alpha_k$ is a geometric progression.