Srednjoeuropska matematička olimpijada 2015

Zadaci

MEMO 2015 pojedinačno problem 1

Find all surjective functions $f : \mathbb{N} \to \mathbb{N}$ such that for all positive integers $a$ and $b$ , exactly one of the following equations is true: $f(a) = f(b),$ $f(a + b) = \min\{f(a), f(b)\}$ .

Remarks: $\mathbb{N}$ denotes the set of all positive integers. A function $f : X \to Y$ is said to be surjective if for every $y \in Y$ there exists $x \in X$ such that $f(x) = y$ .

MEMO 2015 pojedinačno problem 2

Let $n \geqslant 3$ be an integer. An inner diagonal of a simple n-gon is a diagonal that is contained in the $n$ -gon. Denote by $D(P)$ the number of all inner diagonals of a simple $n$ -gon $P$ and by $D(n)$ the least possible value of $D(Q)$ , where $Q$ is a simple $n$ -gon. Prove that no two inner diagonals of $P$ intersect (except possibly at a common endpoint) if and only if $D(P) = D(n)$ .

Remark: A simple $n$ -gon is a non-self-intersecting polygon with $n$ vertices. A polygon is not necessarily convex.

MEMO 2015 pojedinačno problem 3

Let $ABCD$ be a cyclic quadrilateral. Let $E$ be the intersection of lines parallel to $AC$ and $BD$ passing through points $B$ and $A$ , respectively. The lines $EC$ and $ED$ intersect the circumcircle of $AEB$ again at $F$ and $G$ , respectively. Prove that points $C$ , $D$ , $F$ , and $G$ lie on a circle.

MEMO 2015 pojedinačno problem 4

Find all pairs of positive integers $(m, n)$ for which there exist relatively prime integers $a$ and $b$ greater than $1$ such that $\frac{a^m + b^m}{a^n+b^n}$ is an integer.

MEMO 2015 ekipno problem 1

Prove that for all positive real numbers $a, b, c$ such that $abc = 1$ the following inequality holds: $\frac{a}{2b + c^2} + \frac{b}{2c + a^2} + \frac{c}{2a + b^2} \leqslant \frac{a^2 + b^2 +c^2}{3}$

MEMO 2015 ekipno problem 2

Determine all functions $f : \mathbb{R}\backslash \{0\} \to \mathbb{R}\backslash \{0\}$ such that $f(x^2yf(x)) + f(1) = x^2f(x) + f(y)$ holds for all nonzero real numbers $x$ and $y$ .

MEMO 2015 ekipno problem 3

There are $n$ students standing in line in positions $1$ to $n$ . While the teacher looks away, some students change their positions. When the teacher looks back, they are standing in line again. If a student who was initially in position $i$ is now in position $j$ , we say the student moved for $|i - j|$ steps. Determine the maximal sum of steps of all students that they can achieve.

MEMO 2015 ekipno problem 4

Let $N$ be a positive integer. In each of the $N^2$ unit squares of an $N \times N$ board, one of the two diagonals is drawn. The drawn diagonals divide the $N \times N$ board into $K$ regions. For each $N$ , determine the smallest and the largest possible values of $K$ .

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MEMO 2015 ekipno problem 5

Let $ABC$ be an acute triangle with $|AB| > |AC|$ . Prove that there exists a point $D$ with the following property: whenever two distinct points $X$ and $Y$ lie in the interior of $ABC$ such that the points $B$ , $C$ , $X$ , and $Y$ lie on a circle and $|\angle{AXB}| - |\angle{ACB}| =|\angle{CYA}| - |\angle{CBA}|$ holds, the line $XY$ passes through $D$ .

MEMO 2015 ekipno problem 6

Let $I$ be the incentre of triangle $ABC$ with $|AB| > |AC|$ and let the line $AI$ intersect the side $BC$ at $D$ . Suppose that point $P$ lies on the segment $BC$ and satisfies $|PI| = |PD|$ . Further, let $J$ be the point obtained by reflecting $I$ over the perpendicular bisector of $BC$ , and let $Q$ be the other intersection of the circumcircles of the triangles $ABC$ and $APD$ . Prove that $|\angle{BAQ}| = |\angle{CAJ}|$ .