Determine all pairs $(f, g)$ of functions $f, g: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x, y \in \mathbb{R}$
- $f(x) \geq 0$,
- $f(x+g(y))=f(x)+f(y)+2 y g(x)-f(y-g(y))$.
David and Linus play the following game: David chooses a subset $Q$ of the set $\{1, \ldots, 2018\}$. Then Linus chooses a natural number $a_{1}$ and computes the numbers $a_{2}, \ldots, a_{2018}$ recursively, where $a_{n+1}$ is the product of the positive divisors of $a_{n}$.
Let $P$ be the set of natural numbers $k \in\{1, \ldots, 2018\}$ for which $a_{k}$ is a perfect square. Linus wins if $P=Q$. Otherwise David wins. Who has a winning strategy?
All cells of an $8 \times 8$ square are initially coloured white. In one move one may recolour all cells of a horizontal or vertical $1 \times 3$ rectangle (all white cells become black and all black cells become white). Is it possible that after a finite number of moves all cells are coloured black?
Let $a$, $b$ and $c$ be natural numbers. Find the smallest value that the following expression can take:
$$
\frac{a}{\operatorname{gcd}(a+b, a-c)}+\frac{b}{\operatorname{gcd}(b+c, b-a)}+\frac{c}{\operatorname{gcd}(c+a, c-b)}
$$
Find all natural numbers $n$ for which there is no triple of natural numbers $(a, b, c)$ such that the following equation holds:
$$
n=\frac{a \cdot \operatorname{lcm}(b, c)+b \cdot \operatorname{lcm}(c, a)+c \cdot \operatorname{lcm}(a, b)}{\operatorname{lcm}(a, b, c)}
$$
Let $D$ be a point in the interior of an acute triangle $A B C$ such that $\angle B A D = \angle D B C$ and $\angle D A C = \angle B C D$. Let $P$ be a point on the circumcircle of triangle $A D B$. Assume that $P$ lies outside triangle $A B C$. A line through $P$ meets the ray $B A$ at $X$ and the ray $C A$ at $Y$ such that $\angle X P B = \angle P D B$. Show that $B Y$ and $C X$ intersect on $A D$.
Show that there is no function $f: \mathbb{R}_{\gt 0} \rightarrow \mathbb{R}_{\gt 0}$ such that for all $x, y \in \mathbb{R}_{\gt 0}$:
$$
f(x f(x)+y f(y))=x y
$$
Let $k$ be the incircle of triangle $A B C$ with incentre $I$. The circle $k$ touches the sides $B C$, $C A$ and $A B$ at the points $D$, $E$ and $F$, respectively. Let $G$ be the intersection point of the line $A I$ and the circle $k$ that lies between $A$ and $I$. Assume that $B E$ and $F G$ are parallel. Show that $B D=E F$.
Let $n$ be a natural number. Let $k$ be the number of ways to write $n$ as a sum of one or more consecutive natural numbers. Show that $k$ equals the number of odd positive divisors of $n$.
Let $a$, $b$, $c$, $d$ and $e$ be positive real numbers. Determine the largest value that the following expression can take:
$$
\frac{a b+b c+c d+d e}{2 a^{2}+b^{2}+2 c^{2}+d^{2}+2 e^{2}}
$$
Let $n$ be a natural number and $G$ the set of points $(x, y)$ in the plane such that $x$ and $y$ are integers with $1 \leq x, y \leq n$. A subset of $G$ is called parallelogram-free if it does not contain four non-collinear points that are the vertices of a parallelogram. What is the largest number of elements a parallelogram-free subset of $G$ can contain?
Let $p \geq 2$ be a prime number. Louis and Arnaud take turns choosing an index $i \in \{0,1, \ldots, p-1\}$ that has not yet been chosen, together with a digit $a_{i} \in \{0,1, \ldots, 9\}$. Louis starts. When all indices have been chosen, they compute the following sum:
$$
a_{0}+a_{1} \cdot 10+\ldots+a_{p-1} \cdot 10^{p-1}=\sum_{i=0}^{p-1} a_{i} \cdot 10^{i}
$$
If this sum is divisible by $p$, Louis wins; otherwise Arnaud wins. Show that Louis has a winning strategy.
Let $ABC$ be a triangle and let $\gamma = \angle ACB$ be such that $\gamma / 2 \lt \angle BAC$ and $\gamma / 2 \lt \angle CBA$. Let $D$ be the point on the segment $BC$ such that $\angle BAD = \gamma / 2$. Let $E$ be the point on the segment $CA$ such that $\angle EBA = \gamma / 2$. Furthermore, let $F$ be the intersection point of the bisector of $\angle ACB$ and the segment $AB$. Show that $EF + FD = AB$.
SMO-Land has $1111$ inhabitants. The eleven players of the Liechtenstein national team hand out autographs to all inhabitants, where no inhabitant receives the same autograph twice (i.e. each inhabitant receives from each player either no autograph or one autograph).
a. How many possibilities are there for which autographs an inhabitant can receive?
b. After the distribution, the inhabitants notice that no two of them have received autographs from exactly the same players. Show that there are two inhabitants who together possess exactly one autograph from each player.
A skyscraper has 7 lifts, but each of them stops at only 6 floors. Nevertheless, for any two floors there is always a lift that connects the two floors directly.
Show that the skyscraper can have at most 14 floors, and that such a skyscraper with 14 floors can actually be realised.
Let $n \geq 2$ be a natural number. Let $d_{1}, \ldots, d_{r}$ be all the distinct positive divisors of $n$ that are smaller than $n$ itself. Determine all $n$ for which:
$$
\operatorname{lcm}\left(d_{1}, \ldots, d_{r}\right) \neq n
$$
Let $m$ and $n$ be natural numbers and $p$ a prime number such that $m \lt n \lt p$. Furthermore:
$$
p \mid m^{2} + 1 \quad \text{ and } \quad p \mid n^{2} + 1
$$
Show that:
$$
p \mid m n - 1
$$
In the plane there are a triangle $\triangle ABC$ and a point $O$; $\Gamma$ is the circumcircle of $\triangle ABC$. Let the line $CO$ meet the line $AB$ at the point $D$, and let the line $BO$ meet the line $CA$ at the point $E$. Let the line $AO$ meet $\Gamma$ again at the point $F$. Let $I$ be the second intersection point of $\Gamma$ and the circumcircle of $\triangle ADE$, let $Y$ be the second intersection point of the line $BE$ with the circumcircle of $\triangle CEI$, and let $Z$ be the second intersection point of the line $CD$ with the circumcircle of $\triangle BDI$. The two tangents to $\Gamma$ at $B$ and $C$ meet at the point $T$. Let the line $TF$ meet $\Gamma$ again at the point $U$, and let $G$ be the reflection of $U$ in the line $BC$.
Prove that the six points $F, I, G, O, Y, Z$ are concyclic.
Let $a_1, a_2, \dots, a_n$ be positive integers ($a_1 \lt a_2 \lt \dots \lt a_n$), and let $k$ be a positive real number with $k \ge 1$.
Prove that:
$$
\sum_{i=1}^{n} a_i^{2k+1} \ge \left( \sum_{i=1}^{n} a_i^k \right)^2 .
$$
Let $A$, $B$, $C$ be the midpoints of the sides $B'C'$, $C'A'$, $A'B'$ of $\triangle A'B'C'$, respectively. Let the point $P$ lie inside $\triangle ABC$, and let $AP$, $BP$, $CP$ meet $BC$, $CA$, $AB$ at the points $P_a$, $P_b$, $P_c$, respectively.
The lines $P_aP_b$, $P_aP_c$ meet $B'C'$ at $R_b$, $R_c$, respectively; the lines $P_bP_c$, $P_bP_a$ meet $C'A'$ at $S_c$, $S_a$, respectively; the lines $P_cP_a$, $P_cP_b$ meet $A'B'$ at $T_a$, $T_b$, respectively. It is known that $S_c$, $S_a$, $T_a$, $T_b$ all lie on a circle with centre $O$.
Prove that $OR_b = OR_c$.