Let $ABC$ be an acute-angled triangle with $AB \lt AC$. Tangent to its circumcircle $\Omega$ at $A$ intersects the line $BC$ at $D$. Let $G$ be the centroid of $ABC$ and let $AG$ meet $\Omega$ again at $H \neq A$. Suppose the line $DG$ intersects the lines $AB$ and $AC$ at $E$ and $F$, respectively. Prove that $\angle EHG = \angle GHF$.
We consider 7 islands $A_{1}, \ldots, A_{7}$. We are allowed to build bridges, either between an island $A_{i}$ and the next island $A_{i+1}$ (for $i \in \{1,2, \ldots, 6\}$), or between an island $A_{i}$ and the last one $A_{7}$ (for $i \in \{1,2, \ldots, 6\}$). In how many ways can these constructions be carried out with as few bridges as possible so that one can travel from any island to any other?
Example for 3 islands instead of 7: the three possible constructions using two bridges are
1) a bridge between $A_{1}$ and $A_{2}$, and a bridge between $A_{1}$ and $A_{3}$
2) a bridge between $A_{1}$ and $A_{2}$, and a bridge between $A_{2}$ and $A_{3}$
3) a bridge between $A_{1}$ and $A_{3}$, and a bridge between $A_{2}$ and $A_{3}$.
Determine all functions $f$ from $\mathbb{R}$ to $\mathbb{R}$ such that, for all real $x$ and $y$, we have the equality
$$
f(x+y)=f(x-y)+f(f(1-x y))
$$
a) Find all integers $m \geqslant 1$ and $n \geqslant 1$ such that $\frac{5^{m}+2^{n+1}}{5^{m}-2^{n+1}}$ is the square of an integer.
b) More generally, find all integers $m \geqslant 1$ and $n \geqslant 1$, together with the primes $p$, such that $\frac{5^{m}+2^{n} p}{5^{m}-2^{n} p}$ is the square of an integer.
Let $I$ be the incenter of a triangle $ABC$. Let $D$ be the point diametrically opposite to $A$ on the circumcircle. Suppose that the point $E$ on the ray $[BA)$ and the point $F$ on the ray $[CA)$ satisfy the condition
$$
BE = CF = \frac{AB + BC + CA}{2}.
$$
Show that $(EF) \perp (DI)$.
Let $n \geq 5$ be an integer, and $E_{1}, E_{2}, \ldots, E_{2 n-1}$ be distinct two-element subsets of $\{1,2, \ldots, n\}$.
Prove that, among the $2 n-1$ subsets $E_{i}$, one can choose $n$ of them so that the union of these $n$ subsets contains at most $\frac{2}{3} n+1$ elements.
Let $ABC$ be a triangle. Denote by $P$ the reflection of $B$ in $(AC)$ and by $Q$ the reflection of $C$ in $(AB)$.
Let $T$ be the intersection of $(PQ)$ with the tangent at $A$ to the circumcircle of $(APQ)$.
Show that the reflection of $T$ in $A$ lies on $(BC)$.
Determine all integers $a\gt 0$ for which there exist positive integers $n, s, m_{1}, \cdots, m_{n}, k_{1}, \cdots, k_{s}$ such that
$$
\left(a^{m_{1}}-1\right) \cdots\left(a^{m_{n}}-1\right)=\left(a^{k_{1}}+1\right) \cdots\left(a^{k_{s}}+1\right)
$$
Prove that, for every integer $n \geq 2$, we have:
$$
\sum_{k=2}^{n} \frac{1}{\sqrt[k]{(2 k)!}} \geq \frac{n-1}{2 n+2}
$$
N.B. If $a\gt 0$, we denote by $\sqrt[k]{a}$ the unique real number $b\gt 0$ such that $b^{k}=a$.
Let $ABC$ be a non-right triangle such that $AB \lt AC$. Denote by $H$ the projection of $A$ onto $(BC)$, and by $E, F$ the projections of $H$ onto $(AB)$ and $(AC)$ respectively. The line $(EF)$ meets $(BC)$ at the point $D$. Consider the semicircle with diameter $[CD]$ lying in the same half-plane bounded by $(CD)$ as $A$. Let $K$ be the point of this semicircle that projects onto $B$. Show that $(DK)$ is tangent to the circle $KEF$.
Let $n \geq 1$ be an integer. A group of $2n$ people meets. Each of these people has at least $n$ friends in this group (in particular, if $A$ is a friend of $B$ then $B$ is a friend of $A$, and nobody is a friend of themselves). Prove that these $2n$ people can be arranged around a round table so that each of them sits between two of their friends.
Determine all positive integers $m$ with the following property:
The sequence $a_{0}, a_{1}, a_{2}, \ldots$ with $a_{0}=\frac{2m+1}{2}$ and $a_{k+1}=a_{k}\left\lfloor a_{k}\right\rfloor$ for $k=0,1,2, \ldots$ contains at least one integer.
Note: $\lfloor x\rfloor$ denotes the floor of $x$ (the greatest integer not exceeding $x$).
Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ that satisfy the equation
$$
f(x-f(y)) = f(f(x)) - f(y) - 1
$$
for all $x, y \in \mathbb{Z}$.
Find all injective functions $f : \mathbb{R} \to \mathbb{R}$ such that for every real number $x$ and every positive integer $n$,
$$
\left| \sum_{i=1}^{n} i \left( f(x+i+1) - f(x+i) \right) \right| \lt 2016
$$
The integers $p$, $q$ and $r$ are primes and their product is equal to $n$. By increasing $p$ and $q$ by 1, then the product $(p+1)(q+1)r$ is equal to $n+138$. Determine all the possible values of $n$.
The real numbers $x, y, z$, with $x \neq z$, are mutually different and nonzero and they satisfy the following equations:
$$
(x+y)^2 + (2-xy) = 9, \\
(y+z)^2 - (3+yz) = 4.
$$
Determine the value of the expression
$$
A = \left( \frac{x}{y} + \frac{y^2}{x^2} + \frac{z^3}{x^2y} \right) \left( \frac{y}{z} + \frac{z^2}{y^2} + \frac{x^3}{y^2z} \right) \left( \frac{z}{x} + \frac{x^2}{z^2} + \frac{y^3}{z^2x} \right).
$$
Let $AB\Gamma\Delta$ ($A\Delta\parallel B\Gamma$) be a trapezium with $\hat{A} = \hat{B} = 90^\circ$ and $A\Delta \lt B\Gamma$. Denote by $E$ the point of intersection of the two non parallel sides $AB$ and $\Gamma\Delta$, $Z$ the symmetric point of $A$ with respect to the line $B\Gamma$ and $M$ the midpoint of $EZ$. It is given that the line $\Gamma M$ is perpendicular to the line $\Delta Z$. Prove that the line $Z\Gamma$ is perpendicular to the line $\Gamma E$.
Find the number of ordered 6-tuples $(\alpha_1, \alpha_2, \alpha_3, \alpha_4, \alpha_5, \alpha_6)$ can be created, if the numbers $\alpha_1, \alpha_2, \alpha_3, \alpha_4, \alpha_5, \alpha_6$ can take the values $0$, $1$ and $2$ and the sum $\alpha_1 + \alpha_2 + \alpha_3 + \alpha_4 + \alpha_5 + \alpha_6$ is even.