Let $ABC$ be a triangle and $\Gamma$ its circumcircle. Let $\omega_{A}$ be the circle internally tangent to $(AB)$, $(AC)$ and to $\Gamma$. Denote by $T_{A}$ the point of tangency of $\Gamma$ with $\omega_{A}$. Define $T_{B}$ and $T_{C}$ similarly.
Prove that $\left(AT_{A}\right),\left(BT_{B}\right)$ and $\left(CT_{C}\right)$ are concurrent.
Let $O$ be the centre of a regular polygon with 18 sides and vertices $A_{1}, \ldots, A_{18}$. Let $B$ be the point of $[OA_{1}]$ such that $\widehat{BA_{2}O}=20^{\circ}$ and $C$ the point of $[OA_{2}]$ such that $\widehat{CA_{1}O}=10^{\circ}$. Prove that $B, C, A_{2}, A_{3}$ are concyclic.
Let $ABC$ be a triangle with $AB \neq AC$. Let $E$ be such that $AE = BE$ and $(BE)$ is perpendicular to $(BC)$, and let $F$ be such that $AF = CF$ and $(CF)$ is perpendicular to $(BC)$. Let $D$ be the point of $(BC)$ such that $(AD)$ is tangent to the circumcircle of $ABC$ at $A$.
Prove that the points $D$, $E$, $F$ are collinear.
For a given point $P$ of $(BC)$, denote by $E(P)$ and $F(P)$ the second intersection points of the lines $(AB)$ and $(AC)$ with the circle of diameter $[AP]$. Let $T(P)$ be the intersection of the tangents to this circle at $E(P)$ and $F(P)$.
Prove that as $P$ varies on $(BC)$, the locus of $T(P)$ is a line.
Let $ABC$ be a triangle and $O$ the centre of its circumcircle. Let $P$ be a point on $(AO)$ and $D, E, F$ the orthogonal projections of $P$ onto $(AB)$, $(BC)$ and $(CA)$. Let $X$ and $Y$ be the intersections of the circumcircles of $DEF$ and of $BCP$.
Let $\Gamma_{1}$ and $\Gamma_{2}$ be two circles meeting at two distinct points $A$ and $B$. Let $O$ be the centre of $\Gamma_{1}$. Let $C$ be a point of $\Gamma_{1}$, and let $D, E$ be the respective intersections of $(AC)$ and of $(BC)$ with $\Gamma_{2}$.
Let $ABCD$ be a convex quadrilateral. We are given two points $E, F$ such that $E, B, C, F$ are collinear in this order. Assume moreover that $\widehat{BAE} = \widehat{CDF}$ and $\widehat{EAF} = \widehat{FDE}$. Prove that $\widehat{FAC} = \widehat{EDB}$.
Let $[AB]$ be the diameter of a semicircle on which two points $C$ and $D$ are taken. Let $S$ be the intersection of $(AC)$ and $(BD)$, and $T$ the foot of the perpendicular from $S$ to $[AB]$.
Prove that $(ST)$ is the bisector of the angle $\widehat{CTD}$.
Find all quadruples $ (p, q, r, n) $ of positive integers satisfying the following three conditions:
- $p$ and $q$ are prime,
- $p+q$ is not divisible by $3$,
- $p+q = r (p-q)^n$.
Let $\omega_{1}$, $\omega_{2}$ be two circles tangent to each other at a point $T$, such that $\omega_{1}$ is inside $\omega_{2}$. Let $M$ and $N$ be two distinct points on $\omega_{1}$, different from $T$. Let $[AB]$ and $[CD]$ be two chords of the circle $\omega_{2}$ passing through $M$ and $N$ respectively. Assume that the segments $[BD]$, $[AC]$ and $[MN]$ intersect at a point $K$.
Prove that $(TK)$ is the bisector of the angle $\widehat{MTN}$.
Real numbers are written in the cells of a rectangular table with $n$ rows and $m$ columns. Assume that for every row or column, the sum of the numbers written in that row or column is an integer. Prove that it is possible to replace each real number $x$ by the integer $\lfloor x\rfloor$ or $\lceil x\rceil$ so that the sums of each column and of each row remain unchanged.
Remark: Recall that if $x$ is a real number, $\lfloor x\rfloor$ is the unique integer such that $\lfloor x\rfloor \leqslant x\lt\lfloor x\rfloor+1$, and $\lceil x\rceil$ is the unique integer such that $\lceil x\rceil-1\lt x \leqslant\lceil x\rceil$.
Let $k$ and $n$ be positive integers with $n\gt k$. By a binary word of length $n$ we mean a sequence of $n$ terms, all of which are 0 or 1. Anja chooses one of all possible binary words of length $n$. Then she writes on a board all binary words of length $n$ that differ from her chosen word in exactly $k$ positions. Afterwards Bernhard enters the room. Anja tells him the value of $k$, and then he looks at the binary words on the board. He now tries to guess the binary word originally chosen by Anja. What is (as a function of $k$ and $n$) the minimal number of attempts Bernhard needs in order to guess the binary word with certainty?
Let $ABCD$ be a convex quadrilateral. The diagonal $BD$ bisects the angle $\measuredangle CBA$. The circumcircle of triangle $ABC$ meets the segments $\overline{CD}$ and $\overline{DA}$ at the interior points $P$ and $Q$ respectively. The parallel to the line $AC$ through the point $D$ meets the lines $BA$ and $BC$ at the points $R$ and $S$ respectively. Prove that the four points $P, Q, R$ and $S$ lie on a common circle.
Let $\mathbb{N}$ denote the set of positive integers. Determine all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ with the following property: for all positive integers $m$ and $n$, the number $f(m)+f(n)-m n$ is nonzero and is a divisor of the number $m f(m)+n f(n)$.
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into rectangles which all have integer side lengths. Prove that for at least one of these rectangles, the distances to each of the four sides of $\mathcal{R}$ are all even or all odd.
Let $a_{1}, a_{2}, \ldots, a_{n}, k$ and $M$ be positive integers with the properties
$$
\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{n}}=k \quad \text{ and } \quad a_{1} a_{2} \ldots a_{n}=M .
$$
Prove: if $M\gt 1$, the polynomial
$$
P(x)=M(x+1)^{k}-\left(x+a_{1}\right)\left(x+a_{2}\right) \ldots\left(x+a_{n}\right)
$$
has no positive roots.
Determine the smallest real constant $C$ with the following property:
For any five positive real numbers $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$, not necessarily distinct, one can always find pairwise distinct indices $i, j, k, l$ such that
$$
\left|\frac{a_{i}}{a_{j}}-\frac{a_{k}}{a_{l}}\right| \leq C
$$
holds.
Let $n$ be a positive integer coprime to $6$. We colour the vertices of a regular $n$-gon with three colours in such a way that for each colour the number of vertices coloured with it is odd.
Prove that there is always an isosceles triangle whose vertices are vertices of the $n$-gon and are all coloured differently.
Let $A B C D E$ be a convex pentagon with the properties
$$
\overline{A B}=\overline{B C}=\overline{C D}, \angle B A E=\angle D C B \text{ and } \angle E D C=\angle C B A .
$$
Prove that the perpendicular from $E$ to $B C$ passes through the intersection point of $A C$ and $B D$.
Let a positive integer $d$ and a sequence $\left(a_{i}\right)_{i=1,2,3, \ldots}$ of positive integers be given. Assume that the following two conditions are satisfied:
- Every positive integer occurs exactly once in the sequence.
- For all indices $i \geq 10^{100}$ we have $\left|a_{i+1}-a_{i}\right| \leq 2 d$.
Prove that there exist infinitely many indices $j$ for which $\left|a_{j}-j\right|\lt d$.