Let $\Gamma$ be the circumcircle of triangle $ABC$, isosceles at $C$. Let $M$ lie in the interior of side $BC$. Suppose $N$ lies on ray $AM$, with $M$ between $A,N$, and $AN=AC$. The circumcircle of $CMN$ meets $\Gamma$ at distinct points $C,P$. Let lines $AB,CP$ meet at $Q$. Prove $\angle BMQ=\angle QMN$.
In a triangle $ABC$, denote $D$ the orthogonal projection of the point $C$ onto the line $AB$, denote $E$ the orthogonal projection of the point $D$ onto the line $AC$, and denote $P$ the midpoint of the line segment $CD$. Let $K_1$ be the circumscribed circle of the triangle $ABC$, and let $K_2$ be the circle of radius $CD$ and center $C$. Prove that the line $EP$ passes through the intersection points of the circles $K_1$ and $K_2$ if and only if the angle $\angle ACB$ is a right angle.
The circumcenter of the cyclic quadrilateral $ABCD$ is $O$. The second intersection point of the circles $ABO$ and $CDO$, other than $O$, is $P$, which lies in the interior of the triangle $DAO$. Choose a point $Q$ on the extension of $OP$ beyond $P$, and a point $R$ on the extension of $OP$ beyond $O$. Prove that $\angle QAP = \angle OBR$ holds if and only if $\angle PDQ = \angle RCO$.
Let $ABC$ be a triangle and $X$ an interior point. Lines $XA,XB,XC$ meet the circumcircle again at $P,Q,R$, respectively. Let $U$ lie on ray $XP$ (on line $XP$, with $P,U$ on the same side of $X$). The line through $U$ parallel to $AB$ meets $BQ$ at $V$; the line through $U$ parallel to $AC$ meets $CR$ at $W$. Prove that $Q,R,V,W$ lie on one circle.
There are $n$ chameleons standing in a circle. Some of them are colored red and the others are colored green. Every minute, some of the chameleons change their color from red to green or vice versa in accordance with the following rule: a chameleon changes its color if and only if both its neighbours have the same color. Suppose that after $2n$ minutes all chameleons have the same color as at the beginning. Prove that there are two chameleons in the circle that have changed their color the same number of times.
Let $ABC$ be a triangle. An interior point $P$ of $ABC$ is said to be good if we can find exactly $27$ rays emanating from $P$ intersecting the sides of the triangle $ABC$ such that the triangle is divided by these rays into $27$ smaller triangles of equal area. Determine the number of good points for a given triangle $ABC$.
In an acute triangle $A B C$ the points $D$, $E$ and $F$ are the feet of the altitudes through $A$, $B$ and $C$ respectively. The incenters of the triangles $A E F$ and $B D F$ are $I_{1}$ and $I_{2}$ respectively; the circumcenters of the triangles $A C I_{1}$ and $B C I_{2}$ are $O_{1}$ and $O_{2}$ respectively. Prove that $I_{1} I_{2}$ and $O_{1} O_{2}$ are parallel.
Let $ABCD$ be a convex quadrilateral (that is, all its interior angles are less than $180^\circ$), such that there exist a point $M$ on segment $AB$ and a point $N$ on segment $BC$ for which $AN$ divides the quadrilateral into two equal-area pieces, and $CM$ does so as well. Prove that $MN$ bisects diagonal $BD$.
In a $999 \times 999$ square table some cells are white and the remaining ones are red. Let $T$ be the number of triples $(C_{1}, C_{2}, C_{3})$ of cells, the first two in the same row and the last two in the same column, with $C_{1}$ and $C_{3}$ white and $C_{2}$ red. Find the maximum value $T$ can attain.
We have two boxes of balls, containing $m$ and $n$ balls, where $m,n\gt 0$. Two operations are allowed:
(i) Remove the same number of balls from each box.
(ii) Multiply the number of balls in one box by $k$.
Is it always possible to empty both boxes using these operations,
a) if $k=2$?
b) if $k=3$?
For a function $f : \mathcal{P}(\mathbb{N}) \to \mathbb{N}$ and a subset $A \subseteq \mathbb{N}$, we say $f$ is *A-predictor* if the set $\{x \in \mathbb{N} \mid x \notin A, f(A \cup \{x\}) \neq x\}$ is finite. Prove that there exists a function that for every subset $A$ of natural numbers is *A-predictor*.
Find all natural numbers $n \ge 2$ such that for every pair of integers $i, j \in [0, n]$, $i + j$ and $\begin{pmatrix} n \\ i \end{pmatrix} + \begin{pmatrix} n \\ j \end{pmatrix}$ have the same parity.
Let $ABC$ be a triangle with $\angle BCA = 90^{\circ}$, and let $C_0$ be the foot of the altitude from $C$. Choose a point $X$ in the interior of the segment $CC_0$, and let $K, L$ be the points on the segments $AX, BX$ for which $BK = BC$ and $AL = AC$ respectively. Denote by $M$ the intersection of $AL$ and $BK$. Show that $MK = ML$.
Let $S$ be a nonempty set of primes satisfying the property that for each proper subset $P$ of $S$, all the prime factors of the number $\left(\prod_{p \in P} p\right) - 1$ are also in $S$. Determine all possible such sets $S$.
Let $g$ be a polynomial of degree at least $2$ with nonnegative coefficients. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that for every $x, y \in \mathbb{R}^{+}$
Let $ABC$ be a triangle. The bisector of $\angle CAB$ meets $BC$ at $L$. Points $M,N$ lie in the interiors of $AC,AB$, respectively, such that $AL,BM,CN$ are concurrent and $\angle AMN=\angle ALB$. Prove that $\angle NML=90^\circ$.