Let $a$ and $b$ be relatively prime positive integers such that $(a, b) \ne (2, 1)$. Show that
$$
\text{rad}(a^n + b^n) \ne \text{rad}(a^m + b^m),
$$
for any distinct positive integers $m$ and $n$. Here $\text{rad}(c)$ denotes the distinct primes dividing the integer $c$.
A circle with center $I$ is inscribed in a hexagon $ABCDEF$. Let $A', B', C', D', E', F'$ be the midpoints of the diagonals $BF, AC, BD, CE, DF, EA$, respectively. Suppose that the lines $AA', CC', EE'$ intersect at $X$, the lines $BB', DD', FF'$ intersect at $Y$. If the triangle residing between the lines $AB, CD, EF$ is not similar to the triangle residing between the lines $BC, DE, FA$, then prove that the points $X, I, Y$ are collinear.
(Batzaya G.)
Let $ABC$ be a scalene triangle. The midpoints of the sides $AB$, $BC$ and $CA$ are denoted $C_0$, $A_0$ and $B_0$ respectively. Let $l_A$, $l_B$ and $l_C$ denote the bisectors of the interior angles of $A$, $B$ and $C$ respectively. If $M$ is the intersection of the perpendicular from $C_0$ to $l_C$ and the perpendicular from $A_0$ to $l_A$, then show that $MB_0$ is parallel to $l_B$.
Let $n$ and $a$ be positive integers satisfying $a^n \equiv 1 \pmod{n}$. Show that there exists a positive integer $s$ such that $a^s + s \equiv 0 \pmod{n}$.
Given a triangle $ABC$, let the bisectors of the angles $\angle BAC$, $\angle CBA$, $\angle ACB$ intersect the circumcircle of the triangle $ABC$ at the points $M$, $N$, $K$ respectively. Let $P$ be the intersection of the segments $AB$ and $MK$ and $Q$ be the intersection of the segments $AC$ and $MN$. Prove that the lines $PQ$ and $BC$ are parallel.
$51$ distinct integers are placed on a circle in such a way that each number is greater than the sum of the next three numbers in clockwise direction. What is the maximal number of numbers greater than or equal to $1$?
The diagonals $AC$, $BD$ of a convex quadrilateral $ABCD$ intersect at the point $O$. Let $M$, $N$ be the midpoints of the sides $AB$, $CD$ respectively. The point $P$ is chosen so that the quadrilateral $MONP$ is a parallelogram. Prove that the areas of the triangles $APD$, $BPC$ are equal.
Let $H$ be the intersection point of the altitudes $AD$ and $BE$ of an acute triangle $ABC$. The circumcircle of the triangle $ABC$ intersects the circle with diameter $CH$ at the point $K$ other than $C$. Prove that
$$
\frac{DK}{KE} = \frac{DH}{HE}.
$$
Count the number of ways to fill unit squares of a $4 \times 4$ table with two colours, red and blue, such that no two rows and no two columns are painted the same.
Let $M$ be the centroid and $O$ be the circumcenter of a triangle $ABC$. Take a point $K$ on the line $OM$ in such a way that $\angle KAB = \angle ABC$ and the points $K$, $C$ are on the same side of the line $AB$. Prove that $\angle KCB = \angle ABC$.
A sequence of length $n$, consisting of positive integers not exceeding $n - 1$, is given. If there exists exactly one subsequence with sum divisible by $n$, then show that the sequence is constant.
Let $ABCDEF$ be a circumscribed hexagon. Let $AB \cap CD = K$, $CD \cap EF = L$, $DE \cap AF = M$ and $AF \cap BC = N$. Prove that the lines $KM$, $LN$ and $BE$ are concurrent.
Find all maps $f: \mathbb{Z}_{\ge 1} \to \mathbb{Z}_{\ge 1}$ such that for any positive integers $a$ and $b$, the number
$$
af(a)^2 + bf(b)^2 + 3ab(f(a) + f(b))
$$
is a perfect cube.
Let $a$, $b$, $c$ and $d$ be non-negative real numbers satisfying $a + b + c + d = 2$. Prove that
$$
(a^2 + b^2)(b^2 + c^2)(c^2 + d^2)(d^2 + a^2) \le 1.
$$
Let $ABC$ be a triangle with $2 \cdot \angle A \gt \angle B$. A point $D$ is chosen on the segment $BC$ so that $2 \cdot \angle CAD = \angle ABC$. Let $H$ be the orthocenter of the triangle $ABC$ and $AD$ intersects the circumcircle of the triangle $AHC$ at $K$, differently from $A$. $AB$ intersects the circumcircle of the triangle $BDK$ at $E$, differently from $B$. Prove that $AE = CD$.
(Purevsuren D.)