Trapezium $ABCD$ with a longer base $AB$ is inscribed in the circle $k$. Let $A_0$, $B_0$ be respectively the midpoints of segments $BC$, $CA$. Let $N$ be the foot of the altitude from the point $C$ to $AB$, and $G$ the centroid of the triangle $ABC$. Circle $k_1$ goes through $A_0$ and $B_0$ and touches the circle $k$ in the point $X$, different than $C$. Prove that the points $D$, $G$, $N$ and $X$ are collinear.
Let $ABC$ be a right triangle with the right angle at vertex $C$. Let $D$ be a point on the segment $AC$ and let $E$ be a point on the segment $BD$ such that $\angle ABC = \angle DAE = \angle AED$. Prove that $|BE| = 2|CD|$. (Lithuania 2010)
Positive real numbers $a$, $b$, $c$, $d$ satisfy the relations
$$
abcd = 4, \quad a^2 + b^2 + c^2 + d^2 = 10.
$$
Determine the largest possible value of the expression $ab + bc + cd + da$.
An equilateral triangle with side length $3$ is divided into $9$ equilateral triangles with side length $1$. An integer from $1$ to $10$ is written into every point that is a vertex of a small triangle (colored vertices on the figure), such that all numbers are written exactly once. For every small triangle, the sum of the numbers in its three vertices is written inside it. Prove that at least three of those sums are greater than $11$.
An information station employs four different codes, $A$, $B$, $C$ and $D$, for communication, but each week uses only one of them. The code used in a definite week is randomly selected with equal chance among the three ones that have not been used in the last week. Suppose the code used in the first week is $A$. Then the probability that $A$ is also used in the seventh week is _______. (expressed as an irreducible fraction)
The incircle of a scalene triangle $ABC$ touches the sides $\overline{AB}$ and $\overline{CA}$ at the points $M$ and $N$ respectively. The excircles opposite to vertices $B$ and $C$ touch the line $BC$ at points $P$ and $Q$ respectively. Prove that the quadrilateral $MNPQ$ is cyclic if and only if $\angle CAB = 90^\circ$.
Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that for all real numbers $x$ and $y$ holds
$$
f(y + f(x)) - f(x + f(y)) = f(x - y)(f(x + y) - 1).
$$
Along the coast of an island there are 20 villages. Each village has 20 fighters. Every fighter fights all the fighters from all the other villages. No two fighters have equal strength and the stronger fighter wins the fight. We say that the village $A$ is stronger than the village $B$ if in at least $k$ fights among the fighters from $A$ and $B$ a fighter from the village $A$ wins. It turned out that every village is stronger than its neighbour (in the clockwise direction). Show that the maximal possible $k$ is 290.
Each digit of a positive integer $n$ (except the first one) is larger than the digit next left to it. Determine the sum of all digits of the number $9n$. (Russia 1999)
Let $P(x) = x^{2012} + a_{2011}x^{2011} + a_{2010}x^{2010} + \dots + a_1x + a_0$ be a polynomial of degree $2012$ of real coefficients with $1$ as its leading coefficient. Find the minimum of real number $c$ such that $|\operatorname{Im} z| \le c |\operatorname{Re} z|$, where $\operatorname{Re} z$ and $\operatorname{Im} z$ are, respectively, the real and the imaginary parts of any root of a polynomial obtained by changing some of the coefficients of $P(x)$ to their opposite numbers. (posed by Zhu Huawei)
Let the focus and directrix of parabola $y^2 = 2px$ ($p \gt 0$) be $F$ and $l$, respectively. $A$ and $B$ are moving points on the parabola satisfying $\angle AFB = \frac{\pi}{3}$. Let the projection of $M$, the midpoint of segment $AB$, on $l$ be $N$. Then the maximum value of $\frac{|MN|}{|AB|}$ is ______.
Let $ABCD$ be a cyclic quadrilateral and $\omega$ its circumcircle. Let $I$, $J$, and $K$ be the incenters of the triangles $ABC$, $ACD$, and $ABD$, respectively. Also, let $E$ be the midpoint of the arc $DB$ of $\omega$ containing $A$. The line $EK$ intersects $\omega$ at $F$ ($F \neq E$). Prove that the points $C$, $F$, $I$, and $J$ lie on the same circle.
Let $A B C$ be a triangle with circumcentre $O$. The points $D, E$ and $F$ lie in the interiors of the sides $B C, C A$ and $A B$ respectively, such that $D E$ is perpendicular to $C O$ and $D F$ is perpendicular to $B O$. (By interior we mean, for example, that the point $D$ lies on the line $B C$ and $D$ is between $B$ and $C$ on that line.)
Let $K$ be the circumcentre of triangle $A F E$. Prove that the lines $D K$ and $B C$ are perpendicular.
Prove that every positive integer $d$ has a positive multiple $n$, such that one can delete one of the non-zero digits of $n$ to obtain another multiple of $d$.
Let $I$ be the incentre of triangle $ABC$. A line through $I$ meets the interior of segment $AB$ at $M$ and the interior of segment $BC$ at $N$. Assume that $BMN$ is an acute triangle. Let $K,L$ be points on segment $AC$ such that $\angle BMI=\angle ILA$ and $\angle BNI=\angle IKC$. Prove that $|AM|+|KL|+|CN|=|AC|$.
Let $P_0, P_1, P_2, \dots, P_n$ be $n+1$ points on a plane, and the minimum distance between each two points of them is $d$ ($d \gt 0$). Prove
$$
| P_0 P_1 | \cdot | P_0 P_2 | \cdots | P_0 P_n | \gt \left( \frac{d}{3} \right)^n \sqrt{(n+1)!}
$$
