A trapezoid $ABCD$ with bases $AB$ and $CD$ is such that the circumcircle of the triangle $BCD$ intersects the line $AD$ in a point $E$, distinct from $A$ and $D$. Prove that the circumcircle oF the triangle $ABE$ is tangent to the line $BC$.

In a country there are $100$ cities, some of which are connected by roads. For each four cities there are at least two roads between them. Also, there is no path that passes through each city exactly one time. Prove that one can choose two cities among those $100$, such that each of the $98$ remaining cities would be connected by a road with at least one of the two chosen cities.

Given five real numbers $u_0, u_1, u_2, u_3, u_4$, prove that it is always possible to find five real numbers $v0, v_1, v_2, v_3, v_4$ that satisfy the following conditions: $(i)$ $u_i-v_i \in \mathbb N, \quad 0 \leq i \leq 4$ $(ii)$ $\sum_{0 \leq i<j \leq 4} (v_i - v_j)^2 < 4.$ [i]Proposed by Netherlands.[/i]

Consider the set $M=\{1,2,3,...,2020\}.$ Find the smallest positive integer $k$ such that for any subset $A$ of $M$ with $k$ elements, there exist $3$ distinct numbers $a,b,c$ from $M$ such that $a+b, b+c$ and $c+a$ are all in $A.$

Positive numbers $b_1, b_2,..., b_n$ are given so that $b_1 + b_2 + ...+ b_n \le 10$. Further, $a_1 = b_1$ and $a_m = sa_{m-1} + b_m$ for $m > 1$, where $0 \le s < 1$. Show that $a^2_1 + a^2_2 + ... + a^2_n \le \frac{100}{1 - s^2} $

How many three-digit positive integers have digits which sum to a multiple of $10$?

Let $ABCD$ be a parallelogram with $\angle BAD<90^o$ and $AB> BC$ . The angle bisector of $BAD$ intersects line $CD$ at point $P$ and line $BC$ at point $Q$. Prove that the center of the circle circumscirbed around the triangle $CPQ$ is equidistant from points $B$ and $D$.

Given are $10$ distinct real numbers. Kyle wrote down the square of the difference for each pair of those numbers in his notebook, while Peter wrote in his notebook the absolute value of the differences of the squares of these numbers. Is it possible for the two boys to have the same set of $45$ numbers in their notebooks?

Let $p(x) = 3x^2 + 1$. Compute the largest prime divisor of $p(100) - p(3)$

A plane has no vertex of a regular dodecahedron on it,try to find out how many edges at most may the plane intersect the regular dodecahedron?

Prove the following theorem: If there are $n$ pairs of different points $P_i$, $i = 1, 2, ..., n$, $n > 2$ in three dimensions space, such that each of them is at a smaller distance from one and the same point $Q$ than any other $P_i$, then $n < 15$.

Prove: if $2^{2^n-1}-1$ is a prime, then $n$ is a prime.

Consider the set of all permutations, $\mathcal{P}$, of $\{1,2,\ldots,2022\}$. For permutation $P\in \mathcal{P}$, let $P_1$ denote the first element in $P$. Let $\text{sgn}(P)$ denote the sign of the permutation. Compute the following number modulo 1000: $$\displaystyle\sum_{P\in\mathcal{P}}\dfrac{P_1\cdot\text{sgn}(P)^{P_1}}{2020!}.$$ (The [i]sign[/i] of a permutation $P$ is $(-1)^k$, where $k$ is the minimum number of two-element swaps needed to reach that permutation). [i]Proposed by Nairit Sarkar[/i]

In acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$, let $D$ be an arbitrary point on the circumcircle of triangle $ABC$ such that $D$ does not lie on line $OB$ and that line $OD$ is not parallel to line $BC$. Let $E$ be the point on the circumcircle of triangle $ABC$ such that $DE$ is perpendicular to $BC$, and let $F$ be the point on line $AC$ such that $FA = FE$. Let $P$ and $R$ be the points on the circumcircle of triangle $ABC$ such that $PE$ is a diameter, and $BH$ and $DR$ are parallel. Let $M$ be the midpoint of $DH$. (a) Show that $AP$ and $BR$ are perpendicular. \\ (b) Show that $FM$ and $BM$ are perpendicular.

A real valued function $f$ is defined on the interval $(-1,2)$. A point $x_0$ is said to be a fixed point of $f$ if $f(x_0)=x_0$. Suppose that $f$ is a differentiable function such that $f(0)>0$ and $f(1)=1$. Show that if $f'(1)>1$, then $f$ has a fixed point in the interval $(0,1)$.

Compute the radius of the sphere inscribed in the tetrahedron with coordinates $(2,0,0)$, $(4,0,0)$, $(0,1,0)$, and $(0,0,3)$.

Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region. [asy]unitsize(.3cm); defaultpen(linewidth(.8pt)); path c=Circle((0,2),1); filldraw(Circle((0,0),3),grey,black); filldraw(Circle((0,0),1),white,black); filldraw(c,white,black); filldraw(rotate(60)*c,white,black); filldraw(rotate(120)*c,white,black); filldraw(rotate(180)*c,white,black); filldraw(rotate(240)*c,white,black); filldraw(rotate(300)*c,white,black);[/asy]$ \textbf{(A)}\ \pi \qquad \textbf{(B)}\ 1.5\pi \qquad \textbf{(C)}\ 2\pi \qquad \textbf{(D)}\ 3\pi \qquad \textbf{(E)}\ 3.5\pi$

Arash and Babak play the following game, taking turns alternatively, on a $1400\times 1401$ table. Arash starts and in his turns he colors $k$, $L$-corners (any three cell of a square). Babak in his turn colors one $2\times 2$ square. Neither player is allowed to recolor any cell. Find all positive integers $k$ for which Arash has a winning strategy.

Two players, A and B, play the following game: they retire coins of a pile which contains initially 2006 coins. The players play removing alternatingly, in each move, from 1 to 7 coins, each player keeps the coins that retires. If a player wishes he can pass(he doesn't retire any coin), but to do that he must pay 7 coins from the ones he retired from the pile in past moves. These 7 coins are taken to a separated box and don't interfere in the game any more. The winner is the one who retires the last coin, and A starts the game. Determine which player can win for sure, it doesn't matter how the other one plays. Show the winning strategy and explain why it works.

Let $S$ be the solid in three-dimensional space consisting of all points $(x,y,z)$ satisfying the following six simultaneous conditions: $$ x,y,z \geq 0, \;\; x+y+z\leq 11, \;\; 2x+4y+3z \leq 36, \;\; 2x+3z \leq 44.$$ a) Determine the number $V$ of vertices of $S.$ b) Determine the number $E$ of edges of $S.$ c) Sketch in the $bc$-plane the set of points $(b, c)$ such that $(2,5,4)$ is one of the points $(x, y, z)$ at which the linear function $bx + cy + z$ assumes its maximum value on $S.$

The inscribed circle of the non-isosceles triangle $ABC$ touches sides $AB, BC$ and $AC$ at points $C_1, A_1$ and $B_1$, respectively. The circumscribed circle of the triangle $A_1BC_1$ intersects the lines $B_1A_1$ and $B_1C_1$ at the points $A_0$ and $C_0$, respectively. Prove that the orthocenter of triangle $A_0BC_0$, the center of the inscribed circle of triangle $ABC$ and the midpoint of the $AC$ lie on one straight line.

Let $a,b,c$ be the side lengths of any triangle. Prove that $$\frac{a}{\sqrt{2b^2+2c^2-a^2}}+\frac{b}{\sqrt{2c^2+2a^2-b^2 }}+\frac{c}{\sqrt{2a^2+2b^2-c^2}}\ge \sqrt{3}.$$ [i](Zhuge Liang)[/i]

Let $0 < u < 1$ and define \[u_1 = 1 + u, \quad u_2 = \frac{1}{u_1} + u, \quad \dots, \quad u_{n + 1} = \frac{1}{u_n} + u, \quad n \ge 1.\] Show that $u_n > 1$ for all values of $n = 1$, 2, 3, $\dots$.

The set of three-digit natural numbers formed from digits $1,2, 3, 4, 5, 6$ is called [i]nice [/i] if it satisfies the following condition: for any two different digits from $1, 2, 3, 4, 5, 6$ there exists a number from the set which contains both of them. For any nice set we calculate the sum of all its elements. Determine the smallest possible value of these sums. (E. Barabanov)

The trapezoid is inscribed in a circle. Prove that the sum of distances from any point of the circle to the midpoints of the lateral sides are not less than the diagonal of the trapezoid.