Given a triangle $ABC$, the perpendicular bisector of the side $AC$ intersects the angle bisector of the triangle $AK$ at the point $P$, $M$ - such a point that $\angle MAC = \angle PCB$, $\angle MPA = \angle CPK$, and points $M$ and $K$ lie on opposite sides of the line $AC$. Prove that the line $AK$ bisects the segment $BM$. (Anton Trygub)

$3n$ ($n \ge 2, n \in N$) people attend a gathering, in which any two acquaintances have exactly $n$ common acquaintances, and any two unknown people have exactly $2n$ common acquaintances. If three people know each other, it is called a [i]Taoyuan Group[/i]. (1) Find the number of all Taoyuan groups; (2) Prove that these $3n$ people can be divided into three groups, with $n$ people in each group, and the three people obtained by randomly selecting one person from each group constitute a Taoyuan group. Note: Acquaintance means that two people know each other, otherwise they are not acquaintances. Two people who know each other are called acquaintances.

If $\sin x+\cos x=1/5$ and $0\le x<\pi$, then $\tan x$ is $\textbf{(A) }-\frac{4}{3}\qquad\textbf{(B) }-\frac{3}{4}\qquad\textbf{(C) }\frac{3}{4}\qquad\textbf{(D) }\frac{4}{3}\qquad$ $\textbf{(E) }\text{not completely determined by the given information}$

Let $ABC$ be a non-isosceles triangle with incenter $I$. Let $D$ be a point on the segment $BC$ such that the circumcircle of $BID$ intersects the segment $AB$ at $E\neq B$, and the circumcircle of $CID$ intersects the segment $AC$ at $F\neq C$. The circumcircle of $DEF$ intersects $AB$ and $AC$ at the second points $M$ and $N$ respectively. Let $P$ be the point of intersection of $IB$ and $DE$, and let $Q$ be the point of intersection of $IC$ and $DF$. Prove that the three lines $EN, FM$ and $PQ$ are parallel. [i]Proposed by Saudi Arabia[/i]

Let $ABCD$ be a circumscribed quadrilateral with incircle $\gamma$. Let $AB\cap CD=E, AD\cap BC=F, AC\cap EF=K, BD\cap EF=L$. Let a circle with diameter $KL$ intersect $\gamma$ at one of the points $X$. Prove that $(EXF)$ is tangent to $\gamma$.

Let $ABC$ be an isosceles triangle with $AB = AC$ and let $n$ be a natural number, $n>1$. On the side $AB$ we consider the point $M$ such that $n \cdot AM = AB$. On the side $BC$ we consider the points $P_1, P_2, ....., P_ {n-1}$ such that $BP_1 = P_1P_2 = .... = P_ {n-1} C = \frac{1}{n} BC$. Show that: $\angle {MP_1A} + \angle {MP_2A} + .... + \angle {MP_ {n-1} A} = \frac{1} {2} \angle {BAC}$.

Let $\omega$ denote the circumscribed circle of triangle $ABC$, $I$ be its incenter, and $K$ be any point on arc $AC$ of $\omega$ not containing $B$. Point $P$ is symmetric to $I$ with respect to point $K$. Point $T$ on arc $AC$ of $\omega$ containing point $B$ is such that $\angle KCT = \angle PCI$. Show that the bisectors of angles $AKC$ and $ATC$ meet on line $CI$. [i](Proposed by Anton Trygub)[/i]

We know that raw wheat has $70\%$ moisture and dry wheat has $10\%$ moisture. One miller bought $3$ tons of raw wheat with price of $0.4 \$$ per kilo. At which price miller has to sell dry wheat, so he gets $80\%$ profit?

Find the value of the expression $$f\left( \frac{1}{2000} \right)+f\left( \frac{2}{2000} \right)+...+ f\left( \frac{1999}{2000} \right)+f\left( \frac{2000}{2000} \right)+f\left( \frac{2000}{1999} \right)+...+f\left( \frac{2000}{1} \right)$$ assuming $f(x) =\frac{x^2}{1 + x^2}$ .

Call a number $n$ [i]good[/i] if it can be expressed as $2^x+y^2$ for where $x$ and $y$ are nonnegative integers. (a) Prove that there exist infinitely many sets of $4$ consecutive good numbers. (b) Find all sets of $5$ consecutive good numbers. [i]Proposed by Michael Ma[/i]

$A_1, A_2,\dots A_k$ are different subsets of the set $\{1,2,\dots ,2016\}$. If $A_i\cap A_j$ forms an arithmetic sequence for all $1\le i <j\le k$, what is the maximum value of $k$?

Suppose we have a $n$-gon. Some $n-3$ diagonals are coloured black and some other $n-3$ diagonals are coloured red (a side is not a diagonal), so that no two diagonals of the same colour can intersect strictly inside the polygon, although they can share a vertex. Find the maximum number of intersection points between diagonals coloured differently strictly inside the polygon, in terms of $n$. [i]Proposed by Alexander Ivanov, Bulgaria[/i]

Let $a_1$, $a_2$, $\cdots$ be a sequence such that $a_1=a_2=\frac 15$, and for $n \ge 3$, $$a_n=\frac{a_{n-1}+a_{n-2}}{1+a_{n-1}a_{n-2}}.$$ Find the smallest integer $n$ such that $a_n>1-5^{-2022}$.

The decimal expression of the natural number $a$ consists of $n$ digits, while that of $a^3$ consists of $m$ digits. Can $n + m$ be equal to 2001?

Let $(x_n)_{n\ge 1}$ and $(y_n)_{n\ge 1}$ a sequence of positive real numbers, such that: \[x_{n+1}\ge \frac{x_n+y_n}{2},\ y_{n+1}\ge \sqrt{\frac{x_n^2+y_n^2}{2}},\ (\forall)n\in \mathbb{N}^*\] a) Prove that the sequences $(x_n+y_n)_{n\ge 1}$ and $(x_ny_n)_{n\ge 1}$ have limit. b) Prove that the sequences $(x_n)_{n\ge 1}$ and $(y_n)_{n\ge 1}$ have limit and that their limits are equal.

A car is driving along a straight highway at a speed of $60$ km per hour. Not far from the highway there is a parallel to him a $100$-meter fence. Every second, the passenger of the car measures the angle at which the fence is visible. Prove that the sum of all the angles he measured is less than $1100^o$

Let $T$ be a point inside a square $ABCD$. The lines $TA,TB,TC,TD$ meet the circumcircle of $ABCD$ again at $A',B',C',D'$, respectively. Prove that $A'B'\cdot C'D'=A'D'\cdot B'C'$.

Given an acute $\vartriangle ABC$ with incenter $I$. Let $I'$ be the symmetric point $I$ with respect to the midpoint of $B,C$ and $D$ is the foot of $A$. If $DI$ and the circumcircle of vartriangle $BI'C$ intersect at $T$ and $TI' $ intersects the circumcircle of $\vartriangle ATI$ at $X$. Furthermore, $E,F$ are tangent points of the incircle and $AB,AC, P$ is the another intersection of the circumcircles of $\vartriangle ABC, \vartriangle AEF$. Show that $AX \parallel PI$. [img]https://3.bp.blogspot.com/-tj9A8HIR6Vw/XndLEPMRvnI/AAAAAAAALfk/2vw_pZbhpnkTKIc1BcKf4K7SNZ11vu4TACK4BGAYYCw/s1600/2018%2Bimoc%2Bg4.png[/img]

There is a right triangle $\triangle ABC$ in which $\angle A$ is the right angle. On side $AB$, there are three points $X$, $Y$, and $Z$ that satisfy $\angle ACX=\angle XCY=\angle YCZ=\angle ZCB$ and $BZ=2AX$. The smallest angle of $\triangle ABC$ is $\tfrac ab$ degrees, where $a,b$ are positive integers such that $\gcd(a,b)=1$. Find $a+b$.

Let $P(x)=x^3+x^2-r^2x-2020$ be a polynomial with roots $r,s,t$. What is $P(1)$? [i]Proposed by James Lin.[/i]

Player $A$ and player $B$ play the next game on an $8$ by $8$ square chessboard. They in turn color a field that is not yet colored. One player uses red and the other blue. Player $A$ starts. The winner is the first person to color the four squares of a square of $2$ by $2$ squares with his color somewhere on the board. Prove that player $B$ can always prevent player $A$ from winning.

In $ xy \minus{}$plane, there are $ b$ blue and $ r$ red rectangles whose sides are parallel to the axis. Any parallel line to the axis can intersect at most one rectangle with same color. For any two rectangle with different colors, there is a line which is parallel to the axis and which intersects only these two rectangles. $ (b,r)$ cannot be ? $\textbf{(A)}\ (1,7) \qquad\textbf{(B)}\ (2,6) \qquad\textbf{(C)}\ (3,4) \qquad\textbf{(D)}\ (3,3) \qquad\textbf{(E)}\ \text{None}$

What is the area of a square in square feet, if each of its diagonals is $4$ feet long?

Right triangle $ABC$ has its right angle at $A$. A semicircle with center $O$ is inscribed inside triangle $ABC$ with the diameter along $AB$. Let $D$ be the point where the semicircle is tangent to $BC$. If $AD=4$ and $CO=5$, find $\cos\angle{ABC}$. [asy] import olympiad; pair A, B, C, D, O; A = (1,0); B = origin; C = (1,1); O = incenter(C, B, (1,-1)); draw(A--B--C--cycle); dot(O); draw(arc(O, 0.41421356237,0,180)); D = O+0.41421356237*dir(135); label("$A$",A,SE); label("$B$",B,SW); label("$C$",C,NE); label("$D$",D,NW); label("$O$",O,S); [/asy] $\text{(A) }\frac{\sqrt{5}}{4}\qquad\text{(B) }\frac{3}{5}\qquad\text{(C) }\frac{12}{25}\qquad\text{(D) }\frac{4}{5}\qquad\text{(E) }\frac{2\sqrt{5}}{5}$

The $T_0$ set consists of all the numbers, representable as $(2k)!, k = 0, 1, 2, ... , n, ...$. The $T_m$ set is obtained from $T_{m-1}$ by adding all the finite sums of different numbers, that belong to $T_{m-1}$. Prove that there is a natural number, that doesn't belong to $T_{1987}$.