Let $ABC$ be a triangle such that $AC$ is its shortest side. A point $P$ is inside it and satisfies that $BP = AC$. Let $R$ be the midpoint of $BC$ and let $M$ be the midpoint of $AP$. Let $E$ be the intersection of $BP$ and $AC$. Prove that the bisector of angle $\angle BE A$ is perpendicular to segment $MR$.

Jerry has red blocks, yellow blocks, and blue blocks. He builds a tower $5$ blocks high, without any $2$ blocks of the same color touching each other. Also, if the tower is flipped upside-down, it still looks the same. Compute the number of ways Jerry could have built this tower.

$a,b,c$ are positive numbers.$a+b+c=3$ Prove that: $\sum \frac{a^2+6}{2a^2+2b^2+2c^2+2a-1}\leq 3 $

Given 2005 distinct numbers $a_1,\,a_2,\dots,a_{2005}$. By one question, we may take three different indices $1\le i<j<k\le 2005$ and find out the set of numbers $\{a_i,\,a_j,\,a_k\}$ (unordered, of course). Find the minimal number of questions, which are necessary to find out all numbers $a_i$.

Suppose $a,b$ and $c$ are real numbers that satisfy $a+b+c=5$ and $\tfrac{1}{a}+\tfrac{1}{b}+\tfrac{1}{c}=\tfrac15$. Find the greatest possible value of $a^3+b^3+c^3$.

Three circles, each of radius 3, are drawn with centers at $(14,92)$, $(17,76)$, and $(19,84)$. A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the absolute value of the slope of this line?

Given a point $P$ lying on a line $g,$ and given a circle $K.$ Construct a circle passing through the point $P$ and touching the circle $K$ and the line $g.$

At the beginning of a month a shop has $10$ different products for sale, each with equal prices. Every day the price of each product is either doubled or trebled. By the beginning of the following month all the prices have become different. Prove that the ratio (the maximal price) /(the minimal price) is greater than $27$. (D. Fomin and Stanislav Smirnov, St Petersburg)

Each entry in an $n\times n$ table is either $+$ or $-$. At each step, one can choose a row or a column and reverse all signs in it. From the initial position, it is possible to obtain the table in which all signs are $+$. Prove that this can be accomplished in at most $n$ steps.

Circles $\omega_1$ and $\omega_2$ intersect at points $X$ and $Y$. Line $\ell$ is tangent to $\omega_1$ and $\omega_2$ at $A$ and $B$, respectively, with line $AB$ closer to point $X$ than to $Y$. Circle $\omega$ passes through $A$ and $B$ intersecting $\omega_1$ again at $D \neq A$ and intersecting $\omega_2$ again at $C \neq B$. The three points $C$, $Y$, $D$ are collinear, $XC = 67$, $XY = 47$, and $XD = 37$. Find $AB^2$.

$2017$ engineers attend a conference. Any two engineers if they converse, converse with each other in either Chinese or English. No two engineers converse with each other more than once. It is known that within any four engineers, there was an even number of conversations and furthermore within this even number of conversations: i) At least one conversation is in Chinese. ii) Either no conversations are in English or the number of English conversations is at least that of Chinese conversations. Show that there exists $673$ engineers such that any two of them conversed with each other in Chinese.

Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is $32$. If the least integer is $S$ is [i]also[/i] removed, then the average value of the integers remaining is $35$. If the greatest integer is then returned to the set, the average value of the integers rises to $40$. The greatest integer in the original set $S$ is $72$ greater than the least integer in $S$. What is the average value of all the integers in the set $S$? $\textbf{(A)} ~36.2 \qquad\textbf{(B)} ~36.4 \qquad\textbf{(C)} ~36.6 \qquad\textbf{(D)} ~36.8 \qquad\textbf{(E)} ~37$

In rhombus $ABCD$, let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $AD$. If $CN = 7$ and $DM = 24$, compute $AB^2$. [i]Proposed by Andy Liu[/i]

In an acute-angled triangle $ABC, D$ is the foot of the altitude from $A$. Let $D_1$ and $D_2$ be the symmetric points of $D$ wrt $AB$ and $AC$, respectively. Let $E_1$ be the intersection of $BC$ and the line through $D_1$ parallel to $AB$ . Let $E_2$ be the intersection of$ BC$ and the line through $D_2$ parallel to $AC$. Prove that $D_1, D_2, E_1$ and $E_2$ on one circle whose center lies on the circumscribed circle of $\vartriangle ABC$.

$100$ fractions are written on a board, their numerators are numbers from $1$ to $100$ (each once) and denominators are also numbers from $1$ to $100$ (also each once). It appears that the sum of these fractions equals to $a/2$ for some odd $a$. Prove that it is possible to interchange numerators of two fractions so that sum becomes a fraction with odd denominator. [i]N. Agakhanov, I. Bogdanov [/i]

Let $a_0$ be a positive integer. Define the sequence $\{a_n\}_{n \geq 0}$ as follows: if \[ a_n = \sum_{i = 0}^jc_i10^i \] where $c_i \in \{0,1,2,\cdots,9\}$, then \[ a_{n + 1} = c_0^{2005} + c_1^{2005} + \cdots + c_j^{2005}. \] Is it possible to choose $a_0$ such that all terms in the sequence are distinct?

Circle $\omega_1$ is centered at $O_1$ with radius $3$, and circle $\omega_2$ is centered at $O_2$ with radius $2$. Line $\ell$ is tangent to $\omega_1$ and $\omega_2$ at $X$, $Z$, respectively, and intersects segment $\overline{O_1O_2}$ at $Y$ . The circle through $O_1$, $X$, $Y$ has center $O_3$, and the circle through $O_2$, $Y$ , $Z$ has center $O_4$. Given that $O_1O_2 = 13$, find $O_3O_4$.

Angles of a given triangle $ABC$ are all smaller than $120^\circ$. Equilateral triangles $AFB, BDC$ and $CEA$ are constructed in the exterior of $ABC$. (a) Prove that the lines $AD, BE$, and $CF$ pass through one point $S.$ (b) Prove that $SD + SE + SF = 2(SA + SB + SC).$

Prove that every infinite arithmetic progression $a$, $a+d$, $a+2d$,... where $a$ and $d$ are positive integers, contains infinte geometric progression $b$, $bq$, $bq^2$,... where $b$ and $q$ are also positive integers

Let $P$ be a point inside a triangle $ABC$ such that $\angle PAC= \angle PCB$. Let the projections of $P$ onto $BC$, $CA$, and $AB$ be $X,Y,Z$ respectively. Let $O$ be the circumcenter of $\triangle XYZ$, $H$ be the foot of the altitude from $B$ to $AC$, $N$ be the midpoint of $AC$, and $T$ be the point such that $TYPO$ is a parallelogram. Show that $\triangle THN$ is similar to $\triangle PBC$. [i]Proposed by Sammy Luo[/i]

Given a right triangle $ABC$ with $ \angle C=90^o$. On its hypotenuse $AB$ is arbitrary mark the point$ P$. The point $Q$ is symmetric to the point $P$ wrt $AC$. Let the lines $PQ$ and $BQ$ intersect $AC$ at points $O$ and $R$ respectively. Denote by $S$ the foot of the perpendicular from the point $R$ on the line $AB$ ($S \ne P$), and let $T$ be the intersection point of lines $OS$ and $BR$. Prove that $R$ is the center of the circle inscribed in the triangle $CST$.

In a 14 team baseball league, each team played each of the other teams 10 times. At the end of the season, the number of games won by each team differed from those won by the team that immediately followed it by the same amount. Determine the greatest number of games the last place team could have won, assuming that no ties were allowed.

It is given the equation $x^2+px+1=0$, with roots $x_1$ and $x_2$; (a) find a second-degree equation with roots $y_1,y_2$ satisfying the conditions $y_1=x_1(1-x_1)$, $y_2=x_2(1-x_2)$; (b) find all possible values of the real parameter $p$ such that the roots of the new equation lies between $-2$ and $1$.

The side $AD$ of a square $ABCD$ is the base of an obtuse-angled isosceles triangle $AED$ with vertex $E$ lying inside the square. Let $AF$ be a diameter of the circumcircle of this triangle, and $G$ be a point on $CD$ such that $CG = DF$. Prove that angle $BGE$ is less than half of angle $AED$.

Christian Reiher and Reid Barton want to open a security box, they already managed to discover the algorithm to generate the key codes and they obtained the following information: $i)$ In the screen of the box will appear a sequence of $n+1$ numbers, $C_0 = (a_{0,1},a_{0,2},...,a_{0,n+1})$ $ii)$ If the code $K = (k_1,k_2,...,k_n)$ opens the security box then the following must happen: a) A sequence $C_i = (a_{i,1},a_{i,2},...,a_{i,n+1})$ will be asigned to each $k_i$ defined as follows: $a_{i,1} = 1$ and $a_{i,j} = a_{i-1,j}-k_ia_{i,j-1}$, for $i,j \ge 1$ b) The sequence $(C_n)$ asigned to $k_n$ satisfies that $S_n = \sum_{i=1}^{n+1}|a_i|$ has its least possible value, considering all possible sequences $K$. The sequence $C_0$ that appears in the screen is the following: $a_{0,1} = 1$ and $a_0,i$ is the sum of the products of the elements of each of the subsets with $i-1$ elements of the set $A =$ {$1,2,3,...,n$}, $i\ge 2$, such that $a_{0, n+1} = n!$ Find a sequence $K = (k_1,k_2,...,k_n)$ that satisfies the conditions of the problem and show that there exists at least $n!$ of them.