The incircle of triangle $ABC$ touches sides $BC$, $AC$, and $AB$ at points $A_1$, $B_1$, and $C_1$. The line through the midpoints of segments $AB_1$ and $AC_1$ intersects the tangent at $A$ to the circumcircle of triangle $ABC$ at point $A_2$. Points $B_2$ and $C_2$ are defined similarly. Prove that points $A_2$, $B_2$, and $C_2$ lie on a line.

Let $ A \equal{} (a_{ij})$, where $ i,j \equal{} 1,2,\ldots,n$, be a square matrix with all $ a_{ij}$ non-negative integers. For each $ i,j$ such that $ a_{ij} \equal{} 0$, the sum of the elements in the $ i$th row and the $ j$th column is at least $ n$. Prove that the sum of all the elements in the matrix is at least $ \frac {n^2}{2}$.

Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice. $\textbf{(A) }\dfrac18\hspace{14em}\textbf{(B) }\dfrac3{16}\hspace{14em}\textbf{(C) }\dfrac38$ $\textbf{(D) }\dfrac12$

In a country of $n$ cities, an express train runs both ways between any two cities. For any train, ticket prices either direction are equal, but for any different routes these prices are different. Prove that the traveler can select the starting city, leave it and go on, successively, $n-1$ trains, such that each fare is smaller than that of the previous fare. (A traveler can enter the same city several times.)

Let $ABC$ be an acute-angled triangle, $H$ the intersection point of its altitudes , and $AA'$ the diameter of the circumcircle of triangle $ABC$. Prove that the quadrilateral $HB A'C$ is a parallelogram.

Let $a,b,c,d \in \mathbb{N}$ such that $a \ge b \ge c \ge d $. Show that the equation $x^4 - ax^3 - bx^2 - cx -d = 0$ has no integer solution.

There are $2007$ boys and $2007$ girls in a middle school,every student can attend no more than $100$ academic meetings,if we know any pair of students with different sex attend at least one common meeting.prove that there must be a meeting with at least $11$ boys and $11$ girls attend.

Let $P$ be the probability that the product of $2020$ real numbers chosen independently and uniformly at random from the interval $[-1, 2]$ is positive. The value of $2P - 1$ can be written in the form $\left(\frac{m}{n}\right)^b$ , where $m, n$ and $b$ are positive integers such that $m$ and $n$ are relatively prime and $b$ is as large as possible. Compute $m + n + b$.

Let $P$ be a fourth degree polynomial, with derivative $P^\prime$, such that $P(1)=P(3)=P(5)=P^\prime (7)=0$. Find the real number $x\neq 1,3,5$ such that $P(x)=0$.

Let's call a checkered polygon a [i]strip[/i], which can be traversed entirely, starting from some of its cells and then moving only in two directions - up or to the right. Several such strips can be inserted into each other by shifting by a vector $(-1.1)$. Prove that for any strip consisting of an even number of cells, there is such an odd $k$ that if you combine $k$ of the same strips by inserting them sequentially into each other, then the resulting polygon can be divided along the grid lines into two equal parts. [i]Proposed by I. Markelov, S. Markelov[/i]

In every cell of an $n\times n$ board is written $1$ or $-1$. At each step we may choose any of the $4n-2$ diagonals of the board and change the signs of all the numbers on that diagonal. Determine the number of initial configurations from which, after a finite number of steps, we may arrive at a configuration where all products of numbers on rows and columns equal to $1$. [i]Proposed by Pavel Ciurea[/i]

Galatasaray and Fenerbahce have qualified last $16$ in the Europen Champions League. Aftar a random draw, eight matches are regulated in that knock-out phase. The winners of the eight matches will qualify for the next round - round of $8$. Knock-out phase continues until one team remains. If each team has equal chance to win, what is the propability of having a Galatasaray-Fenerbahce match? $ \textbf{(A)}\ \dfrac {1}{32} \qquad\textbf{(B)}\ \dfrac {1}{16} \qquad\textbf{(C)}\ \dfrac {1}{8} \qquad\textbf{(D)}\ \dfrac {1}{4} \qquad\textbf{(E)}\ \text{None of the preceding} $

Find the value of real parameter $ a$ such that $ 2$ is the smallest integer solution of \[ \frac{x\plus{}\log_2 (2^x\minus{}3a)}{1\plus{}\log_2 a} >2.\]

A uniform rectangular wood block of mass $M$, with length $b$ and height $a$, rests on an incline as shown. The incline and the wood block have a coefficient of static friction, $\mu_s$. The incline is moved upwards from an angle of zero through an angle $\theta$. At some critical angle the block will either tip over or slip down the plane. Determine the relationship between $a$, $b$, and $\mu_s$ such that the block will tip over (and not slip) at the critical angle. The box is rectangular, and $a \neq b$. [asy] draw((-10,0)--(0,0)--20/sqrt(3)*dir(150)); label("$\theta$",(0,0),dir(165)*6); real x = 3; fill((0,0)*dir(60)+(-x*sqrt(3),x)--(3,0)*dir(60)+(-x*sqrt(3),x)--(3,3)*dir(60)+(-x*sqrt(3),x)--(0,3)*dir(60)+(-x*sqrt(3),x)--cycle,grey); draw((0,0)*dir(60)+(-x*sqrt(3),x)--(3,0)*dir(60)+(-x*sqrt(3),x)--(3,3)*dir(60)+(-x*sqrt(3),x)--(0,3)*dir(60)+(-x*sqrt(3),x)--cycle); label("$a$",(0,0)*dir(60)+(-x*sqrt(3),x)--(3,0)*dir(60)+(-x*sqrt(3),x)); label("$b$",(3,3)*dir(60)+(-x*sqrt(3),x)--(3,0)*dir(60)+(-x*sqrt(3),x),dir(60)); [/asy] (A) $\mu_s > a/b$ (B) $\mu_s > 1-a/b$ (C) $\mu_s >b/a$ (D) $\mu_s < a/b$ (E) $\mu_s < b/a-1$

Let $ABC$ be a triangle. The points $K, L,$ and $M$ lie on the segments $BC, CA,$ and $AB,$ respectively, such that the lines $AK, BL,$ and $CM$ intersect in a common point. Prove that it is possible to choose two of the triangles $ALM, BMK,$ and $CKL$ whose inradii sum up to at least the inradius of the triangle $ABC$. [i]Proposed by Estonia[/i]

$ \forall n > 0, n \in \mathbb{Z},$ there exists uniquely determined integers $ a_n, b_n, c_n \in \mathbb{Z}$ such \[ \left(1 \plus{} 4 \cdot \sqrt[3]{2} \minus{} 4 \cdot \sqrt[3]{4} \right)^n \equal{} a_n \plus{} b_n \cdot \sqrt[3]{2} \plus{} c_n \cdot \sqrt[3]{4}.\] Prove that $ c_n \equal{} 0$ implies $ n \equal{} 0.$