The incircle of a triangle $ABC$ has centre $I$ and touches sides $AB, BC, CA$ in points $C_1, A_1, B_1$ respectively. Denote by $L$ the foot of a bissector of angle $B$, and by $K$ the point of intersecting of lines $B_1I$ and $A_1C_1$. Prove that $KL\parallel BB_1$. [b]proposed by L. Emelyanov, S. Berlov[/b]

Solve: $a, b, m, n\in \mathbb{N}$ $a^2+b^2=m^2-n^2, ab=2mn$

If \[2011^{2011^{2012}} = x^x\] for some positive integer $x$, how many positive integer factors does $x$ have? [i]Author: Alex Zhu[/i]

A frog is at the center of a circular shaped island with radius $r$. The frog jumps $1/2$ meters at first. After the first jump, it turns right or left at exactly $90^\circ$, and it always jumps one half of its previous jump. After a finite number of jumps, what is the least $r$ that yields the frog can never fall into the water? $ \textbf{(A)}\ \frac{\sqrt 5}{3} \qquad\textbf{(B)}\ \frac{\sqrt {13}}{5} \qquad\textbf{(C)}\ \frac{\sqrt {19}}{6} \qquad\textbf{(D)}\ \frac{1}{\sqrt 2} \qquad\textbf{(E)}\ \frac34 $

In the triangle $ABC$, the altitude $BH$ and the angle bisector $BL$ are drawn, the inscribed circle $w$ touches the side of the $AC$ at the point $K$. It is known that $\angle BKA = 45^o$. Prove that the circle with diameter $HL$ touches the circle $w$. (Anton Trygub)

Every positive integer can be represented as a sum of one or more consecutive positive integers. For each $n$ , find the number of such represententation of $n$.

A convex quadrilateral $ABCD$ is such that $\angle B = \angle D$ and are both acute angles. $E$ is on $AB$ such that $CB = CE$ and $F$ is on $AD$ such that $CF = CD$. If the circumcenter of $CEF$ is $O_1$ and the circumcenter of $ABD$ is $O_2$. Prove that $C,O_1,O_2$ are collinear. [i]Proposed by Kapil Pause[/i]

Left focal point and right focal point of a hyperbola are $F_1,F_2$, left focal point and right focal point of a hyperbola are $M,N$. If $P$ is a point on the hyperbola, then the tangent point of inscribed circle of $\triangle PF_1F_2$ on $F_1F_2$ is $\text{(A)}$a point on segment $MN$ $\text{(B)}$a point on segment $F_1M$ or $F_2N$ $\text{(C)}$point $M$ or $N$ $\text{(D)}$not sure

Let $P, Q$ be points taken on the side $BC$ of a triangle $ABC$, in the order $B, P, Q, C$. Let the circumcircles of $\vartriangle PAB$, $\vartriangle QAC$ intersect at $M$ ($\ne A$) and those of $\vartriangle PAC, \vartriangle QAB$ at N. Prove that $A, M, N$ are collinear if and only if $P$ and $Q$ are symmetric in the midpoint $A' $ of $BC$.

Given real numbers $0=x_1<x_2<\ldots<x_{2n}<x_{2n+1}=1$ such that $x_{i+1}-x_i\le h$ for $1\le i\le2n$, show that $$\frac{1-h}2<\sum_{i=1}^nx_{2i}(x_{2i+1}-x_{2i-1})<\frac{1+h}2.$$

A circle is divided into $n$ sectors ($n \geq 3$). Each sector can be filled in with either $1$ or $0$. Choose any sector $\mathcal{C}$ occupied by $0$, change it into a $1$ and simultaneously change the symbols $x, y$ in the two sectors adjacent to $\mathcal{C}$ to their complements $1-x$, $1-y$. We repeat this process as long as there exists a zero in some sector. In the initial configuration there is a $0$ in one sector and $1$s elsewhere. For which values of $n$ can we end this process?

Let $0 \leq k < n$ be integers and $A=\{a \: : \: a \equiv k \pmod n \}.$ Find the smallest value of $n$ for which the expression \[ \frac{a^m+3^m}{a^2-3a+1} \] does not take any integer values for $(a,m) \in A \times \mathbb{Z^+}.$

When simplified, $ (x^{ \minus{} 1} \plus{} y^{ \minus{} 1})^{ \minus{} 1}$ is equal to: $ \textbf{(A)}\ x \plus{} y \qquad\textbf{(B)}\ \frac {xy}{x \plus{} y} \qquad\textbf{(C)}\ xy \qquad\textbf{(D)}\ \frac {1}{xy} \qquad\textbf{(E)}\ \frac {x \plus{} y}{xy}$

Determine all positive integers $n$ for which there exists an integer $m$ such that ${2^{n}-1}$ is a divisor of ${m^{2}+9}$.

A circle moves so that it is continually in the contact with all three coordinate planes of an ordinary rectangular system. Find the locus of the center of the circle.

In a $m\times n$ square grid, with top-left corner is $A$, there is route along the edges of the grid starting from $A$ and visits all lattice points (called "nodes") exactly once and ending also at $A$. a. Prove that this route exists if and only if at least one of $m,\ n$ is odd. b. If such a route exists, then what is the least possible of turning points? *A turning point is a node that is different from $A$ and if two edges on the route intersect at the node are perpendicular.

Two polynomials $P(x)=x^4+ax^3+bx^2+cx+d$ and $Q(x)=x^2+px+q$ have real coefficients, and $I$ is an interval on the real line of length greater than $2$. Suppose $P(x)$ and $Q(x)$ take negative values on $I$, and they take non-negative values outside $I$. Prove that there exists a real number $x_0$ such that $P(x_0)<Q(x_0)$.

Let $O$, $A$, $B$, and $C$ be points in space such that $\angle AOB=60^{\circ}$, $\angle BOC=90^{\circ}$, and $\angle COA=120^{\circ}$. Let $\theta$ be the acute angle between planes $AOB$ and $AOC$. Given that $\cos^2\theta=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $100m+n$. [i]Proposed by Michael Ren[/i]

Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is $ 4 : 3$. The horizontal length of a “$ 27$-inch” television screen is closest, in inches, to which of the following? [asy]import math; unitsize(7mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); draw((0,0)--(4,0)--(4,3)--(0,3)--(0,0)--(4,3)); fill((0,0)--(4,0)--(4,3)--cycle,mediumgray); label(rotate(aTan(3.0/4.0))*"Diagonal",(2,1.5),NW); label(rotate(90)*"Height",(4,1.5),E); label("Length",(2,0),S);[/asy]$ \textbf{(A)}\ 20 \qquad \textbf{(B)}\ 20.5 \qquad \textbf{(C)}\ 21 \qquad \textbf{(D)}\ 21.5 \qquad \textbf{(E)}\ 22$

Let $X_1,X_2,\ldots$ be a sequence of independent random variables distributed exponentially with mean $1$. Suppose $\mathbb N$ is a random variable independent of $X_i$'s that has a Poisson distribution with mean $\lambda>0$. What is the expected value of $X_1+X_2+\ldots+X_N$? $\textbf{(A)}~N^2$ $\textbf{(B)}~\lambda+\lambda^2$ $\textbf{(C)}~\lambda^2$ $\textbf{(D)}~\lambda$

In a triangle $ABC ~(\overline{AB} < \overline{AC})$, points $D (\neq A, B)$ and $E (\neq A, C)$ lies on side $AB$ and $AC$ respectively. Point $P$ satisfies $\overline{PB}=\overline{PD}, \overline{PC}=\overline{PE}$. $X (\neq A, C)$ is on the arc $AC$ of the circumcircle of triangle $ABC$ not including $B$. Let $Y (\neq A)$ be the intersection of circumcircle of triangle $ADE$ and line $XA$. Prove that $\overline{PX} = \overline{PY}$.

$f(x)$ and $g(x)$ are linear functions such that for all $x$, $f(g(x)) = g(f(x)) = x$. If $f(0) = 4$ and $g(5) = 17$, compute $f(2006)$.

Two digits one are written at the ends of a segment. In the middle, their sum is written, the number 2. Then, in the middle between each two neighboring numbers written, their sum is written again, and so on, 1973 times. How many times will the number 1973 be written? [i]Proposed by G. Halperin[/i]

Suppose that $f$ is a function with two continuous derivatives 2and $f(0) = 0.$ Prove that the function $g,$ defined by $g(0) = f '(0), g(x) = f(x) / x$ for $x \ne 0, $ has a continuous derivative.

In $\triangle ABC$ we consider the points $A',B',C'$ in sides $BC,AC,AB$ such that $$3BA'=CA',~3CB'=AB',~3AC'=BA'$$$\triangle DEF$ is defined by the intersections of $AA',BB',CC'$. If the are of $\triangle ABC$ is $2020$ find the area of $\triangle DEF$. [i]Proposed by Alejandro Madrid, Valle[/i]