The bisector of $\angle BAC$ in $\triangle ABC$ intersects $BC$ in point $L$. The external bisector of $\angle ACB$ intersects $\overrightarrow{BA}$ in point $K$. If the length of $AK$ is equal to the perimeter of $\triangle ACL$, $LB=1$, and $\angle ABC=36^\circ$, find the length of $AC$.

A coin that comes up heads with probability $ p > 0$ and tails with probability $ 1\minus{}p > 0$ independently on each flip is flipped eight times. Suppose the probability of three heads and five tails is equal to $ \frac{1}{25}$ of the probability of five heads and three tails. Let $ p \equal{} \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.

The points $A, B$ and $C$ lie on the surface of a sphere with center $O$ and radius 20. It is given that $AB=13, BC=14, CA=15,$ and that the distance from $O$ to triangle $ABC$ is $\frac{m\sqrt{n}}k,$ where $m, n,$ and $k$ are positive integers, $m$ and $k$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+k.$

Is it possible to arrange a permutation of Integers on the integer lattice infinite from both sides such that each row is increasing from left to right and each column increasing from up to bottom?

Let $x, y$ and $z$ be positive real numbers such that $x + y + z = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ . Prove that $xy + yz + zx \ge 3$.

The function $f(x)$ satisfies $f(2+x) = f(2-x)$ for all real numbers $x$. If the equation $f(x) = 0$ has exactly four distinct real roots, then the sum of these roots is A. 0 B. 2 C. 4 D. 6 E. 8

The product $(8)(888\dots8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$. What is $k$? ${ \textbf{(A)}\ 901\qquad\textbf{(B)}\ 911\qquad\textbf{(C)}\ 919\qquad\textbf{(D)}}\ 991\qquad\textbf{(E)}\ 999 $

Let $k$ be a positive integer, and let $a$ be an integer such that $a-2$ is a multiple of $7$ and $a^6-1$ is a multiple of $7^k$. Prove that $(a + 1)^6-1$ is also a multiple of $7^k$.

Define the function $f:(0,1)\to (0,1)$ by \[\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.\] Let $a$ and $b$ be two real numbers such that $0 < a < b < 1$. We define the sequences $a_n$ and $b_n$ by $a_0 = a, b_0 = b$, and $a_n = f( a_{n -1})$, $b_n = f (b_{n -1} )$ for $n > 0$. Show that there exists a positive integer $n$ such that \[(a_n - a_{n-1})(b_n-b_{n-1})<0.\] [i]Proposed by Denmark[/i]

Isosceles triangle $ABC$, with $AB=AC$, is inscribed in circle $\omega$. Let $D$ be an arbitrary point inside $BC$ such that $BD\neq DC$. Ray $AD$ intersects $\omega$ again at $E$ (other than $A$). Point $F$ (other than $E$) is chosen on $\omega$ such that $\angle DFE = 90^\circ$. Line $FE$ intersects rays $AB$ and $AC$ at points $X$ and $Y$, respectively. Prove that $\angle XDE = \angle EDY$. [i]Proposed by Anton Trygub[/i]

Assume that the two triangles $ABC$ and $A'B'C'$ have the common incircle and the common circumcircle. Let a point $P$ lie inside both the triangles. Prove that the sum of the distances from $P$ to the sidelines of triangle $ABC$ is equal to the sum of distances from $P$ to the sidelines of triangle $A'B'C'$.

The sequence $\{a_n\}$ is defined by: $a_1=\frac{21}{16}$, and for $n\ge2$,\[ 2a_n-3a_{n-1}=\frac{3}{2^{n+1}}. \]Let $m$ be an integer with $m\ge2$. Prove that: for $n\le m$, we have\[ \left(a_n+\frac{3}{2^{n+3}}\right)^{\frac{1}{m}}\left(m-\left(\frac{2}{3}\right)^{{\frac{n(m-1)}{m}}}\right)<\frac{m^2-1}{m-n+1}. \]

Let $ABCD$ be a square, point $E$ be the midpoint of the side $BC$. On the side $AB$ mark a point $F$ such that $FE \perp DE$. Prove that $AF + BE = DF$. (Ercole Suppa, Italy)

For a non-negative integer $n$, call a one-variable polynomial $F$ with integer coefficients $n$-[i]good [/i] if: (a) $F(0) = 1$ (b) For every positive integer $c$, $F(c) > 0$, and (c) There exist exactly $n$ values of $c$ such that $F(c)$ is prime. Show that there exist infinitely many non-constant polynomials that are not $n$-good for any $n$.

A natural number $n$ is called "$k$-squared" if it can be written as a sum of $k$ perfect squares not equal to 0. a) Prove that 2020 is "$2$-squared" , "$3$-squared" and "$4$-squared". b) Determine all natural numbers not equal to 0 ($a, b, c, d ,e$) $a<b<c<d<e$ that verify the following conditions simultaneously : 1) $e-2$ , $e$ , $e+4$ are all prime numbers. 2) $a^2+ b^2 + c^2 + d^2 + e^2$ = 2020.

A box contains $900$ cards numbered from $100$ to $999$. Paulo randomly takes a certain number of cards from the box and calculates, for each card, the sum of the digits written on it. How many cards does Paulo need to take out of the box to be sure of finding at least three cards whose digit sums are the same?

Suppose $\triangle ABC$ has $D$ as the midpoint of $BC$ and orthocenter $H$. Let $P$ be an arbitrary point on the nine point circle of $ABC$. The line through $P$ perpendicular to $AP$ intersects $BC$ at $Q$. The line through $A$ perpendicular to $AQ$ intersects $PQ$ at $X$. If $M$ is the midpoint of $AQ$, show that $HX \perp DM$.

Let $n = 2^{p-1} (2^p - 1)$, and let $2^p- 1$ be a prime number. Prove that the sum of all (positive) divisors of $n$ (not including $n$ itself) is exactly $n$.

Let $n \ge 2$ be a positive integer. A grasshopper is moving along the sides of an $n \times n$ square net, which is divided on $n^2$ unit squares. It moves so that а) in every $1 \times 1$ unit square of the net, it passes only through one side b) when it passes one side of $1 \times1$ unit square of the net, it jumps on a vertex on another arbitrary $1 \times 1$ unit square of the net, which does not have a side on which the grasshopper moved along. The grasshopper moves until the conditions can be fulfilled. What is the shortest and the longest path that the grasshopper can go through if it moves according to the condition of the problem? Calculate its length and draw it on the net.

Let $I_n=\frac{1}{n+1}\int_0^{\pi} x(\sin nx+n\pi\cos nx)dx\ \ (n=1,\ 2,\ \cdots).$ Answer the questions below. (1) Find $I_n.$ (2) Find $\sum_{n=1}^{\infty} I_n.$

Prove that there is no real number $x$ satisfying both equations \begin{align*}2^x+1=2\sin x \\ 2^x-1=2\cos x.\end{align*}

Let $p$, $q$, and $r$ be distinct positive prime numbers. Show that if \[pqr\mid (pq)^r+(qr)^p+(rp)^q-1,\] then \[(pqr)^3\mid 3((pq)^r+(qr)^p+(rp)^q-1).\]

Let $O$ be the circumcenter of an acute triangle $ABC$. Suppose that the circumradius of the triangle is $R = 2p$, where $p$ is a prime number. The lines $AO,BO,CO$ meet the sides $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. Given that the lengths of $OA_1,OB_1,OC_1$ are positive integers, find the side lengths of the triangle.

Let $\Omega$ be the circumcircle of the triangle $ABC$. The circle $\omega$ is tangent to the sides $AC$ and $BC$, and it is internally tangent to the circle $\Omega$ at the point $P$. A line parallel to $AB$ intersecting the interior of triangle $ABC$ is tangent to $\omega$ at $Q$. Prove that $\angle ACP = \angle QCB$.

What is the last three digits of $49^{303}\cdot 3993^{202}\cdot 39^{606}$? $ \textbf{(A)}\ 001 \qquad\textbf{(B)}\ 081 \qquad\textbf{(C)}\ 561 \qquad\textbf{(D)}\ 721 \qquad\textbf{(E)}\ 961 $