Let $\triangle{ABC}$ be a triangle with circumcircle $\tau$. The tangentlines to $\tau$ through $A$ and $B$ intersect at $T$. The circle through $A$, $B$ and $T$ intersects $BC$ and $AC$ again at $D$ and $E$, respectively; $CT$ and $BE$ intersect at $F$. Suppose $D$ is the midpoint of $BC$. Calculate the ratio $BF:BE$. [i](Swiss Mathematical Olympiad 2011, Final round, problem 5)[/i]

Show that if $a$ and $b$ are integer numbers, and $a^2 + b^2 + 9ab$ is divisible by $11$, then $a^2-b^2$ divisible by $11$.

In the equation $ \frac {x(x \minus{} 1) \minus{} (m \plus{} 1)}{(x \minus{} 1)(m \minus{} 1)} \equal{} \frac {x}{m}$ the roots are equal when $ \textbf{(A)}\ m \equal{} 1 \qquad\textbf{(B)}\ m \equal{} \frac {1}{2} \qquad\textbf{(C)}\ m \equal{} 0 \qquad\textbf{(D)}\ m \equal{} \minus{} 1 \qquad\textbf{(E)}\ m \equal{} \minus{} \frac {1}{2}$

A four-pointed star is formed by placing for equilateral triangles of side length $4$ in a coordinate grid. The triangles are placed such that their bases lie along one of the coordinate axes, with the midpoint of the bases lying at the origin, and such that the vertices opposite the bases lie at four distinct points. Compute the area contained within the star.

Determine all (not necessarily finite) sets $S$ of points in the plane such that given any four distinct points in $S$, there is a circle passing through all four or a line passing through some three. [i]Carl Lian.[/i]

What maximal number of the men in checkers game can be put on the chess-board $8\times 8$ so, that every man can be taken by at least one other man ?

Find the range value of $a$, satisfyin that $\forall x\in\mathbb{R},\theta\in\left[0,\frac{\pi}{2}\right]$, $$(x+3+2\sin\theta\cos\theta)^2+(x+a\sin\theta+a\cos\theta)^2\geq\frac{1}{8}.$$

Show that given any nine integers, we can find four, $a, b, c, d$ such that $a + b - c - d$is divisible by $20$. Show that this is not always true for eight integers.

Let $ABC$ be an acute-angled, nonisosceles triangle. Altitudes $AA'$ and $BB' $meet at point $H$, and the medians of triangle $AHB$ meet at point $M$. Line $CM$ bisects segment $A'B'$. Find angle $C$. (D. Krekov)

Let $I$ be the incenter of a triangle $ABC$ with $AB = 20$, $BC = 15$, and $BI = 12$. Let $CI$ intersect the circumcircle $\omega_1$ of $ABC$ at $D \neq C $. Alice draws a line $l$ through $D$ that intersects $\omega_1$ on the minor arc $AC$ at $X$ and the circumcircle $\omega_2$ of $AIC$ at $Y$ outside $\omega_1$. She notices that she can construct a right triangle with side lengths $ID$, $DX$, and $XY$. Determine, with proof, the length of $IY$.

A circular track with diameter $500$ is externally tangent at a point A to a second circular track with diameter $1700.$ Two runners start at point A at the same time and run at the same speed. The first runner runs clockwise along the smaller track while the second runner runs clockwise along the larger track. There is a first time after they begin running when their two positions are collinear with the point A. At that time each runner will have run a distance of $\frac{m\pi}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n. $

Let $ABC$ be the triangle with vertices located at the center of masses of Vincent Huang's house, Tristan Shin's house, and Edward Wan's house; here, assume the three are not collinear. Let $N = 2017$, and define the $A$-[i]ntipodes[/i] to be the points $A_1,\dots, A_N$ to be the points on segment $BC$ such that $BA_1 = A_1A_2 = \cdots = A_{N-1}A_N = A_NC$, and similarly define the $B$, $C$-ntipodes. A line $\ell_A$ through $A$ is called a [i]qevian[/i] if it passes through an $A$-ntipode, and similarly we define qevians through $B$ and $C$. Compute the number of ordered triples $(\ell_A, \ell_B, \ell_C)$ of concurrent qevians through $A$, $B$, $C$, respectively. [i]Proposed by Brandon Wang[/i]

You have $14$ coins, dated $1901$ through $1914$. Seven of these coins are real and weigh $1.000$ ounce each. The other seven are counterfeit and weigh $0.999$ ounces each. You do not know which coins are real or counterfeit. You also cannot tell which coins are real by look or feel. Fortunately for you, Zoltar the Fortune-Weighing Robot is capable of making very precise measurements. You may place any number of coins in each of Zoltar's two hands and Zoltar will do the following: [list][*] If the weights in each hand are equal, Zoltar tells you so and returns all of the coins. [*] If the weight in one hand is heavier than the weight in the other, then Zoltar takes one coin, at random, from the heavier hand as tribute. Then Zoltar tells you which hand was heavier, and returns the remaining coins to you.[/list] Your objective is to identify a single real coin that Zoltar has not taken as tribute. Is there a strategy that guarantees this? If so, then describe the strategy and why it works. If not, then prove that no such strategy exists.

Find all functions $f : N\rightarrow N - \{1\}$ satisfying $f (n)+ f (n+1)= f (n+2) +f (n+3) -168$ for all $n \in N$ .

A convex $N\text{-gon}$ is divided by diagonals into triangles so that no two diagonals intersect inside the polygon. The triangles are painted in black and white so that any two triangles are painted in black and white so that any two triangles with a common side are painted in different colors. For each $N$ find the maximal difference between the numbers of black and white triangles.

Find the sum of the real roots of the polynomial \[ \prod_{k=1}^{100} \left( x^2-11x+k \right) = \left( x^2-11x+1 \right)\left( x^2-11x+2 \right)\dots\left(x^2-11x+100\right). \][i]Proposed by Evan Chen[/i]

Suppose that the differentiable functions $a, b, f, g:\mathbb{R} \rightarrow \mathbb{R} $ satisfy \[ f(x)\geq 0, f'(x) \geq 0,g(x)\geq 0, g'(x) \geq 0 \text{ for all } x \in \mathbb{R}, \] \[\lim_{x\rightarrow \infty} a(x)=A\geq 0,\lim_{x\rightarrow \infty} b(x)=B\geq 0, \lim_{x\rightarrow \infty} f(x)=\lim_{x\rightarrow \infty} g(x)=\infty,\] and \[\frac{f'(x)}{g'(x)}+a(x)\frac{f(x)}{g(x)}=b(x).\] Prove that $\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=\frac{B}{A+1}$.

How many distinct positive integers can be formed by choosing their digits from the string $04072019$?

Two concentric circles $\omega, \Omega$ with radii $8,13$ are given. $AB$ is a diameter of $\Omega$ and the tangent from $B$ to $\omega$ touches $\omega$ at $D$. What is the length of $AD$.

A [i]palindrome[/i] is a number that has the same value when read from left to right or from right to left. (For example, 12321 is a palindrome). Let $N$ be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of $N$? $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6$

Given a permutation $\sigma = (a_1,a_2,a_3,...a_n)$ of $(1,2,3,...n)$ , an ordered pair $(a_j,a_k)$ is called an inversion of $\sigma$ if $a \leq j < k \leq n$ and $a_j > a_k$. Let $m(\sigma)$ denote the no. of inversions of the permutation $\sigma$. Find the average of $m(\sigma)$ as $\sigma$ varies over all permutations.

The diagonals of a quadrilateral $ABCD$ are perpendicular: $AC\perp BD$. Four squares, $ABEF,BCGH,CDIJ,DAKL$, are erected externally on its sides. The intersection points of the pairs of straight lines $CL,DF; DF,AH; AH,BJ; BJ,CL$ are denoted by $P_1,Q_1,R_1, S_1$, respectively, and the intersection points of the pairs of straight lines $AI,BK; BK,CE;$ $ CE,DG; DG,AI$ are denoted by $P_2,Q_2,R_2, S_2$, respectively. Prove that $P_1Q_1R_1S_1 \cong P_2Q_2R_2S_2.$

The value of $ x \minus{} y^{x \minus{} y}$ when $ x \equal{} 2$ and $ y \equal{} \minus{}2$ is: $ \textbf{(A)}\ \minus{}18 \qquad \textbf{(B)}\ \minus{}14\qquad \textbf{(C)}\ 14\qquad \textbf{(D)}\ 18\qquad \textbf{(E)}\ 256$

Consider an acute triangle $ABC$ with $AB>AC$ inscribed in a circle of center $O$. From the midpoint $D$ of side $BC$ we draw line $(\ell)$ perpendicular to side $AB$ that intersects it at point $E$. If line $AO$ intersects line $(\ell)$ at point $Z$, prove that points $A,Z,D,C$ are concyclic.

What is the minimum value of $ f(x) \equal{} |x \minus{} 1| \plus{} |2x \minus{} 1| \plus{} |3x \minus{} 1| \plus{} \cdots \plus{} |119x \minus{} 1|$? $ \textbf{(A)}\ 49 \qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 51 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 53$