On a straight track are several runners, each running at a different constant speed. They start at one end of the track at the same time. When a runner reaches any end of the track, he immediately turns around and runs back with the same speed (then he reaches the other end and turns back again, and so on). Some time after the start, all runners meet at the same point. Prove that this will happen again.

A zerg player can produce one zergling every minute and a protoss player can produce one zealot every $2.1$ minutes. Both players begin building their respective units immediately from the beginning of the game. In a ght, a zergling army overpowers a zealot army if the ratio of zerglings to zealots is more than $3$. What is the total amount of time (in minutes) during the game such that at that time the zergling army would overpower the zealot army?

$\frac1{1+\sqrt3}+\frac1{\sqrt3+\sqrt5}+\frac1{\sqrt5+\sqrt7}+\ldots+\frac1{\sqrt{2017}+\sqrt{2019}}=?$ $\textbf{(A)}~\frac{\sqrt{2019}-1}2$ $\textbf{(B)}~\frac{\sqrt{2019}+1}2$ $\textbf{(C)}~\frac{\sqrt{2019}-1}4$ $\textbf{(D)}~\text{None of the above}$

In the classroom of at least four students the following holds: no matter which four of them take seats around a round table, there is always someone who either knows both of his neighbours, or does not know either of his neighbours. Prove that it is possible to divide the students into two groups such that in one of them, all students know one another, and in the other, none of the students know each other. (Note: if student A knows student B, then student B knows student A as well.)

prove that if graph $G$ is a tree, then there is a vertex that is common between all of the longest paths. [i]proposed by Sina Rezayi[/i]

The diameter \( AD \) of the circumcircle of triangle \( ABC \) intersects line \( BC \) at point \( K \). Point \( D \) is reflected symmetrically with respect to point \( K \), resulting in point \( L \). On line \( AB \), a point \( F \) is chosen such that \( FL \perp AC \). Prove that \( FK \perp AD \). [i]Proposed by Matthew Kurskyi[/i]

Given positive integers $k$ and $m$, show that $m$ and $\binom{n}{k}$ are coprime for infinitely many integers $n\geq k$.

The digits 1, 2, 3, 4 and 9 are each used once to form the smallest possible even five-digit number. The digit in the tens place is $ \text{(A)}\ 1\qquad\text{(B)}\ 2\qquad\text{(C)}\ 3\qquad\text{(D)}\ 4\qquad\text{(E)}\ 9 $

On the coordinate plane, let $C$ be a circle centered $P(0,\ 1)$ with radius 1. let $a$ be a real number $a$ satisfying $0<a<1$. Denote by $Q,\ R$ intersection points of the line $y=a(x+1) $ and $C$. (1) Find the area $S(a)$ of $\triangle{PQR}$. (2) When $a$ moves in the range of $0<a<1$, find the value of $a$ for which $S(a)$ is maximized. [i]2011 Tokyo University entrance exam/Science, Problem 1[/i]

In $\triangle ABC$, $D,E,F$ are respectively the midpoints of $AB, BC, and CA$. Futhermore $AB=10$, $CD=9$, $CD\perp AE$. Find $BF$.

A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two different boxes and announces the sum of numbers on those cards. Given this information, the magician locates the box from which no card has been drawn. How many ways are there to put the cards in the three boxes so that the trick works?

A [b] bijective[/b] mapping from a plane to itself maps every circle to a circle. Prove that it maps every line to a line.

Let $ABCDE$ be a pentagon inscribe in a circle $(O)$. Let $ BE \cap AD = T$. Suppose the parallel line with $CD$ which passes through $T$ which cut $AB,CE$ at $X,Y$. If $\omega$ be the circumcircle of triangle $AXY$ then prove that $\omega$ is tangent to $(O)$.

Let $k$ be a positive integer. Lexi has a dictionary $\mathbb{D}$ consisting of some $k$-letter strings containing only the letters $A$ and $B$. Lexi would like to write either the letter $A$ or the letter $B$ in each cell of a $k \times k$ grid so that each column contains a string from $\mathbb{D}$ when read from top-to-bottom and each row contains a string from $\mathbb{D}$ when read from left-to-right. What is the smallest integer $m$ such that if $\mathbb{D}$ contains at least $m$ different strings, then Lexi can fill her grid in this manner, no matter what strings are in $\mathbb{D}$?

What is the last digit of $4^{3^{2019}}$? [b]A.[/b] $0$ [b]B.[/b] $2$ [b]C.[/b] $4$ [b]D.[/b] $6$ [b]E.[/b] $8$

Prove that for any positive integer $n$ there exists a positive integer $k$ such that $3^k+4^k-1 \vdots 12^n$

Let $p_1, p_2, p_3$, and $p$ be prime numbers. Prove that there exist $x,y\in \mathbb{Z}$ such that $y^2\equiv p_1 x^4-p_1 p_2^2 p_3^2\, (mod\, p)$.

Prove that if all four vertices of a parallelogram are lattice points and there are some other lattice points in or on the parallelogram, then its area exceeds $1$.

Find all positive integer solutions $a, b, c$ such that $(a - 1)\cdot(b - 2)\cdot(c - 3) = abc$ [i]Lightning 1.2[/i]

Alice and Bob play the following game: Alice picks a set $A = \{1, 2, ..., n \}$ for some natural number $n \ge 2$. Then, starting from Bob, they alternatively choose one number from the set $A$, according to the following conditions: initially Bob chooses any number he wants, afterwards the number chosen at each step should be distinct from all the already chosen numbers and should differ by $1$ from an already chosen number. The game ends when all numbers from the set $A$ are chosen. Alice wins if the sum of all the numbers that she has chosen is composite. Otherwise Bob wins. Decide which player has a winning strategy. Proposed by [i]Demetres Christofides, Cyprus[/i]

Real numbers $a$, $b$, $c$, and $d$ satisfy the system of equations \begin{align*} -a-27b-8d &= 1, \\ 8a+64b+c+27d &= 0, \\ 27a+125b+8c+64d &= 1, \\ 64a+216b+27c+125d &= 8. \end{align*} Find $12a+108b+48d$. [i]Proposed by firebolt360[/i]

Let $u_{jk}$ be a real number for each $j=1,2,3$ and each $k=1,2$ and let $N$ be an integer such that \[\max_{1\le k \le 2} \sum_{j=1}^3 |u_{jk}| \leq N\] Let $M$ and $l$ be positive integers such that $l^2 <(M+1)^3$. Prove that there exist integers $\xi_1,\xi_2,\xi_3$ not all zero, such that \[\max_{1\le j \le 3}\xi_j \le M\ \ \ \ \text{and} \ \ \ \left|\sum_{j=1}^3 u_{jk}\xi_k\right| \le \frac{MN}{l} \ \ \ \ \text{for k=1,2}\]

The sequence $a_i$ is defined as $a_1 = 2, a_2 = 3$, and $a_{n+1} = 2a_{n-1}$ or $a_{n+1} = 3a_n - 2a_{n-1}$ for all integers $n \ge 2$. Prove that no term in $a_i$ is in the range $[1612, 2012]$.

Show that the center of gravity of a convex region in the plane halves at least three chords of the region. [Gy. Hajos]

As the Kubiks head out of town for vacation, Jerry takes the first driving shift while Hannah and most of the kids settle down to read books they brought along. Tony does not feel like reading, so Alexis gives him one of her math notebooks and Tony gets to work solving some of the problems, and struggling over others. After a while, Tony comes to a problem he likes from an old AMC 10 exam: \begin{align*}&\text{Four distinct circles are drawn in a plane. What is the maximum}\\&\quad\,\,\text{number of points where at least two of the circles intersect?}\end{align*} Tony realizes that he can draw the four circles such that each pair of circles intersects in two points. After careful doodling, Tony finds the correct answer, and is proud that he can solve a problem from late on an AMC 10 exam. "Mom, why didn't we all get Tony's brain?" Wendy inquires before turning he head back into her favorite Harry Potter volume (the fifth year). Joshua leans over to Tony's seat to see his brother's work. Joshua knows that Tony has not yet discovered all the underlying principles behind the problem, so Joshua challenges, "What if there are a dozen circles?" Tony gets to work on Joshua's problem of finding the maximum number of points of intersections where at least two of the twelve circles in a plane intersect. What is the answer to this problem?