Let $a_i, b_i$ be coprime positive integers for $i = 1, 2, \ldots , k$, and $m$ the least common multiple of $b_1, \ldots , b_k$. Prove that the greatest common divisor of $a_1 \frac{m}{b_1} , \ldots, a_k \frac{m}{b_k}$ equals the greatest common divisor of $a_1, \ldots , a_k.$

Two players, $B$ and $R$, play the following game on an infinite grid of unit squares, all initially colored white. The players take turns starting with $B$. On $B$'s turn, $B$ selects one white unit square and colors it blue. On $R$'s turn, $R$ selects two white unit squares and colors them red. The players alternate until $B$ decides to end the game. At this point, $B$ gets a score, given by the number of unit squares in the largest (in terms of area) simple polygon containing only blue unit squares. What is the largest score $B$ can guarantee? (A [i]simple polygon[/i] is a polygon (not necessarily convex) that does not intersect itself and has no holes.) [i]Proposed by David Torres[/i]

Restore the isosceles triangle $ABC$ ($AB = AC$) if the common points $I, M, H$ of bisectors, medians and altitudes respectively are given.

Sequence $\{a_n\}$ defined by recurrence relation $a_{n+1} = 1+\frac{n^2}{a_n}$. Given $a_1>1$, find the value of $\lim\limits_{n\to\infty} \frac{a_n}{n}$ with proof.

The values of $ a$ in the equation: $ \log_{10}(a^2 \minus{} 15a) \equal{} 2$ are: $ \textbf{(A)}\ \frac {15\pm\sqrt {233}}{2} \qquad\textbf{(B)}\ 20, \minus{} 5 \qquad\textbf{(C)}\ \frac {15 \pm \sqrt {305}}{2}$ $ \textbf{(D)}\ \pm20 \qquad\textbf{(E)}\ \text{none of these}$

Find $x$ such that trigonometric \[\frac{\sin 3x \cos (60^\circ -4x)+1}{\sin(60^\circ - 7x) - \cos(30^\circ + x) + m}=0\] where $m$ is a fixed real number.

Let $f:(0,+\infty)\to(0,+\infty)$ be a decreasing function which satisfies $\int^\infty_0f(x)\text dx<+\infty$. Prove that $\lim_{x\to+\infty}xf(x)=0$.

Let $a_0, a_1, a_2, a_3, a_4$ be five positive numbers in the arithmetic progression with a difference $d$. Prove that $a^3_2 \leq \frac{1}{10}(a^3_0 + 4a^3_1 + 4a^3_3 + a^3_4).$

Let $p$ be a positive integer, $p>1.$ Find the number of $m\times n$ matrices with entries in the set $\left\{ 1,2,\dots,p\right\} $ and such that the sum of elements on each row and each column is not divisible by $p.$

Prove for each non-negative integer $n$ and real number $x$ the inequality \[ \sin{x} \cdot(n \sin{x}-\sin{nx}) \geq 0 \]

Determine all integers $n>1$ such that every power of $n$ has an odd number of digits.

In isosceles triangle $ABC$ with base $BC$, let $M$ be the midpoint of $BC$. Let $P$ be the intersection of the circumcircle of $\vartriangle ACM$ with the circle with center $B$ passing through $M$, such that $P \ne M$. If $\angle BPC = 135^o$, then $\frac{CP}{AP}$ can be written as $a +\sqrt{b}$ for positive integers $a$ and $b$, where $b$ is not divisible by the square of any prime. Find $a + b$.

In $\triangle ABC$ with $\angle BAC = 60^{\circ}$ and circumcircle $\omega$, the angle bisector of $\angle BAC$ intersects side $\overline{BC}$ at point $D$, and line $AD$ is extended past $D$ to a point $A'$. Let points $E$ and $F$ be the feet of the perpendiculars of $A'$ onto lines $AB$ and $AC$, respectively. Suppose that $\omega$ is tangent to line $EF$ at a point $P$ between $E$ and $F$ such that $\tfrac{EP}{FP} = \tfrac{1}{2}$. Given that $EF=6$, the area of $\triangle ABC$ can be written as $\tfrac{m\sqrt{n}}{p}$, where $m$ and $p$ are relatively prime positive integers, and $n$ is a positive integer not divisible by the square of any prime. Find $m+n+p$. [i]Proposed by Taiki Aiba[/i]

Determine, with proof, all positive integers $k$ such that $$\frac{1}{n+1} \sum_{i=0}^n \binom{n}{i}^k$$ is an integer for every positive integer $n.$

Twenty numbers $a_1,a_2,..,a_{20}$ satisfy: $$a_k \ge 7k \,\,\,\,\, for \,\,\,\,\, k = 1,2,..., 20$$ $$a_1+a_2+...+a_{20}=1518$$ Prove that among the numbers $k = 1,2,... ,20$ there are no more than seventeen, for which $a_k \ge 20k -2k^2$.

Determine the largest natural number that has all its digits different and is a multiple of $5$, $8$ and $11$.

Construct a triangle $ABC$ with the right angle at vertex $C$ given lengths of its medians $m_a$, $m_b$. Discuss conditions of solvability.

Prove that if non-zero complex numbers $\alpha_1,\alpha_2,\alpha_3$ are distinct and noncollinear on the plane, and satisfy $\alpha_1+\alpha_2+\alpha_3=0$, then there holds \[\sum_{i=1}^{3}\left(\frac{|\alpha_{i+1}-\alpha_{i+2}|}{\sqrt{|\alpha_i|}}\left(\frac{1}{\sqrt{|\alpha_{i+1}|}}+\frac{1}{\sqrt{|\alpha_{i+2}|}}-\frac{2}{\sqrt{|\alpha_{i}|}}\right)\right)\leq 0......(*)\] where $\alpha_4=\alpha_1, \alpha_5=\alpha_2$. Verify further the sufficient and necessary condition for the equality holding in $(*)$.

Determine the sum of all possible positive integers $n, $ the product of whose digits equals $n^2 -15n -27$.

Show that if the distance between opposite edges of a tetrahedron is at least $1$, then its volume is at least $\frac{1}{3}$. [i]Proposed by Simeon Kiflie[/i]

Let $x$ be a real number that satisfies $x^1 + x^{-1} = 3$ Prove that $x^n + x^{-n}$ is an positive integer , then prove that the positive integer $x^{3^{1437}}+x^{3^{-1437}}$ is divisible by at least $1439 \times 2^{1437}$ positive integers

Prove that a graph containing a copy of each possible tree on $n$ vertices as a subgraph has at least $n(\ln n - 2)$ edges.

Solve the equation $|x-m|+|x+m|=x$ depending on the value of the parameter $m$.

Commercial vinegar is a $5.00\%$ by mass aqueous solution of acetic acid, $\ce{CH3CO2H}$ $(M=60.0)$. What is the molarity of acetic acid in vinegar? [density of vinegar = 1.00g/mL) $ \textbf{(A)}\hspace{.05in}0.833 M\qquad\textbf{(B)}\hspace{.05in}1.00 M\qquad\textbf{(C)}\hspace{.05in}1.20 M\qquad\textbf{(D)}\hspace{.05in}3.00 M\qquad$

For any positive $ a_1 ,a_2 ,...,a_n$ prove that $ \frac {{a_1 }} {{a_2 \plus{} a_3 }} \plus{} \frac {{a_2 }} {{a_3 \plus{} a_4 }} \plus{} ... \plus{} \frac {{a_n }} {{a_1 \plus{} a_2 }} > \frac {n} {4}$ holds.