Determine all $(m,n) \in \mathbb{Z}^+ \times \mathbb{Z}^+$ which satisfy $3^m-7^n=2.$

Let $A,B,C,D$ be points on a circle with radius $r$ in this order such that $|AB|=|BC|=|CD|=s$ and $|AD|=s+r$. Find all possible values of the interior angles of the quadrilateral $ABCD$.

Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every $ 90$ seconds, and Robert runs clockwise and completes a lap every $ 80$ seconds. Both start from the start line at the same time. At some random time between $ 10$ minutes and $ 11$ minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture? $ \textbf{(A)}\ \frac{1}{16}\qquad \textbf{(B)}\ \frac18\qquad \textbf{(C)}\ \frac{3}{16} \qquad \textbf{(D)}\ \frac14\qquad \textbf{(E)}\ \frac{5}{16}$

Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.

Consider $A,B\in\mathcal{M}_n(\mathbb{C})$ for which there exist $p,q\in\mathbb{C}$ such that $pAB-qBA=I_n$. Prove that either $(AB-BA)^n=O_n$ or the fraction $\frac{p}{q}$ is well-defined ($q \neq 0$) and it is a root of unity. [i](Sergiu Novac)[/i]

Given is a convex quadrilateral $ABCD$ with $AB=CD$. Draw the triangles $ABE$ and $CDF$ outside $ABCD$ so that $\angle{ABE} = \angle{DCF}$ and $\angle{BAE}=\angle{FDC}$. Prove that the midpoints of $\overline{AD}$, $\overline{BC}$ and $\overline{EF}$ are collinear.

21 people take a test with 15 true or false questions. It is known that every 2 people have at least 1 correct answer in common. What is the minimum number of people that could have correctly answered the question which the most people were correct on?

Suppose $$\frac{1}{3}\frac{(x+1)(x-3)}{(x+2)(x-4)} + \frac{1}{4}\frac{(x+3)(x-5)}{(x+4)(x-6)} - \frac{2}{11}\frac{(x+5)(x-7)}{(x+6)(x-8)} = \frac{53}{132}.$$ Also, suppose $x > 0$. Then $x$ can be written as $a + \sqrt{b}$ where $a$ and $b$ are integers. Find $a + b$.

Let $\Delta ABC$ be a triangle with angle $\angle CAB=60^{\circ}$, let $D$ be the intersection point of the angle bisector at $A$ and the side $BC$, and let $r_B,r_C,r$ be the respective radii of the incircles of $ABD$, $ADC$, $ABC$. Let $b$ and $c$ be the lengths of sides $AC$ and $AB$ of the triangle. Prove that \[ \frac{1}{r_B} +\frac{1}{r_C} ~=~ 2\cdot\left( \frac1r +\frac1b +\frac1c\right)\]

Let $G$ be an undirected simple graph. Let $f(G)$ be the number of ways to orient all of the edges of $G$ in one of the two possible directions so that the resulting directed graph has no directed cycles. Show that $f(G)$ is a multiple of $3$ if and only if $G$ has a cycle of odd length.

Find all pairs $(x,y)$ of positive real numbers such that $xy$ is an integer and $x+y = \lfloor x^2 - y^2 \rfloor$.

Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]

Show that the equation $z^{n}+z+1=0$ has a solution with $|z|=1$ if and only if $n-2$ is divisble by $3$.

Let $a,b,c$ be sides of a triangle. Show that $a^3+b^3+3abc>c^3$.

Let $a,b$ and $c$ be positive real numbers satisfying $\min(a+b,b+c,c+a) > \sqrt{2}$ and $a^2+b^2+c^2=3.$ Prove that \[\frac{a}{(b+c-a)^2} + \frac{b}{(c+a-b)^2} + \frac{c}{(a+b-c)^2} \geq \frac{3}{(abc)^2}.\] [i]Proposed by Titu Andreescu, Saudi Arabia[/i]

Let point $T$ be on side $AB$ of $\Delta ABC$ be such that $AT-BT=AC-BC$. The perpendicular from point $T$ to $AB$ intersects $AC$ in point $E$ and the angle bisectors of $\angle B$ and $\angle C$ intersect the circumscribed circle $k$ of $ABC$ in points $M$ and $L$. If $P$ is the second intersection point of the line $ME$ with $k$, then prove that $P,T,L$ are collinear.

Let $S$ be a subset of the set $\{1, 2, 3, \dots, 2015\}$ such that for any two elements $a, b \in S$, the difference $a - b$ does not divide the sum $a + b$. Find the maximum possible size of $S$.

Find the least real number $m$ such that with all $5$ equilaterial triangles with sum of areas $m$ we can cover an equilaterial triangle with side 1. [i]O. Mushkarov, N. Nikolov[/i]

Find the least positive integer $n$ ($n\geq 3$), such that among any $n$ points (no three are collinear) in the plane, there exist three points which are the vertices of a non-isoscele triangle.

Each square in a $3 \times 3$ grid is randomly filled with one of the $4$ gray-and-white tiles shown below on the right.[asy] size(5.663333333cm); draw((0,0)--(3,0)--(3,3)--(0,3)--cycle,gray); draw((1,0)--(1,3)--(2,3)--(2,0),gray); draw((0,1)--(3,1)--(3,2)--(0,2),gray); fill((6,.33)--(7,.33)--(7,1.33)--cycle,mediumgray); draw((6,.33)--(7,.33)--(7,1.33)--(6,1.33)--cycle,gray); fill((6,1.67)--(7,2.67)--(6,2.67)--cycle,mediumgray); draw((6,1.67)--(7,1.67)--(7,2.67)--(6,2.67)--cycle,gray); fill((7.33,.33)--(8.33,.33)--(7.33,1.33)--cycle,mediumgray); draw((7.33,.33)--(8.33,.33)--(8.33,1.33)--(7.33,1.33)--cycle,gray); fill((8.33,1.67)--(8.33,2.67)--(7.33,2.67)--cycle,mediumgray); draw((7.33,1.67)--(8.33,1.67)--(8.33,2.67)--(7.33,2.67)--cycle,gray); [/asy] What is the probability that the tiling will contain a large gray diamond in one of the smaller $2\times 2$ grids? Below is an example of one such tiling. [asy] size(2cm); fill((1,0)--(0,1)--(0,2)--(1,1)--cycle,mediumgray); fill((2,0)--(3,1)--(2,2)--(1,1)--cycle,mediumgray); fill((1,2)--(1,3)--(0,3)--cycle,mediumgray); fill((1,2)--(2,2)--(2,3)--cycle,mediumgray); fill((3,2)--(3,3)--(2,3)--cycle,mediumgray); draw((0,0)--(3,0)--(3,3)--(0,3)--cycle,gray); draw((1,0)--(1,3)--(2,3)--(2,0),gray); draw((0,1)--(3,1)--(3,2)--(0,2),gray); [/asy] $\textbf{(A) } \frac{1}{1024} \qquad \textbf{(B) } \frac{1}{256} \qquad \textbf{(C) } \frac{1}{64} \qquad \textbf{(D) } \frac{1}{16} \qquad \textbf{(E) } \frac{1}{4}$

Let be given $n$ points, no three of which are on a line. All the segments with endpoints in these points are colored so that two segments with a common endpoint are of different colors. Determine the least number of colors for which this is possible

Square $ ABCD$ has area $ 36,$ and $ \overline{AB}$ is parallel to the x-axis. Vertices $ A,$ $ B,$ and $ C$ are on the graphs of $ y \equal{} \log_{a}x,$ $ y \equal{} 2\log_{a}x,$ and $ y \equal{} 3\log_{a}x,$ respectively. What is $ a?$ $ \textbf{(A)}\ \sqrt [6]{3}\qquad \textbf{(B)}\ \sqrt {3}\qquad \textbf{(C)}\ \sqrt [3]{6}\qquad \textbf{(D)}\ \sqrt {6}\qquad \textbf{(E)}\ 6$

Let $a_1,a_2,\cdots,a_n$ be a permutation of $1,2,\cdots,n$. Among all possible permutations, find the minimum of $$\sum_{i=1}^n \min \{ a_i,2i-1 \}.$$

Call a polynomial $P$ [i]prime-covering[/i] if for every prime $p$, there exists an integer $n$ for which $p$ divides $P(n)$. Determine the number of ordered triples of integers $(a,b,c)$, with $1\leq a < b < c \leq 25$, for which $P(x)=(x^2-a)(x^2-b)(x^2-c)$ is prime-covering.

In an acute triangle $ABC$ the angle $\angle C$ is greater than $\angle A$. Let $E$ be such that $AE$ is a diameter of the circumscribed circle $\Gamma$ of \vartriangle ABC. Let $K$ be the intersection of $AC$ and the tangent line at $B$ to $\Gamma$. Let $L$ be the orthogonal projection of $K$ on $AE$ and let $D$ be the intersection of $KL$ and $AB$. Prove that $CE$ is the bisector of $\angle BCD$.