Denote by $ M$ midpoint of side $ BC$ in an isosceles triangle $ \triangle ABC$ with $ AC = AB$. Take a point $ X$ on a smaller arc $ \overarc{MA}$ of circumcircle of triangle $ \triangle ABM$. Denote by $ T$ point inside of angle $ BMA$ such that $ \angle TMX = 90$ and $ TX = BX$. Prove that $ \angle MTB - \angle CTM$ does not depend on choice of $ X$. [i]Author: Farzan Barekat, Canada[/i]

Given a positive integer $n$. (a) If $P$ is a polynomial of degree $n$ where $P(x) \in \mathbb{Z}$ for every $x \in \mathbb{Z}$, prove that for every $a,b \in \mathbb{Z}$ where $P(a) \neq P(b)$, \[\text{lcm}(1, 2, \ldots, n) \ge \left| \dfrac{a-b}{P(a) - P(b)} \right|\] (b) Find one $P$ (for each $n$) such that the equality case above is achieved for some $a,b \in \mathbb{Z}$.

Frieda the frog begins a sequence of hops on a $3 \times 3$ grid of squares, moving one square on each hop and choosing at random the direction of each hop up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around'' and jumps to the opposite edge. For example if Frieda begins in the center square and makes two hops "up'', the first hop would place her in the top row middle square, and the second hop would cause Frieda to jump to the opposite edge, landing in the bottom row middle square. Suppose Frieda starts from the center square, makes at most four hops at random, and stops hopping if she lands on a corner square. What is the probability that she reaches a corner square on one of the four hops? $\textbf{(A) }\frac{9}{16}\qquad\textbf{(B) }\frac{5}{8}\qquad\textbf{(C) }\frac{3}{4}\qquad\textbf{(D) }\frac{25}{32}\qquad\textbf{(E) }\frac{13}{16}$

$2(81+83+85+87+89+91+93+95+97+99)=$ $\text{(A)}\ 1600 \qquad \text{(B)}\ 1650 \qquad \text{(C)}\ 1700 \qquad \text{(D)}\ 1750 \qquad \text{(E)}\ 1800$

What is the least integer $n\geq 100$ such that $77$ divides $1+2+2^2+2^3+\dots + 2^n$ ? $ \textbf{(A)}\ 101 \qquad\textbf{(B)}\ 105 \qquad\textbf{(C)}\ 111 \qquad\textbf{(D)}\ 119 \qquad\textbf{(E)}\ \text{None} $

Solve the following system of equations: $$xy^2 = 108$$ $$\frac{x^3}{y}= 10^{10}$$

There are 300 points in space. Four planes $A$, $B$, $C$, and $D$ each have the property that they split the 300 points into two equal sets. (No plane contains one of the 300 points.) What is the maximum number of points that can be found inside the tetrahedron whose faces are on $A$, $B$, $C$, and $D$?

In the diagram below, square $ABCD$ with side length 23 is cut into nine rectangles by two lines parallel to $\overline{AB}$ and two lines parallel to $\overline{BC}$. The areas of four of these rectangles are indicated in the diagram. Compute the largest possible value for the area of the central rectangle. [asy] size(250); defaultpen (linewidth (0.7) + fontsize (10)); draw ((0,0)--(23,0)--(23,23)--(0,23)--cycle); label("$A$", (0,23), NW); label("$B$", (23, 23), NE); label("$C$", (23,0), SE); label("$D$", (0,0), SW); draw((0,6)--(23,6)); draw((0,19)--(23,19)); draw((5,0)--(5,23)); draw((12,0)--(12,23)); label("13", (17/2, 21)); label("111",(35/2,25/2)); label("37",(17/2,3)); label("123",(2.5,12.5));[/asy] [i]Proposed by Lewis Chen[/i]

Show that there exist infinitely many integers that cannot be written as the difference between two perfect squares.

Prove that for each tetrahedron, the three products of pairs of opposite edges are sides of a triangle.

(a) Show that $n(n + 1)(n + 2)$ is divisible by $6$. (b) Show that $1^{2015} + 2^{2015} + 3^{2015} + 4^{2015} + 5^{2015} + 6^{2015}$ is divisible by $7$.

A triangular pyramid $ABCD$ is given. Prove that $\frac Rr > \frac ah$, where $R$ is the radius of the circumscribed sphere, $r$ is the radius of the inscribed sphere, $a$ is the length of the longest edge, $h$ is the length of the shortest altitude (from a vertex to the opposite face).

All the vertices of a convex pentagon are on lattice points. Prove that the area of the pentagon is at least $\frac{5}{2}$. [i]Bogdan Enescu[/i]

Edward’s formula for the stock market predicts correctly that the price of HMMT is directly proportional to a secret quantity $x$ and inversely proportional to $y$, the number of hours he slept the night before. If the price of HMMT is $\$12$ when $x = 8$ and $y = 4$, how many dollars does it cost when $x = 4$ and $y = 8$?

Points $E$ and $F$ lie on diagonal $\overline{AC}$ of square $ABCD$ with side length $24$, such that $AE = CF = 3\sqrt2$. An ellipse with foci at $E$ and $F$ is tangent to the sides of the square. Find the sum of the distances from any point on the ellipse to the two foci.

Suppose $a$ and $b$ are positive integers with a curious property: $(a^3 - 3ab +\tfrac{1}{2})^n + (b^3 +\tfrac{1}{2})^n$ is an integer for at least $3$, but at most finitely many different choices of positive integers $n$. What is the least possible value of $a+b$?

Let $(F_n)$ be the sequence defined recursively by $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ for $n\geq 2$. Find all pairs of positive integers $(x,y)$ such that $$5F_x-3F_y=1.$$

Bus tickets from a transportation company are numbered with six digits, ranging from 000000 to 999999. A ticket is considered "lucky" if the sum of the first three digits equals the sum of the last three digits. For example, ticket 721055 is lucky, whereas 003101 is not. Determine how many consecutive tickets a person must buy to guarantee obtaining at least one lucky ticket, regardless of the starting ticket number.

1) Find a $4$-digit number $\overline{PERU}$ such that $\overline{PERU}=(P+E+R+U)^U$. Also prove that there is only one number satisfying this property.

Can the set of positive rationals be split into two nonempty disjoint subsets $\mathbb Q_1$ and $\mathbb Q_2$, such that both are closed under addition, i.e. $p+q\in\mathbb Q_k$ for every $p,q\in\mathbb Q_k$, $k=1,2$? Can it be done when addition is exchanged for multiplication, i.e. $p\cdot q\in\mathbb Q_k$ for every $p,q\in\mathbb Q_k$, $k=1,2$?

Let $p>2$ be a prime. Define a sequence $(Q_{n}(x))$ of polynomials such that $Q_{0}(x)=1, Q_{1}(x)=x$ and $Q_{n+1}(x) =xQ_{n}(x) + nQ_{n-1}(x)$ for $n\geq 1.$ Prove that $Q_{p}(x)-x^p $ is divisible by $p$ for all integers $x.$

Compute all real numbers $a$ such that the polynomial $x^4 + ax^3 + 1$ has exactly one real root.

The [i]Purple Comet! Math Meet[/i] runs from April 27 through May 3, so the sum of the calendar dates for these seven days is $27 + 28 + 29 + 30 + 1 + 2 + 3 = 120.$ What is the largest sum of the calendar dates for seven consecutive Fridays occurring at any time in any year?

The sequence $a_1, a_2, a_3,...$ is defined so that $a_1 = 1$ and $a_{n+1} =\frac{a_1 + a_2 + ...+ a_n}{n}+1$ for $n \ge 1$. Show that for every positive real number $b$ we can find $a_k$ so that $a_k < bk$.

Let $k$ and $n$ be positive integers with $k < n$. Find the number of subsets of $\{1, 2, . . . , n\}$ such that the difference between the largest and smallest elements in the subset is $k$.