Two circles $C_{1}$ and $C_{2}$ are (externally or internally) tangent at a point $P$. The straight line $D$ is tangent at $A$ to one of the circles and cuts the other circle at the points $B$ and $C$. Prove that the straight line $PA$ is an interior or exterior bisector of the angle $\angle BPC$.

Let $ABC$ be a scalene triangle and $M$ be the midpoint of $BC$. Let $X$ be the point such that $CX \parallel AB$ and $\angle AMX = 90^{\circ}.$ Prove that $AM$ bisects $\angle BAX$.

Let $a$, $b$, and $c$ be real numbers such that $0\le a,b,c\le 5$ and $2a + b + c = 10$. Over all possible values of $a$, $b$, and $c$, determine the maximum possible value of $a + 2b + 3c$. [i]Proposed by Andrew Wen[/i]

For $ n\geq2$ let $ a_1, a_2, \ldots a_n$ be positive real numbers such that \[ (a_1 \plus{} a_2 \plus{} \cdots \plus{} a_n)\left(\frac {1}{a_1} \plus{} \frac {1}{a_2} \plus{} \cdots \plus{} \frac {1}{a_n}\right) \leq \left(n \plus{} \frac {1}{2}\right)^2. \] Prove that $ \max(a_1, a_2, \ldots, a_n)\leq 4\min(a_1, a_2, \ldots, a_n)$.

The Lindelöf number $L(X)$ of a topological space $X$ is the least infinite cardinal $\lambda$ with the property that every open covering of $X$ has a subcovering of cardinality at most $\lambda$. Prove that if evert non-countably infinite subset of a first countable space $X$ has a point of condensation, then $L(X)=\sup L(A)$, where $A$ runs over the separable closed subspaces of $X$. (A point of condensation of a subset $H\subseteq X$ is a point $x\in X$ such that any neighbourhood of $x$ intersects $H$ in a non-countably infinite set.)

Given are three numbers $a, b, c \in R-\{0\}$. The parabola with equation $y = ax^2+bx+c$ lies above the line $y = cx$. Prove that the parabola with equation $y = cx^2 - bx + a$ lies above the line $y = cx - b$.

Find the maximal value of \[S = \sqrt[3]{\frac{a}{b+7}} + \sqrt[3]{\frac{b}{c+7}} + \sqrt[3]{\frac{c}{d+7}} + \sqrt[3]{\frac{d}{a+7}},\] where $a$, $b$, $c$, $d$ are nonnegative real numbers which satisfy $a+b+c+d = 100$. [i]Proposed by Evan Chen, Taiwan[/i]

Suppose $f \colon \mathbb{R} \longrightarrow \mathbb{R}$ is differentiable and $| f'(x)| < \frac{1}{2}$ for all $x \in \mathbb{R}$. Show that for some $x_0 \in \mathbb{R}$, $f \left( x_0 \right) = x_0$.

Let $n > 1$ and $x_i \in \mathbb{R}$ for $i = 1,\cdots, n$. Set \[S_k = x_1^k+ x^k_2+\cdots+ x^k_n\] for $k \ge 1$. If $S_1 = S_2 =\cdots= S_{n+1}$, show that $x_i \in \{0, 1\}$ for every $i = 1, 2,\cdots, n.$

Let $ C$ and $ D$ be two intersection points of circle $ O_1$ and circle $ O_2$. A line, passing through $ D$, intersects the circle $ O_1$ and the circle $ O_2$ at the points $ A$ and $ B$ respectively. The points $ P$ and $ Q$ are on circles $ O_1$ and $ O_2$ respectively. The lines $ PD$ and $ AC$ intersect at $ H$, and the lines $ QD$ and $ BC$ intersect at $ M$. Suppose that $ O$ is the circumcenter of the triangle $ ABC$. Prove that $ OD\perp MH$ if and only if $ P,Q,M$ and $ H$ are concyclic.

There are $n \ge 3$ positive integers written on a board. A [i]move[/i] consists of choosing three numbers $a, b, c$ written from the board such that there exists a non-degenerate non-equilateral triangle with sides $a, b, c$ and replacing those numbers with $a + b - c, b + c - a$ and $c + a - b$. Prove that a sequence of moves cannot be infinite.

Find all triples of positive integers $(m,p,q)$ such that $2^mp^2 + 27 = q^3$ and $p$ is a prime.

On the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ find the point $T=(x_0,y_0)$ such that the triangle bounded by the axes of the ellipse and the tangent at that point has the least area.

Let $\mathbb N$ denote the set of positive integers, and for a function $f$, let $f^k(n)$ denote the function $f$ applied $k$ times. Call a function $f : \mathbb N \to \mathbb N$ [i]saturated[/i] if \[ f^{f^{f(n)}(n)}(n) = n \] for every positive integer $n$. Find all positive integers $m$ for which the following holds: every saturated function $f$ satisfies $f^{2014}(m) = m$. [i]Proposed by Evan Chen[/i]

Assume you have a magical pizza in the shape of an infinite plane. You have a magical pizza cutter that can cut in the shape of an infinite line, but it can only be used $14$ times. To share with as many of your friends as possible, you cut the pizza in a way that maximizes the number of finite pieces (the infinite pieces have infinite mass, so you can’t lift them up). How many finite pieces of pizza do you have?

Let points $A=(0,0,0)$, $B=(1,0,0)$, $C=(0,2,0)$, and $D=(0,0,3)$. Points $E,F,G$, and $H$ are midpoints of line segments $\overline{BD},\overline{AB},\overline{AC}$, and $\overline{DC}$ respectively. What is the area of $EFGH$? $ \textbf{(A)}\ \sqrt2 \qquad\textbf{(B)}\ \frac{2\sqrt5}{3} \qquad\textbf{(C)}\ \frac{3\sqrt5}{4} \qquad\textbf{(D)}\ \sqrt3 \qquad\textbf{(E)}\ \frac{2\sqrt7}{3} $

The first $1997$ natural numbers are written on the blackboard: $1,2,3,\ldots ,1997$. In front of each number, a "$+$" sign or a "$-$" sign will be written in order, from left to right. To decide each sign, a coin is tossed; If it comes up heads, you write "$+$", if it comes up tails, you write "$-$". Once the $1997$ signs are written, the algebraic sum of the expression on the blackboard is carried out and the result is $S$. What is the probability that $S$ is greater than $0$? Clarification: The probability of an event is equal to the number of favorable cases/number of possible cases.

Given a number $k\in \mathbb{N}$. $\{a_{n}\}_{n\geq 0}$ and $\{b_{n}\}_{n\geq 0}$ are two sequences of positive integers that $a_{i},b_{i}\in \{1,2,\cdots,9\}$. For all $n\geq 0$ $$\left.\overline{a_{n}\cdots a_{1}a_{0}}+k \ \middle| \ \overline{b_{n}\cdots b_{1}b_{0}}+k \right. .$$ Prove that there is a number $1\leq t \leq 9$ and $N\in \mathbb{N}$ such that $b_n=ta_n$ for all $n\geq N$.\\ (Note that $(\overline{x_nx_{n-1}\dots x_0}) = 10^n\times x_n + \dots + 10\times x_1 + x_0$)

Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an $11$ on the next quiz, her mean will increase by $1$. If she scores an $11$ on each of the next three quizzes, her mean will increase by $2$. What is the mean of her quiz scores currently? $\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad\textbf{(E) }8$

Let $a_1, a_2, ..., a_m$ be positive integers, none of which is equal to $10$, such that $a_1 + a_2 + ...+ a_m = 10m$. Prove that $(a_1a_2a_3 \cdot ...\cdot a_m)^{1/m} \le 3\sqrt{11}$.

Is there a whole number which becomes exactly $57$ times less than itself when one crosses out its first digit?

On the infinite surface of the sea floats a black and bounded oil slick. After every minute the slick and the sea change according to the following law: at each point $P$ of the sea (or of the slick), a disk $D$ of radius $1$ is considered centered on $ P$. If more than half of the area inside the disk $D$ is black, the $P$ point will remain black for the next minute. If more than half of the area inside the disk $D$ is dark blue, the point $P$ will be dark blue for the next minute. In the event that both the clean and the contaminated area within the disk $D$ are the same, its center $P$ will not change color. Can that stain "live" forever or will it disappear at some point?

On a chessboard with $6$ rows and $9$ columns, the Slow Rook is placed in the bottom-left corner and the Blind King is placed on the top-left corner. Then, $8$ Sleeping Pawns are placed such that no two Sleeping Pawns are in the same column, no Sleeping Pawn shares a row with the Slow Rook or the Blind King, and no Sleeping Pawn is in the rightmost column. The Slow Rook can move vertically or horizontally $1$ tile at a time, the Slow Rook cannot move into any tile containing a Sleeping Pawn, and the Slow Rook takes the shortest path to reach the Blind King. How many ways are there to place the Sleeping Pawns such that the Slow Rook moves exactly $15$ tiles to get to the space containing the Blind King? Proposed by Harry Chen (Extile)

Let $x,y,z$ be three positive integers with $\gcd(x,y,z)=1$. If \[x\mid yz(x+y+z),\] \[y\mid xz(x+y+z),\] \[z\mid xy(x+y+z),\] and \[x+y+z\mid xyz,\] show that $xyz(x+y+z)$ is a perfect square. [i]Proposed by usjl[/i]

Let $ABC$ be an acute triangle with circumcenter $O$. Points $E$ and $F$ are chosen on segments $OB$ and $OC$ such that $BE = OF$. If $M$ is the midpoint of the arc $EOA$ and $N$ is the midpoint of the arc $AOF$, prove that $\sphericalangle ENO + \sphericalangle OMF = 2 \sphericalangle BAC$.