Let $ABC$ be an acute-angled triangle. Points $A_0$ and $C_0$ are the midpoints of minor arcs $BC$ and $AB$ respectively. A circle passing though $A_0$ and $C_0$ meet $AB$ and $BC$ at points $P$ and $S$ , $Q$ and $R$ respectively ([i]all these points are distinct[/i]). It is known that $PQ\parallel AC$. Prove that $A_0P+C_0S=C_0Q+A_0R$.

A circle with radius $r$ is contained within the region bounded by a circle with radius $R$. The area bounded by the larger circle is $a/b$ times the area of the region outside the smaller circle and inside the larger circle. Then $R:r$ equals: $\textbf{(A) }\sqrt a:\sqrt b\qquad \textbf{(B) }\sqrt a:\sqrt{a-b}\qquad \textbf{(C) }\sqrt b:\sqrt{a-b}\qquad$ $\textbf{(D) }a:\sqrt{a-b}\qquad \textbf{(E) }b:\sqrt{a-b}$

Let $a$, $b$, and $c$ be positive integers for which the number \[\frac{a\sqrt2+b}{b\sqrt2+c}\] is rational. Show that the number $ab+bc+ca$ is divisible by $a+b+c$.

The points $D$ and $E$ on the circumcircle of an acute triangle $ABC$ are such that $AD=AE = BC$. Let $H$ be the common point of the altitudes of triangle $ABC$. It is known that $AH^{2}=BH^{2}+CH^{2}$. Prove that $H$ lies on the segment $DE$. [i]Proposed by D. Shiryaev[/i]

In a cyclic quadrilateral $ABCD$, the diagonals intersect at $E$. $F$ and $G$ are on chord $AC$ and chord $BD$ respectively such that $AF = BE$ and $DG = CE$. Prove that, $A, G, F, D$ lie on the same circle.

For every positive integer $n$, find the maximum power of $2$ that divides the number $$1 + 2019 + 2019^2 + 2019^3 +.. + 2019^{n-1}.$$

Alice, Bob, and Carol repeatedly take turns tossing a die. Alice begins; Bob always follows Alice; Carol always follows Bob; and Alice always follows Carol. Find the probability that Carol will be the first one to toss a six. (The probability of obtaining a six on any toss is $ \frac{1}{6}$, independent of the outcome of any other toss.) $ \textbf{(A)}\ \frac{1}{3}\qquad \textbf{(B)}\ \frac{2}{9}\qquad \textbf{(C)}\ \frac{5}{18}\qquad \textbf{(D)}\ \frac{25}{91}\qquad \textbf{(E)}\ \frac{36}{91}$

Let $ABC$ be a triangle. The incircle of triangle $ABC$ touches the side $BC$ at $A^{\prime}$, and the line $AA^{\prime}$ meets the incircle again at a point $P$. Let the lines $CP$ and $BP$ meet the incircle of triangle $ABC$ again at $N$ and $M$, respectively. Prove that the lines $AA^{\prime}$, $BN$ and $CM$ are concurrent.

What is the largest whole number that is equal to the product of its digits?

Find all functions $f: R\to R$ such that for all real numbers $x, y$ is satisfied that $$f (x + y) = (f (x))^{ 2013} + f (y).$$

Eight points are spaced around at intervals of one unit around a $2 \times 2$ square, as shown. Two of the $8$ points are chosen at random. What is the probability that the two points are one unit apart? [asy] size((50)); dot((5,0)); dot((5,5)); dot((0,5)); dot((-5,5)); dot((-5,0)); dot((-5,-5)); dot((0,-5)); dot((5,-5)); [/asy] $ \textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{2}{7}\qquad\textbf{(C)}\ \frac{4}{11}\qquad\textbf{(D)}\ \frac{1}{2}\qquad\textbf{(E)}\ \frac{4}{7} $

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the following relationship for all real numbers $x$ and $y$\[f(xf(y)-f(x))=2f(x)+xy.\]

Let $ p_i \geq 2$, $ i \equal{} 1,2, \cdots n$ be $ n$ integers such that any two of them are relatively prime. Let: \[ P \equal{} \{ x \equal{} \sum_{i \equal{} 1}^{n} x_i \prod_{j \equal{} 1, j \neq i}^{n} p_j \mid x_i \text{is a non \minus{} negative integer}, i \equal{} 1,2, \cdots n \} \] Prove that the biggest integer $ M$ such that $ M \not\in P$ is greater than $ \displaystyle \frac {n \minus{} 2}{2} \cdot \prod_{i \equal{} 1}^{n} p_i$, and also find $ M$.

A rook starts moving on an infinite chessboard, alternating horizontal and vertical moves. The length of the first move is one square, of the second – two squares, of the third – three squares and so on. a) Is it possible for the rook to arrive at its starting point after exactly $2013$ moves? b) Find all $n$ for which it possible for the rook to come back to its starting point after exactly $n$ moves.

The [i]harmonic mean[/i] of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4? $\textbf{(A) }\frac{3}{7}\qquad\textbf{(B) }\frac{7}{12}\qquad\textbf{(C) }\frac{12}{7}\qquad\textbf{(D) }\frac{7}{4}\qquad \textbf{(E) }\frac{7}{3}$

Find all (finite) increasing arithmetic progressions, consisting only of prime numbers, such that the number of terms is larger than the common difference.

Find all real numbers $x$, such that: a) $\lfloor x \rfloor + \lfloor 2x \rfloor +...+ \lfloor 2012x \rfloor = 2013$ b) $\lfloor x \rfloor + \lfloor 2x \rfloor +...+ \lfloor 2013x \rfloor = 2014$

Let $ABC$ be a triangle with incenter $I$ and incircle $\omega$. Let $\ell$ be the tangent to $\omega$ parallel to $BC$ and distinct from $BC$. Let $D$ be the intersection of $\ell$ and $AC$, and let $M$ be the midpoint of $\overline{ID}$. Prove that $\angle AMD = \angle DBC$.

Find all pairs $(a,b)$ of real numbers such that $\lfloor an + b \rfloor$ is a perfect square, for all positive integer $n$.

In acute triangle $\triangle ABC$, $\overline{AB} = 20$ and $\overline{AC} = 21$. Let the feet of the perpendiculars from $A$ to the angle bisectors of $\angle ACB$ and $\angle ABC$ be $X$ and $Y$, respectively. Let $M$ be the midpoint of $\overline{XY}$. Suppose $P$ is the point on side $BC$ such that $MP$ is parallel to the angle bisector of $\angle BAC$. If given that $\overline{BP} = 11$, then the length of side $BC$ can be expressed as a common fraction in the form $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let $S = \{1,2,\dots,2014\}$. For each non-empty subset $T \subseteq S$, one of its members is chosen as its representative. Find the number of ways to assign representatives to all non-empty subsets of $S$ so that if a subset $D \subseteq S$ is a disjoint union of non-empty subsets $A, B, C \subseteq S$, then the representative of $D$ is also the representative of one of $A$, $B$, $C$. [i]Warut Suksompong, Thailand[/i]

$ABC$ is a triangle with $|AB|=6$, $|BC|=7$, and $|AC|=8$. Let the angle bisector of $\angle A$ intersect $BC$ at $D$. If $E$ is a point on $[AC]$ such that $|CE|=2$, what is $|DE|$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ \frac {17}5 \qquad\textbf{(C)}\ \frac 72 \qquad\textbf{(D)}\ 2\sqrt 3 \qquad\textbf{(E)}\ 3\sqrt 2 $

Compute $\frac{x^2 + 8x + 7}{x^2 + 9x + 14}$, if $x = 2015$.

(a) A circle is inscribed in a triangle. The tangent points are the vertices of a second triangle in which another circle is inscribed. Its tangency points are the vertices of a third triangle. The angles of this triangle are identical to those of the first triangle. Find these angles. (b) A circle is inscribed in a scalene triangle. The tangent points are vertices of another triangle, in which a circle is inscribed whose tangent points are vertices of a third triangle, in which a third circle is inscribed, etc. Prove that the resulting sequence does not contain a pair of similar triangles.

Let triangle $\vartriangle ABC$ have side lengths $AB = 6$, $BC = 8$, and $CA = 10$. Let $S_1$ be the largest square fitting inside of $\vartriangle ABC$ (sharing points on edges is allowed). Then, for $i \ge 2$, let $S_i$ be the largest square that fits inside of $\vartriangle ABC$ while remaining outside of all other squares $S_1$,$...$, $S_{i-1}$ (with ties broken arbitrarily). For all $i \ge 1$, let $m_i$ be the side length of $S_i$ and let $S$ be the set of all $m_i$. Let $x$ be the $2023$rd largest value in $S$. Compute $\log_2 \left( \frac{1}{x}\right).$ Submit your answer as a decimal $E$ to at most $3$ decimal places. If the correct answer is $A$, your score for this question will be $\max(0, 25 -2|A - E|)$, rounded to the nearest integer