$AB$ and $CD$ are two fixed parallel chords of the circle $S$. $M$ is a variable point on the circle. $Q$ is the intersection of the lines $MD$ and $AB$. $X$ is the circumcenter of the triangle $MCQ$. Find the locus of $X$. What happens to $X$ as $M$ tends to (1) $D$, (2) $C$? Find a point $E$ outside the plane of $S$ such that the circumcenter of the tetrahedron $MCQE$ has the same locus as $X$.

Let $ABC$ be a triangle with $\angle BAC =60^\circ$. $A'$ is the symmetric point of $A$ wrt $\overline{BC}$. $D$ is the point in $\overline{AC}$ such that $\overline{AB}=\overline{AD}$. $H$ is the orthocenter of triangle $ABC$. $l$ is the external angle bisector of $\angle BAC$. $\{M\}=\overline{A'D}\cap l$,$\{N\}=\overline{CH} \cap l$. Show that $\overline{AM}=\overline{AN}$.

A quadratic polynomial $p(x)$ with integer coefficients satisfies $p(41) = 42$. For some integers $a, b > 41$, $p(a) = 13$ and $p(b) = 73$. Compute the value of $p(1)$. [i]Proposed by Aaron Lin[/i]

Let $n$ be a positive integer. Suppose that the following simultaneous equations $$\begin{cases} \sin x_1 + \sin x_2+ ...+ \sin x_n = 0 \\ \sin x_1 + 2\sin x_2+ ...+ n \sin x_n = 100 \end{cases}$$ has a solution, where $x_1 x_2,.., x_n$ are the unknowns. Find the smallest possible positive integer $n$. Justify your answer.

An integer $a$ is given. Find all real-valued functions $f (x)$ defined on integers $x \ge a$, satisfying the equation $f (x+y) = f (x) f (y)$ for all $x,y \ge a$ with $x + y \ge a$.

Let $A$ be a subset with seven elements of the set $\{1,2,3, ...,26\}$. Show that there are two distinct elements of $A$, having the same sum of their elements.

Let $ P(z) \equal{} z^3 \plus{} az^2 \plus{} bz \plus{} c$, where $ a$, $ b$, and $ c$ are real. There exists a complex number $ w$ such that the three roots of $ P(z)$ are $ w \plus{} 3i$, $ w \plus{} 9i$, and $ 2w \minus{} 4$, where $ i^2 \equal{} \minus{} 1$. Find $ |a \plus{} b \plus{} c|$.

We are given $100$ strictly increasing sequences of positive integers: $A_{i}= (a_{1}^{(i)}, a_{2}^{(i)},...), i = 1, 2,..., 100$. For $1 \leq r, s \leq 100$ we define the following quantities: $f_{r}(u)=$ the number of elements of $A_{r}$ not exceeding $n$; $f_{r,s}(u) =$ the number of elements of $A_{r}\cap A_{s}$ not exceeding $n$. Suppose that $f_{r}(n) \geq\frac{1}{2}n$ for all $r$ and $n$. Prove that there exists a pair of indices $(r, s)$ with $r \not = s$ such that $f_{r,s}(n) \geq\frac{8n}{33}$ for at least five distinct $n-s$ with $1 \leq n < 19920.$

A convex hexagon $ABCDEF$ is given, with each two opposite sides of different lengths and parallel ($AB \parallel DE$, $BC \parallel EF$ and $CD \parallel FA$). If $|AE| = |BD|$ and $|BF| = |CE|$, prove that the hexagon $ABCDEF$ is cyclic.

Let $ABC$ be an acute-angled triangle with $AB \neq AC$, circumcentre $O$ and circumcircle $\Gamma$. Let the tangents to $\Gamma$ at $B$ and $C$ meet each other at $D$, and let the line $AO$ intersect $BC$ at $E$. Denote the midpoint of $BC$ by $M$ and let $AM$ meet $\Gamma$ again at $N \neq A$. Finally, let $F \neq A$ be a point on $\Gamma$ such that $A, M, E$ and $F$ are concyclic. Prove that $FN$ bisects the segment $MD$.

How many ways can you color the squares of a $ 2 \times 2008$ grid in 3 colors such that no two squares of the same color share an edge?

Let $A$ and $B$ be points on the same branch of the hyperbola $xy=1.$ Suppose that $P$ is a point lying between $A$ and $B$ on this hyperbola, such that the area of the triangle $APB$ is as large as possible. Show that the region bounded by the hyperbola and the chord $AP$ has the same area as the region bounded by the hyperbola and the chord $PB.$

Let $n$ be a positive integer. Let $(a, b, c)$ be a random ordered triple of nonnegative integers such that $a + b + c = n$, chosen uniformly at random from among all such triples. Let $M_n$ be the expected value (average value) of the largest of $a$, $b$, and $c$. As $n$ approaches infinity, what value does $\frac{M_n}{n}$ approach?

Determine, with proof, the maximum and minimum among the numbers $$\sqrt5 - \lfloor \sqrt5 \rfloor, 2\sqrt5 - \lfloor 2\sqrt5 \rfloor, 3\sqrt5 - \lfloor 3 \sqrt5\rfloor, ..., 2013\sqrt5 - \lfloor 2013\sqrt5\rfloor, 2014\sqrt5 - \lfloor 2014\sqrt5\rfloor $$

Find all integer solutions to $2 x^4 + 1 = y^2.$

For each prime number $p$, determine the number of residue classes modulo $p$ which can be represented as $a^2+b^2$ modulo $p$, where $a$ and $b$ are arbitrary integers. [i](Daniel Holmes)[/i]

Let $h \ge 3$ be an integer and $X$ the set of all positive integers that are greater than or equal to $2h$. Let $S$ be a nonempty subset of $X$ such that the following two conditions hold: [list] [*]if $a + b \in S$ with $a \ge h, b \ge h$, then $ab \in S$; [*]if $ab \in S$ with $a \ge h, b \ge h$, then $a + b \in S$.[/list] Prove that $S = X$.

Prove that \[ \left(\frac{6}{5}\right)^{\sqrt{3}}>\left(\frac{5}{4}\right)^{\sqrt{2}}. \]

A $15$-inch-long stick has four marks on it, dividing it into five segments of length $1,2,3, 4$, and $5$ inches (although not neccessarily in that order) to make a “ruler.” Here is an example. [img]https://cdn.artofproblemsolving.com/attachments/0/e/065d42b36083453f3586970125bedbc804b8a1.png[/img] Using this ruler, you could measure $8$ inches (between the marks $B$ and $D$) and $11$ inches (between the end of the ruler at $A$ and the mark at $E$), but there’s no way you could measure $12$ inches. Prove that it is impossible to place the four marks on the stick such that the five segments have length $1,2,3, 4$, and $5$ inches, and such that every integer distance from $1$ inch through $15$ inches could be measured.

If $x\neq 0$ or $4$ and $y \neq 0$ or $6$, then $\frac{2}{x}+\frac{3}{y}=\frac{1}{2}$ is equivalent to $ \textbf{(A)}\ 4x+3y=xy \qquad\textbf{(B)}\ y=\frac{4x}{6-y} \qquad\textbf{(C)}\ \frac{x}{2}+\frac{y}{3}=2 \\ \qquad\textbf{(D)}\ \frac{4y}{y-6}=x \qquad\textbf{(E)}\ \text{none of these} $

Let $d$ be the greatest common divisor of $2^{30^{10}}-2$ and $2^{30^{45}}-2$. Find the remainder when $d$ is divided by $2013$.

A mathematical competition was attended by 120 participants from several contingents. At the closing ceremony, each participant gave 1 souvenir each to every other participants from the same contingent, and 1 souvenir to any person from every other contingents. It is known that there are 3840 souvenirs whom were exchanged. Find the maximum possible contingents such that the above condition still holds? [i]Raymond Christopher Sitorus, Singapore[/i]

Find all pairs of positive integers $\left( x;\;y\right) $ satisfying the equation $2^{x}=3^{y}+5.$

In a triangle $\triangle ABC$ points $L, P$ and $Q$ lie on the segments $AB, AC$ and $BC$, respectively, and are such that $PCQL$ is a parallelogram. The circle with center the midpoint $M$ of the segment $AB$ and radius $CM$ and the circle of diameter $CL$ intersect for the second time at the point $T$. Prove that the lines $AQ, BP$ and $LT$ intersect in a point.

Through the midpoint $ S $ of the segment $ MN $ with endpoints lying on the legs of an isosceles triangle, a straight line is drawn parallel to the base of the triangle, intersecting its legs at points $ K $ and $ L $. Prove that the orthogonal projection of the segment $ MN $ onto the base of the triangle is equal to the segment $ KL $.