Given that $x$, $y$, and $z$ are real numbers that satisfy: \[ x=\sqrt{y^2-\frac{1}{16}}+\sqrt{z^2-\frac{1}{16}} \] \[ y=\sqrt{z^2-\frac{1}{25}}+\sqrt{x^2-\frac{1}{25}} \] \[ z=\sqrt{x^2-\frac{1}{36}}+\sqrt{y^2-\frac{1}{36}} \] and that $x+y+z=\frac{m}{\sqrt{n}}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime, find $m+n$.

If $a$ is any number, $\lfloor a \rfloor$ is $a$ rounded down to the nearest integer. For example, $\lfloor \pi \rfloor =$ $3$. Show that the sequence $\lfloor \frac{2^{1}}{17} \rfloor$, $\lfloor \frac{2^{2}}{17} \rfloor$, $\lfloor \frac{2^{3}}{17} \rfloor$, $\dots$ contains infinitely many odd numbers.

Let $ABC$ be an obtuse triangle with circumcenter $O$ such that $\angle ABC = 15^o$ and $\angle BAC > 90^o$. Suppose that $AO$ meets $BC$ at $D$, and that $OD^2 + OC \cdot DC = OC^2$. Find $\angle C$.

On a $ 50 \times 50$ board, the centers of several unit squares are colored black. Find the maximum number of centers that can be colored black in such a way that no three black points form a right-angled triangle.

If one side of a triangle is $ 12$ inches and the opposite angle is $ 30$ degrees, then the diameter of the circumscribed circle is: $ \textbf{(A)}\ 18\text{ inches} \qquad\textbf{(B)}\ 30\text{ inches} \qquad\textbf{(C)}\ 24\text{ inches} \qquad\textbf{(D)}\ 20\text{ inches}\\ \textbf{(E)}\ \text{none of these}$

Big Welk writes the letters of the alphabet in order, and starts again at $A$ each time he gets to $Z.$ What is the $4^3$-rd letter that he writes down?

Polynomial $P(x)$ has integer coefficients. Prove, that if polynomials $P(x)$ and $P(P(P(x)))$ have common real root, they also have a common integer root.

Let $P(x,y)$ be a polynomial with real coefficients in the variables $x,y$ that is not identically zero. Suppose that $P(\lfloor 2a \rfloor, \lfloor 3a\rfloor) = 0$ for all real numbers $a.$ If $P$ has the minimum possible degree and the coefficient of the monomial $y$ is $4,$ find the coefficient of $x^2y^2$ in $P.$ (The [i]degree[/i] of a monomial $x^my^n$ is $m + n.$ The [i]degree[/i] of a polynomial $P(x,y)$ is then the maximum degree of any of its monomials.)

Given a circle and a fixed point $P$ not lying on it. Find the geometrical locus of the orthocenters of the triangles $ABP$, where $AB$ is the diameter of the circle.

Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

In a multiplication table, the entry in the $i$-th row and the $j$-th column is the product $ij$ From an $m\times n$ subtable with both $m$ and $n$ odd, the interior $(m-2) (n-2)$ rectangle is removed, leaving behind a frame of width $1$. The squares of the frame are painted alternately black and white. Prove that the sum of the numbers in the black squares is equal to the sum of the numbers in the white squares.

Let $O$ be the intersection point of the diagonals of a parallelogram $ABCD$ . Prove that if the line $BC$ is tangent to the circle passing through the points $A, B$, and $O$, then the line $CD$ is tangent to the circle passing through the points $B, C$ and $O$. (A Zaslavskiy)

Let $ \mathcal{M} =\left\{ A\in M_2\left( \mathbb{C}\right)\big| \det\left( A-zI_2\right) =0\implies |z| < 1\right\} . $ Prove that: $$ X,Y\in\mathcal{M}\wedge X\cdot Y=Y\cdot X\implies X\cdot Y\in\mathcal{M} . $$

A calculator has a squaring key $\boxed{x^2}$ which replaces the current number displayed with its square. For example, if the display is $\boxed{000003}$ and the $\boxed{x^2}$ key is depressed, then the display becomes $\boxed{000009}$. If the display reads $\boxed{000002}$, how many times must you depress the $\boxed{x^2}$ key to produce a displayed number greater than $500$? $\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 8 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 250$

Find all nonegative integers $ x,y,z$ such that $ 12^x\plus{}y^4\equal{}2008^z$

Let $a$, $b$, and $c$ be nonzero real numbers such that $a + \frac{1}{b} = 5$, $b + \frac{1}{c} = 12$, and $c + \frac{1}{a} = 13$. Find $abc + \frac{1}{abc}$.

Let $a, b, c$ be reals satisfying $a^2+b^2+c^2=6$. Find the maximal values of the expressions a) $(a-b)^2+(b-c)^2+(c-a)^2$; b) $(a-b)^2 \cdot (b-c)^2 \cdot (c-a)^2$. In both cases, describe all triples for which equality holds.

Find the maximum value of real number $k$ such that \[\frac{a}{1+9bc+k(b-c)^2}+\frac{b}{1+9ca+k(c-a)^2}+\frac{c}{1+9ab+k(a-b)^2}\geq \frac{1}{2}\] holds for all non-negative real numbers $a,\ b,\ c$ satisfying $a+b+c=1$.

In the figure below, $E$ is the midpoint of the arc $ABEC$ and the segment $ED$ is perpendicular to the chord $BC$ at $D$. If the length of the chord $AB$ is $l_1$, and that of the segment $BD$ is $l_2$, determine the length of $DC$ in terms of $l_1, l_2$. [asy] unitsize(1 cm); pair A=2dir(240),B=2dir(190),C=2dir(30),E=2dir(135),D=foot(E,B,C); draw(circle((0,0),2)); draw(A--B--C); draw(E--D); draw(rightanglemark(C,D,E,8)); label("$A$",A,.5A); label("$B$",B,.5B); label("$C$",C,.5C); label("$E$",E,.5E); label("$D$",D,dir(-60)); [/asy]

Let $\mathbb{Q}$ denote the set of rational numbers. Determine all functions $f:\mathbb{Q}\longrightarrow\mathbb{Q}$ such that, for all $x, y \in \mathbb{Q}$, $$f(x)f(y+1)=f(xf(y))+f(x)$$ [i]Nicolás López Funes and José Luis Narbona Valiente, Spain[/i]

Prove that $ \binom{m+n}{\min (m,n)}\le \sqrt{\binom{2m}{m}\cdot \binom{2n}{n}} , $ for nonnegative $ m,n. $ [i]Gheorghe Stoica[/i]

Let $x,y,z$ be positive real numbers. Prove that \[\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\geqslant\frac{z(x+y)}{y(y+z)}+\frac{x(y+z)}{z(z+x)}+\frac{y(z+x)}{x(x+y)}.\]

In a triangle $\triangle ABC$ with $\angle ABC < \angle BCA$, we define $K$ as the excenter with respect to $A$. The lines $AK$ and $BC$ intersect in a point $D$. Let $E$ be the circumcenter of $\triangle BKC$. Prove that \[\frac{1}{|KA|} = \frac{1}{|KD|} + \frac{1}{|KE|}.\]

The coefficient of $x^7$ in the polynomial expansion of \[ (1+2x-x^2)^4 \] is $ \textbf{(A)}\ -8 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ -12 \qquad\textbf{(E)}\ \text{none of these} $

On Misha's new phone, a passlock consists of six circles arranged in a $2\times 3$ rectangle. The lock is opened by a continuous path connecting the six circles; the path cannot pass through a circle on the way between two others (e.g. the top left and right circles cannot be adjacent). For example, the left path shown below is allowed but the right path is not. (Paths are considered to be oriented, so that a path starting at $A$ and ending at $B$ is different from a path starting at $B$ and ending at $A$. However, in the diagrams below, the paths are valid/invalid regardless of orientation.) How many passlocks are there consisting of all six circles? [asy] size(270); defaultpen(linewidth(0.8)); real r = 0.3, rad = 0.1, shift = 3.7; pen th = linewidth(5)+gray(0.2); for(int i=0; i<= 2;i=i+1) { for(int j=0; j<= 1;j=j+1) { fill(circle((i,j),r),gray(0.8)); fill(circle((i+shift,j),r),gray(0.8)); } draw((0,1)--(2-rad,1)^^(2,1-rad)--(2,rad)^^(2-rad,0)--(0,0),th); draw(arc((2-rad,1-rad),rad,0,90)^^arc((2-rad,rad),rad,270,360),th); draw((shift+1,0)--(shift+1,1-2*rad)^^(shift+1-rad,1-rad)--(shift+rad,1-rad)^^(shift+rad,1+rad)--(shift+2,1+rad),th); draw(arc((shift+1-rad,1-2*rad),rad,0,90)^^arc((shift+rad,1),rad,90,270),th); } [/asy]