A box with weight $W$ will slide down a $30^\circ$ incline at constant speed under the influence of gravity and friction alone. If instead a horizontal force $P$ is applied to the box, the box can be made to move up the ramp at constant speed. What is the magnitude of $P$? $\textbf{(A) } P = W/2 \\ \textbf{(B) } P = 2W/\sqrt{3}\\ \textbf{(C) } P = W\\ \textbf{(D) } P = \sqrt{3}W \\ \textbf{(E) } P = 2W$

Which of the following statements are true? (A) $X$ implies $Y$, or $Y$ implies $X$, where $X$ is the statement, the lines $L_1, L_2, L_3$ lie in a plane, and $Y$ is the statement, each pair of the lines $L_1, L_2, L_3$ intersect. (B) Every sufficiently large integer $n$ satisfies $n = a^4 + b^4$ for some integers a, b. (C) There are real numbers $a_1, a_2,... , a_n$ such that $a_1 \cos x + a_2 \cos 2x +... + a_n \cos nx > 0$ for all real $x$.

Let $ ABC$ be a triangle with $ \angle A \equal{} 45^\circ$. Let $ P$ be a point on side $ BC$ with $ PB \equal{} 3$ and $ PC \equal{} 5$. Let $ O$ be the circumcenter of $ ABC$. Determine the length $ OP$.

A point $M$ lies on the side $BC$ of square $ABCD$. Let $X$, $Y$ , and $Z$ be the incenters of triangles $ABM$, $CMD$, and $AMD$ respectively. Let $H_x$, $H_y$, and $H_z$ be the orthocenters of triangles $AXB$, $CY D$, and $AZD$. Prove that $H_x$, $H_y$, and $H_z$ are collinear.

Let $ABC$ be a right triangle with $A = 90^{\circ}$ and $D \in (AC)$. Denote by $E$ the reflection of $A$ in the line $BD$ and $F$ the intersection point of $CE$ with the perpendicular in $D$ to $BC$. Prove that $AF, DE$ and $BC$ are concurrent.

Given a regular tetrahedron $OABC$. Take points $P,\ Q,\ R$ on the sides $OA,\ OB,\ OC$ respectively. Note that $P,\ Q,\ R$ are different from the vertices of the tetrahedron $OABC$. If $\triangle{PQR}$ is an equilateral triangle, then prove that three sides $PQ,\ QR,\ RP$ are pararell to three sides $AB,\ BC,\ CA$ respectively. 30 points

Let $ABC$ be an acute-angled triangle with $AB<AC$ and let $O$ be the centre of its circumcircle $\omega$. Let $D$ be a point on the line segment $BC$ such that $\angle BAD = \angle CAO$. Let $E$ be the second point of intersection of $\omega$ and the line $AD$. If $M$, $N$ and $P$ are the midpoints of the line segments $BE$, $OD$ and $AC$, respectively, show that the points $M$, $N$ and $P$ are collinear.

A calculator is broken so that the only keys that still work are the $ \sin$, $ \cos$, and $ \tan$ buttons, and their inverses (the $ \arcsin$, $ \arccos$, and $ \arctan$ buttons). The display initially shows $ 0$. Given any positive rational number $ q$, show that pressing some finite sequence of buttons will yield the number $ q$ on the display. Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians.

Let the circle $ {\omega}_{1}$ be internally tangent to another circle $ {\omega}_{2}$ at $ N$.Take a point $ K$ on $ {\omega}_{1}$ and draw a tangent $ AB$ which intersects $ {\omega}_{2}$ at $ A$ and $ B$. Let $M$ be the midpoint of the arc $ AB$ which is on the opposite side of $ N$. Prove that, the circumradius of the $ \triangle KBM$ doesnt depend on the choice of $ K$.

For a finite set $ X$ of positive integers, let $ \Sigma(X) \equal{} \sum_{x \in X} \arctan \frac{1}{x}.$ Given a finite set $ S$ of positive integers for which $ \Sigma(S) < \frac{\pi}{2},$ show that there exists at least one finite set $ T$ of positive integers for which $ S \subset T$ and $ \Sigma(S) \equal{} \frac{\pi}{2}.$ [i]Kevin Buzzard, United Kingdom[/i]

On a $49\times 69$ rectangle formed by a grid of lattice squares, all $50\cdot 70$ lattice points are colored blue. Two persons play the following game: In each step, a player colors two blue points red, and draws a segment between these two points. (Different segments can intersect in their interior.) Segments are drawn this way until all formerly blue points are colored red. At this moment, the first player directs all segments drawn - i. e., he takes every segment AB, and replaces it either by the vector $\overrightarrow{AB}$, or by the vector $\overrightarrow{BA}$. If the first player succeeds to direct all the segments drawn in such a way that the sum of the resulting vectors is $\overrightarrow{0}$, then he wins; else, the second player wins. Which player has a winning strategy?

Find the smallest positive root of the equation $$\{tg x\}=\sin x. $$ ($\{a\}$ is the fractional part of $a$, $\{a\}$ is equal to the difference between $ a$ and the largest integer not exceeding $a$.)

Right triangle $ABC$ has right angle at $C$ and $\angle BAC=\theta$; the point $D$ is chosen on $AB$ so that $|AC|=|AD|=1$; the point $E$ is chosen on $BC$ so that $\angle CDE=\theta$. The perpendicular to $BC$ at $E$ meets $AB$ at $F$. Evaluate $\lim_{\theta\to 0}|EF|$.

Evaluate the integrals as follows. (1) $\int \frac{x^2}{2-x}\ dx$ (2) $\int \sqrt[3]{x^5+x^3}\ dx$ (3) $\int_0^1 (1-x)\cos \pi x\ dx$

Show that $\cos \frac{\pi}{7}$ is irrational.

Denote $f(m) =\sum_{k=1}^m (-1)^k cos \frac{k\pi}{2 m + 1}$ For which positive integers $m$ is $f(m)$ rational?

In a triangle $ABC$, let $H_a$, $H_b$ and $H_c$ denote the base points of the altitudes on the sides $BC$, $CA$ and $AB$, respectively. Determine for which triangles $ABC$ two of the lengths $H_aH_b$, $H_bH_c$ and $H_aH_c$ are equal.

Let $ H$ be the orthocenter of an acute-angled triangle $ ABC$. The circle $ \Gamma_{A}$ centered at the midpoint of $ BC$ and passing through $ H$ intersects the sideline $ BC$ at points $ A_{1}$ and $ A_{2}$. Similarly, define the points $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$. Prove that the six points $ A_{1}$, $ A_{2}$, $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$ are concyclic. [i]Author: Andrey Gavrilyuk, Russia[/i]

Equilateral triangle $ABC$ has been creased and folded so that vertex $A$ now rests at $A'$ on $\overline{BC}$ as shown. If $BA' = 1$ and $A'C = 2$ then the length of crease $\overline{PQ}$ is [asy] size(170); defaultpen(linewidth(0.7)+fontsize(10)); pair B=origin, A=(1.5,3*sqrt(3)/2), C=(3,0), D=(1,0), P=B+1.6*dir(B--A), Q=C+1.2*dir(C--A); draw(B--P--D--B^^P--Q--D--C--Q); draw(Q--A--P, linetype("4 4")); label("$A$", A, N); label("$B$", B, W); label("$C$", C, E); label("$A'$", D, S); label("$P$", P, W); label("$Q$", Q, E); [/asy] $ \textbf{(A)}\ \frac{8}{5}\qquad\textbf{(B)}\ \frac{7}{20}\sqrt{21}\qquad\textbf{(C)}\ \frac{1+\sqrt{5}}{2}\qquad\textbf{(D)}\ \frac{13}{8}\qquad\textbf{(E)}\ \sqrt{3} $

Let $a,b$ be the smaller sides of a right triangle. Let $c$ be the hypothenuse and $h$ be the altitude from the right angle. Fint the maximal value of $\frac{c+h}{a+b}$.

Find the value of $n\in{\mathbb{N}}$ satisfying the following inequality. \[\left|\int_0^{\pi} x^2\sin nx\ dx\right|<\frac{99\pi ^ 2}{100n}\]

For any square matrix $\mathcal{A}$ we define $\sin {\mathcal{A}}$ by the usual power series. \[ \sin {\mathcal{A}}=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}\mathcal{A}^{2n+1} \] Prove or disprove : $\exists 2\times 2$ matrix $A\in \mathcal{M}_2(\mathbb{R})$ such that \[ \sin{A}=\left(\begin{array}{cc}1 & 1996 \\0 & 1 \end{array}\right) \]

Let O be the circumcentre of an acute triangle ABC and let A′, B′ and C′ be the circumcentres of triangles BCO, CAO and ABO, respectively. Prove that the area of triangle ABC does not exceed the area of triangle A′B′C′.

Consider two circles of radius one, and let $O$ and $O'$ denote their centers. Point $M$ is selected on either circle. If $OO' = 2014$, what is the largest possible area of triangle $OMO'$? [i]Proposed by Evan Chen[/i]

Let $|a|<\frac{\pi}{2}$. Evaluate \[\int_0^{\frac{\pi}{2}} \frac{dx}{\{\sin (a+x)+\cos x\}^2}\]