Which of the following describes the largest subset of values of $y$ within the closed interval $[0,\pi]$ for which $$\sin(x+y)\leq \sin(x)+\sin(y)$$ for every $x$ between $0$ and $\pi$, inclusive? $\textbf{(A) } y=0 \qquad \textbf{(B) } 0\leq y\leq \frac{\pi}{4} \qquad \textbf{(C) } 0\leq y\leq \frac{\pi}{2} \qquad \textbf{(D) } 0\leq y\leq \frac{3\pi}{4} \qquad \textbf{(E) } 0\leq y\leq \pi $

If $n \ge 2$ is an integer, prove the equality $$\prod_{k=1}^n \tan \frac{\pi}{3}\left(1+\frac{3^k}{3^n-1}\right)=\prod_{k=1}^n \cot \frac{\pi}{3}\left(1-\frac{3^k}{3^n-1}\right)$$

In a acute-angled triangle $ABC$, a point $D$ lies on the segment $BC$. Let $O_1,O_2$ denote the circumcentres of triangles $ABD$ and $ACD$ respectively. Prove that the line joining the circumcentre of triangle $ABC$ and the orthocentre of triangle $O_1O_2D$ is parallel to $BC$.

Let $0 < a < \frac{\pi}{2}$ and $x_1,x_2,...,x_n$ be real numbers such that $\sin x_1 + \sin x_2 +... + \sin x_n \ge n \cdot sin a $. Prove that $\sin (x_1 - a) + \sin (x_2 - a) + ... + \sin (x_n - a) \ge 0$ .

Let $f_n(x,\ y)=\frac{n}{r\cos \pi r+n^2r^3}\ (r=\sqrt{x^2+y^2})$, $I_n=\int\int_{r\leq 1} f_n(x,\ y)\ dxdy\ (n\geq 2).$ Find $\lim_{n\to\infty} I_n.$ [i]2009 Tokyo Institute of Technology, Master Course in Mathematics[/i]

Let $D$ be the foot of the internal bisector of the angle $\angle A$ of the triangle $ABC$. The straight line which joins the incenters of the triangles $ABD$ and $ACD$ cut $AB$ and $AC$ at $M$ and $N$, respectively. Show that $BN$ and $CM$ meet on the bisector $AD$.

In a triangle $ABC$, points $A_{1}$ and $A_{2}$ are chosen in the prolongations beyond $A$ of segments $AB$ and $AC$, such that $AA_{1}=AA_{2}=BC$. Define analogously points $B_{1}$, $B_{2}$, $C_{1}$, $C_{2}$. If $[ABC]$ denotes the area of triangle $ABC$, show that $[A_{1}A_{2}B_{1}B_{2}C_{1}C_{2}] \geq 13 [ABC]$.

How many zeros does $ f(x) \equal{} \cos(\log(x)))$ have on the interval $ 0 < x < 1$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ \text{infinitely many}$

For a particular value of the angle $\theta$ we can take the product of the two complex numbers $(8+i)\sin\theta+(7+4i)\cos\theta$ and $(1+8i)\sin\theta+(4+7i)\cos\theta$ to get a complex number in the form $a+bi$ where $a$ and $b$ are real numbers. Find the largest value for $a+b$.

$ABCD$ is a parallelogram in which angle $DAB$ is acute. Points $A, P, B, D$ lie on one circle in exactly this order. Lines $AP$ and $CD$ intersect in $Q$. Point $O$ is the circumcenter of the triangle $CPQ$. Prove that if $D \neq O$ then the lines $AD$ and $DO$ are perpendicular.

Let the curve $ C: y\equal{}x\sqrt{9\minus{}x^2}\ (x\geq 0)$. (1) Find the maximum value of $ y$. (2) Find the area of the figure bounded by the curve $ C$ and the $ x$ axis. (3) Find the volume of the solid generated by rotation of the figure about the $ y$ axis.

Let $ a$, $ b$, $ c$ be the sides of a triangle, and $ S$ its area. Prove: \[ a^{2} \plus{} b^{2} \plus{} c^{2}\geq 4S \sqrt {3} \] In what case does equality hold?

Let $|a|<\frac{\pi}{2}.$ Evaluate the following definite integral. \[\int_{0}^{\frac{\pi}{2}}\frac{dx}{\{\sin (a+x)+\cos x\}^{2}}\]

In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have.

Points $M,\ N,\ K,\ L$ are taken on the sides $AB,\ BC,\ CD,\ DA$ of a rhombus $ABCD,$ respectively, in such a way that $MN\parallel LK$ and the distance between $MN$ and $KL$ is equal to the height of $ABCD.$ Show that the circumcircles of the triangles $ALM$ and $NCK$ intersect each other, while those of $LDK$ and $MBN$ do not.

An acute angle between the diagonals of a cyclic quadrilateral is equal to $\phi$. Prove that an acute angle between the diagonals of any other quadrilateral having the same sidelengths is smaller than $\phi$.

Calculate the following integarls. [1] $\int x\cos ^ 2 x dx$ [2] $\int \frac{x-1}{(3x-1)^2}dx$ [3] $\int \frac{x^3}{(2-x^2)^4}dx$ [4] $\int \left({\frac{1}{4\sqrt{x}}+\frac{1}{2x}}\right)dx$ [5] $\int (\ln x)^2 dx$

Three concentric circles with common center $O$ are cut by a common chord in successive points $A, B, C$. Tangents drawn to the circles at the points $A, B, C$ enclose a triangular region. If the distance from point $O$ to the common chord is equal to $p$, prove that the area of the region enclosed by the tangents is equal to \[\frac{AB \cdot BC \cdot CA}{2p}\]

Let $ A^{\prime}$ be an arbitrary point on the side $ BC$ of a triangle $ ABC$. Denote by $ \mathcal{T}_{A}^{b}$, $ \mathcal{T}_{A}^{c}$ the circles simultanously tangent to $ AA^{\prime}$, $ A^{\prime}B$, $ \Gamma$ and $ AA^{\prime}$, $ A^{\prime}C$, $ \Gamma$, respectively, where $ \Gamma$ is the circumcircle of $ ABC$. Prove that $ \mathcal{T}_{A}^{b}$, $ \mathcal{T}_{A}^{c}$ are congruent if and only if $ AA^{\prime}$ passes through the Nagel point of triangle $ ABC$. ([i]If $ M,N,P$ are the points of tangency of the excircles of the triangle $ ABC$ with the sides of the triangle $ BC$, $ CA$ and $ AB$ respectively, then the Nagel point of the triangle is the intersection point of the lines $ AM$, $ BN$ and $ CP$[/i].)

Let $a_n=\int_0^{\frac{\pi}{2}} \sin 2t\ (1-\sin t)^{\frac{n-1}{2}}dt\ (n=1,2,\cdots)$ Evaluate \[\sum_{n=1}^{\infty} (n+1)(a_n-a_{n+1})\]

Let $ A_1,B_1,C_1 $ be the feet of the heights of an acute triangle $ ABC. $ On the segments $ B_1C_1,C_1A_1,A_1B_1, $ take the points $ X,Y, $ respectively, $ Z, $ such that $$ \left\{\begin{matrix}\frac{C_1X}{XB_1} =\frac{b\cos\angle BCA}{c\cos\angle ABC} \\ \frac{A_1Y}{YC_1} =\frac{c\cos\angle BAC}{a\cos\angle BCA} \\ \frac{B_1Z}{ZA_1} =\frac{a\cos\angle ABC}{b\cos\angle BAC} \end{matrix}\right. . $$ Show that $ AX,BY,CZ, $ are concurrent.

How many real triples $(a,b,c)$ are there such that the polynomial $p(x)=x^4+ax^3+bx^2+ax+c$ has exactly three distinct roots, which are equal to $\tan y$, $\tan 2y$, and $\tan 3y$ for some real number $y$?

Points $ A$ and $ B$ are on a circle of radius $ 5$ and $ AB\equal{}6$. Point $ C$ is the midpoint of the minor arc $ AB$. What is the length of the line segment $ AC$? $ \textbf{(A)}\ \sqrt{10} \qquad \textbf{(B)}\ \frac{7}{2} \qquad \textbf{(C)}\ \sqrt{14} \qquad \textbf{(D)}\ \sqrt{15} \qquad \textbf{(E)}\ 4$

Let $\theta_i \in \left ( 0,\frac{\pi}{4} \right ]$ for $i=1,2,3,4$. Prove that: $\tan \theta _1 \tan \theta _2 \tan \theta _3 \tan \theta _4 \le (\frac{\sin^8 \theta _1+\sin^8 \theta _2+\sin^8 \theta _3+\sin^8 \theta _4}{\cos^8 \theta _1+\cos^8 \theta _2+\cos^8 \theta _3+\cos^8 \theta _4})^\frac{1}{2}$ [hide=edit]@below, fixed now. There were some problems (weird characters) so aops couldn't send it.[/hide]

Prove the following inequality. \[\frac{1}{12}(\pi -6+2\sqrt{3})\leq \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \ln (1+\cos 2x) dx\leq \frac{1}{4}(2-\sqrt{3})\]