$(a)$ Prove that $\sqrt{2}(\sin t + \cos t) \ge 2\sqrt[4]{\sin 2t}$ for $0 \le t \le\frac{\pi}{2}.$ $(b)$ Find all $y, 0 < y < \pi$, such that $1 +\frac{2 \cot 2y}{\cot y} \ge \frac{\tan 2y}{\tan y}$. .

Characterize all triangles $ABC$ s.t. \[ AI_a : BI_b : CI_c = BC: CA : AB \] where $I_a$ etc. are the corresponding excentres to the vertices $A, B , C$

Consider the configuration of six squares as shown on the picture. Prove that the sum of the area of the three outer squares ($ I,II$ and $ III$) equals three times the sum of the areas of the three inner squares ($ IV,V$ and $ VI$).

Show that for any arbitrary triangle $ABC$, we have \[\sin\left(\frac{A}{2}\right) \cdot \sin\left(\frac{B}{2}\right) \cdot \sin\left(\frac{C}{2}\right) \leq \frac{abc}{(a+b)(b+c)(c+a)}.\]

It is given the function $f:\mathbb{R} \to \mathbb{R}$ such that it holds $f(\sin x)=\sin (2011x)$. Find the value of $f(\cos x)$.

Find $$\frac{\tan 1^o}{1 + \tan 1^o }+\frac{\tan 2^o}{1 + \tan 2^o } + ... + \frac{\tan 89^o}{1 + \tan 89^o}$$

Take two points $A\ (-1,\ 0),\ B\ (1,\ 0)$ on the $xy$-plane. Let $F$ be the figure by which the whole points $P$ on the plane satisfies $\frac{\pi}{4}\leq \angle{APB}\leq \pi$ and the figure formed by $A,\ B$. Answer the following questions: (1) Illustrate $F$. (2) Find the volume of the solid generated by a rotation of $F$ around the $x$-axis.

Let $ ABCD$ be a convex quadrilateral. We have that $ \angle BAC\equal{}3\angle CAD$, $ AB\equal{}CD$, $ \angle ACD\equal{}\angle CBD$. Find angle $ \angle ACD$

Let $ABC$ be a triangle. Take points $D$, $E$, $F$ on the perpendicular bisectors of $BC$, $CA$, $AB$ respectively. Show that the lines through $A$, $B$, $C$ perpendicular to $EF$, $FD$, $DE$ respectively are concurrent.

Let $A$ and $B$ are two points on a plane, and let $M$ be the midpoint of $AB$. Let $r$ be a line and let $R$ and $S$ be the projections of $A$ and $B$ onto $r$. Assuming that $A$, $M$, and $R$ are not collinear, prove that the circumcircle of triangle $AMR$ has the same radius as the circumcircle of $BSM$.

Solve the equation $$ tg 7x - \sin 6x=\cos 4x - ctg 7x.$$

In the following, the abbreviation $g \cap h$ will mean the point of intersection of two lines $g$ and $h$. Let $ABCDE$ be a convex pentagon. Let $A^{\prime}=BD\cap CE$, $B^{\prime}=CE\cap DA$, $C^{\prime}=DA\cap EB$, $D^{\prime}=EB\cap AC$ and $E^{\prime}=AC\cap BD$. Furthermore, let $A^{\prime\prime}=AA^{\prime}\cap EB$, $B^{\prime\prime}=BB^{\prime}\cap AC$, $C^{\prime\prime}=CC^{\prime}\cap BD$, $D^{\prime\prime}=DD^{\prime}\cap CE$ and $E^{\prime\prime}=EE^{\prime}\cap DA$. Prove that: \[ \frac{EA^{\prime\prime}}{A^{\prime\prime}B}\cdot\frac{AB^{\prime\prime}}{B^{\prime\prime}C}\cdot\frac{BC^{\prime\prime}}{C^{\prime\prime}D}\cdot\frac{CD^{\prime\prime}}{D^{\prime\prime}E}\cdot\frac{DE^{\prime\prime}}{E^{\prime\prime}A}=1. \] Darij

The parallelogram $ABCD$ has $AB=a,AD=1,$ $\angle BAD=A$, and the triangle $ABD$ has all angles acute. Prove that circles radius $1$ and center $A,B,C,D$ cover the parallelogram if and only \[a\le\cos A+\sqrt3\sin A.\]

Let $ABC$ and $A_{1}B_{1}C_{1}$ be two triangles. Prove that $\frac{a}{a_{1}}+\frac{b}{b_{1}}+\frac{c}{c_{1}}\leq\frac{3R}{2r_{1}}$, where $a = BC$, $b = CA$, $c = AB$ are the sidelengths of triangle $ABC$, where $a_{1}=B_{1}C_{1}$, $b_{1}=C_{1}A_{1}$, $c_{1}=A_{1}B_{1}$ are the sidelengths of triangle $A_{1}B_{1}C_{1}$, where $R$ is the circumradius of triangle $ABC$ and $r_{1}$ is the inradius of triangle $A_{1}B_{1}C_{1}$.

A triangle has area $30$, one side of length $10$, and the median to that side of length $9$. Let $\theta$ be the acute angle formed by that side and the median. What is $\sin{\theta}$? $ \textbf{(A)}\ \frac{3}{10}\qquad\textbf{(B)}\ \frac{1}{3}\qquad\textbf{(C)}\ \frac{9}{20}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{9}{10} $

Let $\alpha$, $\beta$ and $\gamma$ be three acute angles such that $\sin \alpha+\sin \beta+\sin \gamma = 1$. Show that \[\tan^{2}\alpha+\tan^{2}\beta+\tan^{2}\gamma \geq \frac{3}{8}. \]

Find the minimum value of $ \int_0^{\pi} \left(x \minus{} \pi a \minus{} \frac {b}{\pi}\cos x\right)^2dx$.

Find all angles a for which the set of numbers $\sin a$, $\sin 2a$, $\sin 3a$ coincides with the set $cos a$, $cos 2a$, $cos 3a$.

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 $\frac{c+h}{a+b}$.

For $f(x)=x^4+|x|,$ let $I_1=\int_0^\pi f(\cos x)\ dx,\ I_2=\int_0^\frac{\pi}{2} f(\sin x)\ dx.$ Find the value of $\frac{I_1}{I_2}.$

Let $({{x}_{n}}),({{y}_{n}})$ be two positive sequences defined by ${{x}_{1}}=1,{{y}_{1}}=\sqrt{3}$ and \[ \begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \] for all $n=1,2,3,\ldots$. Prove that they are converges and find their limits.

Rhombus $PQRS$ is inscribed in rectangle $ABCD$ so that vertices $P$, $Q$, $R$, and $S$ are interior points on sides $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, and $\overline{DA}$, respectively. It is given that $PB=15$, $BQ=20$, $PR=30$, and $QS=40$. Let $m/n$, in lowest terms, denote the perimeter of $ABCD$. Find $m+n$.

Do either (1) or (2): (1) Show that any solution $f(t)$ of the functional equation $$f(x+y)f(x-y)=f(x)^{2} +f(y)^{2} -1$$ for $x,y\in \mathbb{R}$ satisfies $$f''(t)= \pm c^{2} f(t)$$ for a constant $c$, assuming the existence and continuity of the second derivative. Deduce that $f(t)$ is one of the functions $$ \pm \cos ct, \;\;\; \pm \cosh ct.$$ (2) Let $(a_{i})_{i=1,...,n}$ and $(b_{i})_{i=1,...,n}$ be real numbers. Define an $(n+1)\times (n+1)$-matrix $A=(c_{ij})$ by $$ c_{i1}=1, \; \; c_{1j}= x^{j-1} \; \text{for} \; j\leq n,\; \; c_{1n+1}=p(x), \;\; c_{ij}=a_{i-1}^{j-1} \; \text{for}\; i>1, j\leq n,\;\; c_{in+1}=b_{i-1}\; \text{for}\; i>1.$$ The polynomial $p(x)$ is defined by the equation $\det A=0$. Let $f$ be a polynomial and replace $(b_{i})$ with $(f(b_{i}))$. Then $\det A=0$ defines another polynomial $q(x)$. Prove that $f(p(x))-q(x)$ is a multiple of $$\prod_{i=1}^{n} (x-a_{i}).$$

In isosceles right triangle $ ABC$ a circle is inscribed. Let $ CD$ be the hypotenuse height ($ D\in AB$), and let $ P$ be the intersection of inscribed circle and height $ CD$. In which ratio does the circle divide segment $ AP$?

Consider an isosceles triangle $KL_1L_2$ with $|KL_1|=|KL_2|$ and let $KA, L_1B_1,L_2B_2$ be its angle bisectors. Prove that $\cos \angle B_1AB_2 < \frac35$