Let $f:[0,1)\to (0,1)$ a continous onto (surjective) function. a) Prove that, for all $a\in(0,1)$, the function $f_a:(a,1)\to (0,1)$, given by $f_a(x) = f(x)$, for all $x\in(a,1)$ is onto; b) Give an example of such a function.

Let $\Gamma_i, i = 0, 1, 2, \dots$ , be a circle of radius $r_i$ inscribed in an angle of measure $2\alpha$ such that each $\Gamma_i$ is externally tangent to $\Gamma_{i+1}$ and $r_{i+1} < r_i$. Show that the sum of the areas of the circles $\Gamma_i$ is equal to the area of a circle of radius $r =\frac 12 r_0 (\sqrt{ \sin \alpha} + \sqrt{\text{csc} \alpha}).$

Points $A, B, C$ and $D$ are on a circle of diameter $1$, and $X$ is on diameter $\overline{AD}$. If $BX=CX$ and $3 \angle BAC=\angle BXC=36^{\circ}$, then $AX=$ [asy] draw(Circle((0,0),10)); draw((-10,0)--(8,6)--(2,0)--(8,-6)--cycle); draw((-10,0)--(10,0)); dot((-10,0)); dot((2,0)); dot((10,0)); dot((8,6)); dot((8,-6)); label("A", (-10,0), W); label("B", (8,6), NE); label("C", (8,-6), SE); label("D", (10,0), E); label("X", (2,0), NW); [/asy] $ \textbf{(A)}\ \cos 6^{\circ}\cos 12^{\circ} \sec 18^{\circ} \qquad\textbf{(B)}\ \cos 6^{\circ}\sin 12^{\circ} \csc 18^{\circ} \qquad\textbf{(C)}\ \cos 6^{\circ}\sin 12^{\circ} \sec 18^{\circ} \\ \qquad\textbf{(D)}\ \sin 6^{\circ}\sin 12^{\circ} \csc 18^{\circ} \qquad\textbf{(E)}\ \sin 6^{\circ} \sin 12^{\circ} \sec 18^{\circ} $

Let ABCD be a convex quadrilateral such that $AB + CD = \sqrt{2}AC$ and $BC + DA = \sqrt{2}BD$. Prove that ABCD is a parallelogram.

Given a right $ n $ -angle $ {{A} _ {1}} {{A} _ {2}} \ldots {{A} _ {n}} $, $n \ge 4 $, and a point $ M $ inside it. Prove the inequality $$\sin (\angle {{A} _ {1}} M {{A} _ {2}}) + \sin (\angle {{A} _ {2}} M {{A} _ {3}} ) + \ldots + \sin (\angle {{A} _ {n}} M {{A} _ {1}})> \sin \frac{2 \pi}{n} + (n-2) sin \frac{\pi}{n}$$

The area and the perimeter of the triangle with sides $10,8,6$ are equal. Find all the triangles with integral sides whose area and perimeter are equal.

An arbitrary circle can intersect the graph $ y \equal{} \sin x$ in $ \textbf{(A)} \text{ at most 2 points} \qquad \textbf{(B)} \text{ at most 4 points} \qquad$ $ \textbf{(C)} \text{ at most 6 points} \qquad \textbf{(D)} \text{ at most 8 points} \qquad$ $ \textbf{(E)} \text{ more than 16 points}$

In a triangle $ABC$ with $B = 90^\circ$, $D$ is a point on the segment $BC$ such that the inradii of triangles $ABD$ and $ADC$ are equal. If $\widehat{ADB} = \varphi$ then prove that $\tan^2 (\varphi/2) = \tan (C/2)$.

In triangle $ABC$ we have $\angle ABC = 2 \angle ACB$. Prove that a) $AC^2 = AB^2 + AB \cdot BC$; b) $AB+BC < 2 \cdot AC$.

Let $ABC$, $AC \not= BC$, be an acute triangle with orthocenter $H$ and incenter $I$. The lines $CH$ and $CI$ meet the circumcircle of $\bigtriangleup ABC$ at points $D$ and $L$, respectively. Prove that $\angle CIH = 90^{\circ}$ if and only if $\angle IDL = 90^{\circ}$

In a triangle $ABC$, $\angle BAC = 90^{\circ}$. Let $D$ be the point on $BC$ such that $AB + BD = AC + CD$. Suppose $BD : DC = 2:1$. if $\frac{AC}{AB} = \frac{m + \sqrt{p}}{n}$, Where $m,n$ are relatively prime positive integers and $p$ is a prime number, determine the value of $m+n+p$.

In $\triangle ABC$, prove that $1< \sum_{cyc}{\cos A}\le \frac{3}{2}$.

Let $M$ be a point in the interior of triangle $ABC$. Let $A'$ lie on $BC$ with $MA'$ perpendicular to $BC$. Define $B'$ on $CA$ and $C'$ on $AB$ similarly. Define \[ p(M) = \frac{MA' \cdot MB' \cdot MC'}{MA \cdot MB \cdot MC}. \] Determine, with proof, the location of $M$ such that $p(M)$ is maximal. Let $\mu(ABC)$ denote this maximum value. For which triangles $ABC$ is the value of $\mu(ABC)$ maximal?

Evaluate \[\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{(1+\cos x)\{1-\tan ^ 2 \frac{x}{2}\tan (x+\sin x)\tan (x-\sin x)\}}{\tan (x+\sin x)}\ dx\]

[b]a)[/b] Let be a nonnegative integer $ n. $ Solve in the complex numbers the equation $ z^n\cdot\Re z=\bar z^n\cdot\Im z. $ [b]b)[/b] Let be two complex numbers $ v,d $ satisfying $ v+1/v=d/\bar d +\bar d/d. $ Show that $$ v^n+1/v^n=d^n/\bar d^n + \bar d^n/d^n, $$ for any nonnegative integer $ n. $

[asy] import olympiad; import cse5; size(5cm); pointpen = black; pair A = Drawing((10,17.32)); pair B = Drawing((0,0)); pair C = Drawing((20,0)); draw(A--B--C--cycle); pair X = 0.85*A + 0.15*B; pair Y = 0.82*A + 0.18*C; pair W = (-11,0) + X; pair Z = (19, 9); draw(W--X, EndArrow); draw(X--Y, EndArrow); draw(Y--Z, EndArrow); anglepen=black; anglefontpen=black; MarkAngle("\theta", C,Y,Z, 3); [/asy] The cross-section of a prism with index of refraction $1.5$ is an equilateral triangle, as shown above. A ray of light comes in horizontally from air into the prism, and has the opportunity to leave the prism, at an angle $\theta$ with respect to the surface of the triangle. Find $\theta$ in degrees and round to the nearest whole number. [i](Ahaan Rungta, 5 points)[/i]

Evaluate $\int_0^{\frac{\pi}{12}} \frac{\tan ^ 2 x-3}{3\tan ^ 2 x-1}dx$. created by kunny

In $ \triangle ABC$, $ \angle ABC \equal{} 45^\circ$. Point $ D$ is on $ \overline{BC}$ so that $ 2 \cdot BD \equal{} CD$ and $ \angle DAB \equal{} 15^\circ$. Find $ \angle ACB$. [asy] pair A, B, C, D; A = origin; real Bcoord = 3*sqrt(2) + sqrt(6); B = Bcoord/2*dir(180); C = sqrt(6)*dir(120); draw(A--B--C--cycle); D = (C-B)/2.4 + B; draw(A--D); label("$A$", A, dir(0)); label("$B$", B, dir(180)); label("$C$", C, dir(110)); label("$D$", D, dir(130)); [/asy] $ \textbf{(A)} \ 54^\circ \qquad \textbf{(B)} \ 60^\circ \qquad \textbf{(C)} \ 72^\circ \qquad \textbf{(D)} \ 75^\circ \qquad \textbf{(E)} \ 90^\circ$

Point $O$ is inside triangle $ABC$ such that $\angle AOB = \angle BOC = \angle COA = 120^\circ .$ Prove that \[\frac{AO^2}{BC}+\frac{BO^2}{CA}+\frac{CO^2}{AB} \geq \frac{AO+BO+CO}{\sqrt 3}.\]

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.

Let $ ABC$ be a triangle such that \[ \frac{BC}{AB \minus{} BC}\equal{}\frac{AB \plus{} BC}{AC}\] Determine the ratio $ \angle A : \angle C$.

Let $0 \leq \alpha , \beta , \gamma \leq \frac{\pi}{2}$, such that $\cos ^{2} \alpha + \cos ^{2} \beta + \cos ^{2} \gamma = 1$. Prove that $2 \leq (1 + \cos ^{2} \alpha ) ^{2} \sin^{4} \alpha + (1 + \cos ^{2} \beta ) ^{2} \sin ^{4} \beta + (1 + \cos ^{2} \gamma ) ^{2} \sin ^{4} \gamma \leq (1 + \cos ^{2} \alpha )(1 + \cos ^{2} \beta)(1 + \cos ^{2} \gamma ).$

Consider a right-angled triangle with $AB=1,\ AC=\sqrt{3},\ \angle{BAC}=\frac{\pi}{2}.$ Let $P_1,\ P_2,\ \cdots\cdots,\ P_{n-1}\ (n\geq 2)$ be the points which are closest from $A$, in this order and obtained by dividing $n$ equally parts of the line segment $AB$. Denote by $A=P_0,\ B=P_n$, answer the questions as below. (1) Find the inradius of $\triangle{P_kCP_{k+1}}\ (0\leq k\leq n-1)$. (2) Denote by $S_n$ the total sum of the area of the incircle for $\triangle{P_kCP_{k+1}}\ (0\leq k\leq n-1)$. Let $I_n=\frac{1}{n}\sum_{k=0}^{n-1} \frac{1}{3+\left(\frac{k}{n}\right)^2}$, show that $nS_n\leq \frac {3\pi}4I_n$, then find the limit $\lim_{n\to\infty} I_n$. (3) Find the limit $\lim_{n\to\infty} nS_n$.

In $\vartriangle ABC$ with edges $a, b$ and $c$, suppose $b + c = 6$ and the area $S$ is $a^2 - (b -c)^2$. Find the value of $\cos A$ and the largest possible value of $S$.

The parametric equations of a curve are given by $x = 2(1+\cos t)\cos t,\ y =2(1+\cos t)\sin t\ (0\leq t\leq 2\pi)$. (1) Find the maximum and minimum values of $x$. (2) Find the volume of the solid enclosed by the figure of revolution about the $x$-axis.