Let $f:[-1,1]\rightarrow\mathbb{R}$ be a continuous function. Show that: a) if $\int_0^1 f(\sin (x+\alpha ))\, dx=0$, for every $\alpha\in\mathbb{R}$, then $f(x)=0,\ \forall x\in [-1,1]$. b) if $\int_0^1 f(\sin (nx))\, dx=0$, for every $n\in\mathbb{Z}$, then $f(x)=0,\ \forall x\in [-1,1]$.

The inscribed circle of triangle $ABC$ is tangent to $\overline{AB}$ at $P,$ and its radius is 21. Given that $AP=23$ and $PB=27,$ find the perimeter of the triangle.

Assume $A,B,C$ are three collinear points that $B \in [AC]$. Suppose $AA'$ and $BB'$ are to parrallel lines that $A'$, $B'$ and $C$ are not collinear. Suppose $O_1$ is circumcenter of circle passing through $A$, $A'$ and $C$. Also $O_2$ is circumcenter of circle passing through $B$, $B'$ and $C$. If area of $A'CB'$ is equal to area of $O_1CO_2$, then find all possible values for $\angle CAA'$

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$).

Consider a function $f(x)=\frac{\sin x}{9+16\sin ^ 2 x}\ \left(0\leq x\leq \frac{\pi}{2}\right).$ Let $a$ be the value of $x$ for which $f(x)$ is maximized. Evaluate $\int_a^{\frac{\pi}{2}} f(x)\ dx.$ [i]2010 Saitama University entrance exam/Mathematics[/i] Last Edited

Triangle $ABC$ has right angle at $B$, and contains a point $P$ for which $PA = 10$, $PB = 6$, and $\angle APB = \angle BPC = \angle CPA$. Find $PC$. [asy] pair A=(0,5), B=origin, C=(12,0), D=rotate(-60)*C, F=rotate(60)*A, P=intersectionpoint(A--D, C--F); draw(A--P--B--A--C--B^^C--P); dot(A^^B^^C^^P); pair point=P; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$P$", P, NE);[/asy]

Given a circle and a point $A$ inside the circle, but not at its center. Find points $B$, $C$, $D$ on the circle which maximise the area of the quadrilateral $ABCD$.

Without using a calculator, find out what is greater: $\sin 28^o$ or $tg21^o$?

If $A,B,C$ are the angles of a triangle, prove that \[-2 \le \sin{3A}+\sin{3B}+\sin{3C} \le \frac{3\sqrt{3}}{2}\] and determine when equality holds.

Let $AB$ be a diameter of a circle $(O)$ and let $P$ be any point on the tangent drawn at $B$ to $(O).$ Define $AP\cap (O)=C\neq A,$ and let $D$ be the point diametrically opposite to $C.$ If $DP$ meets $(O)$ second time in $E,$ then, [b](i)[/b] Prove that $AE, BC, PO$ concur at $M.$ [b](ii)[/b] If $R$ is the radius of $(O),$ find $P$ such that the area of $\triangle AMB$ is maximum, and calculate the area in terms of $R.$

Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.

Define $\mathcal{A}=\{(x,y)|x=u+v,y=v, u^2+v^2\le 1\}$. Find the length of the longest segment that is contained in $\mathcal{A}$.

Let $ABCD$ be a convex quadrilateral whose diagonals $AC$ and $BD$ intersect in a point $P$. Prove that \[\frac{AP}{PC}=\frac{\cot \angle BAC + \cot \angle DAC}{\cot \angle BCA + \cot \angle DCA}\]

Let $a,b,c$ be three positive real numbers with $a+b+c=3$. Prove that \[ (3-2a)(3-2b)(3-2c) \leq a^2b^2c^2 . \] [i]Robert Szasz[/i]

Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. let $V_1$ be the volume of the cone and $V_2$ be the volume of the cylinder. a) Prove that $V_1 \neq V_2$; b) Find the smallest number $k$ for which $V_1=kV_2$; for this case, construct the angle subtended by a diamter of the base of the cone at the vertex of the cone.

Let $\alpha$ be an arbitrary angle and let $x = cos\alpha, y = cosn\alpha$ ($n \in Z$). i) Prove that to each value $x \in [-1, 1]$ corresponds one and only one value of $y$. Thus we can write $y$ as a function of $x, y = T_n(x)$. Compute $T_1(x), T_2(x)$ and prove that $T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)$. From this it follows that $T_n(x)$ is a polynomial of degree $n$. ii) Prove that the polynomial $T_n(x$) has $n$ distinct roots in $[-1, 1]$.

For every pair of reals $0 < a < b < 1$, we define sequences $\{x_n\}_{n \ge 0}$ and $\{y_n\}_{n \ge 0}$ by $x_0 = 0$, $y_0 = 1$, and for each integer $n \ge 1$: \begin{align*} x_n & = (1 - a) x_{n - 1} + a y_{n - 1}, \\ y_n & = (1 - b) x_{n - 1} + b y_{n - 1}. \end{align*} The [i]supermean[/i] of $a$ and $b$ is the limit of $\{x_n\}$ as $n$ approaches infinity. Over all pairs of real numbers $(p, q)$ satisfying $\left (p - \textstyle\frac{1}{2} \right)^2 + \left (q - \textstyle\frac{1}{2} \right)^2 \le \left(\textstyle\frac{1}{10}\right)^2$, the minimum possible value of the supermean of $p$ and $q$ can be expressed as $\textstyle\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m + n$. [i]Proposed by Lewis Chen[/i]

Let $P$ be a point inside a triangle $ABC$ such that \[\angle P AB = \angle P BC = \angle P CA\] Suppose $AP, BP, CP$ meet the circumcircles of triangles $P BC, P CA, P AB$ at $X, Y, Z$ respectively $(\neq P)$ . Prove that \[[XBC] + [Y CA] + [ZAB] \ge 3[ABC]\]

Fedja used matches to put down the equally long sides of a parallelogram whose vertices are not on a common line. He figures out that exactly 7 or 9 matches, respectively, fit into the diagonals. How many matches compose the parallelogram's perimeter?

$(POL 2)$ Given two segments $AB$ and $CD$ not in the same plane, find the locus of points $M$ such that $MA^2 +MB^2 = MC^2 +MD^2.$

Compute \[\lim_{h\to 0}\dfrac{\sin(\frac{\pi}{3}+4h)-4\sin(\frac{\pi}{3}+3h)+6\sin(\frac{\pi}{3}+2h)-4\sin(\frac{\pi}{3}+h)+\sin(\frac{\pi}{3})}{h^4}.\]

Let $ABC$ be a triangle, let $H$ be the orthocentre and $L,M,N$ the midpoints of the sides $AB, BC, CA$ respectively. Prove that \[HL^{2} + HM^{2} + HN^{2} < AL^{2} + BM^{2} + CN^{2}\] if and only if $ABC$ is acute-angled.

Given a regular pentagon, find the ratio of its diagonal, $d$, to its side, $a$

Given mobile points $P(0,\ \sin \theta),\ Q(8\cos \theta,\ 0)\ \left(0\leq \theta \leq \frac{\pi}{2}\right)$ on the $x$-$y$ plane. Denote by $D$ the part in which line segment $PQ$ sweeps. Find the volume $V$ generated by a rotation of $D$ around the $x$-axis.

Evaluate $\int_{0}^{\infty}\frac{\tan^{-1}(\pi x)-\tan^{-1}x}{x}dx$