Let triangle $ ABC$ acutangle, with $ m \angle ACB\leq\ m \angle ABC$. $ M$ the midpoint of side $ BC$ and $ P$ a point over the side $ MC$. Let $ C_{1}$ the circunference with center $ C$. Let $ C_{2}$ the circunference with center $ B$. $ P$ is a point of $ C_{1}$ and $ C_{2}$. Let $ X$ a point on the opposite semiplane than $ B$ respecting with the straight line $ AP$; Let $ Y$ the intersection of side $ XB$ with $ C_{2}$ and $ Z$ the intersection of side $ XC$ with $ C_{1}$. Let $ m\angle PAX \equal{} \alpha$ and $ m\angle ABC \equal{} \beta$. Find the geometric place of $ X$ if it satisfies the following conditions: $ (a) \frac {XY}{XZ} \equal{} \frac {XC \plus{} CP}{XB \plus{} BP}$ $ (b) \cos(\alpha) \equal{} AB\cdot \frac {\sin(\beta )}{AP}$

Prove that the equation $4x^3-3x+1=2y^2$ has at least $31$ solutions in positive integers $x$ and $y$ with $x\leq 2005$.

Let $a$ and $ b$ be acute corners. Prove that if $\sin a$, $\sin b$, and $\sin (a + b)$ are rational numbers, then $\cos a$, $\cos b$, and $\cos (a + b)$ are also rational numbers.

Consider triangle $ABC$ with $\angle A=2\angle B$. The angle bisectors from $A$ and $C$ intersect at $D$, and the angle bisector from $C$ intersects $\overline{AB}$ at $E$. If $\dfrac{DE}{DC}=\dfrac13$, compute $\dfrac{AB}{AC}$.

Let $K$ be a point inside an acute triangle $ ABC $, such that $ BC $ is a common tangent of the circumcircles of $ AKB $ and $ AKC$. Let $ D $ be the intersection of the lines $ CK $ and $ AB $, and let $ E $ be the intersection of the lines $ BK $ and $ AC $ . Let $ F $ be the intersection of the line $BC$ and the perpendicular bisector of the segment $DE$. The circumcircle of $ABC$ and the circle $k$ with centre $ F$ and radius $FD$ intersect at points $P$ and $Q$. Prove that the segment $PQ$ is a diameter of $k$.

Find the smallest value of a function $$y = \cos 8x + 3\cos 4x +3\cos2x + 2\cos x.$$

Let $ x\in \mathbb R$. Find the minimum and maximum values of the expresion: $ E\equal{}\dfrac{(1\plus{}x)^8\plus{}16x^4}{(1\plus{}x^2)^4}$

Let $ABC$ be a triangle. $I$ is the incenter of $\Delta ABC$ and $D$ is the meet point of $AI$ and the circumcircle of $\Delta ABC$. Let $E,F$ be on $BD,CD$, respectively such that $IE,IF$ are perpendicular to $BD,CD$, respectively. If $IE+IF=\frac{AD}{2}$, find the value of $\angle BAC$.

Given a circle and a point $ C$ not lying on this circle. Consider all triangles $ ABC$ such that points $ A$ and $ B$ lie on the given circle. Prove that the triangle of maximal area is isosceles.

Show that \[ (2a-1) \sin x + (1-a) \sin(1-a)x \geq 0 \] for $0 \leq a \leq 1$ and $0 \leq x \leq \pi$.

Triangle $ABC$ has a right angle at $C$. If $\sin A = \frac{2}{3}$, then $\tan B$ is $ \textbf{(A)}\ \frac{3}{5}\qquad\textbf{(B)}\ \frac{\sqrt 5}{3}\qquad\textbf{(C)}\ \frac{2}{\sqrt 5}\qquad\textbf{(D)}\ \frac{\sqrt 5}{2}\qquad\textbf{(E)}\ \frac{5}{3} $

Show that triangle $ABC$ is right-angled if its angles satisfy the ratio $\cos^2A + \cos ^2B +\ cos ^2C=1$.

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

Consider a sequence of circles $K_1,K_2,K_3,K_4, \ldots$ of radii $r_1, r_2, r_3, r_4, \ldots$ , respectively, situated inside a triangle $ABC$. The circle $K_1$ is tangent to $AB$ and $AC$; $K_2$ is tangent to $K_1$, $BA$, and $BC$; $K_3$ is tangent to $K_2$, $CA$, and $CB$; $K_4$ is tangent to $K_3$, $AB$, and $AC$; etc. (a) Prove the relation \[r_1 \cot \frac 12 A+ 2 \sqrt{r_1r_2} + r_2 \cot \frac 12 B = r \left(\cot \frac 12 A + \cot \frac 12 B \right) \] where $r$ is the radius of the incircle of the triangle $ABC$. Deduce the existence of a $t_1$ such that \[r_1=r \cot \frac 12 B \cot \frac 12 C \sin^2 t_1\] (b) Prove that the sequence of circles $K_1,K_2, \ldots $ is periodic.

Let $n$ be a positive integer having at least two different prime factors. Show that there exists a permutation $a_1, a_2, \dots , a_n$ of the integers $1, 2, \dots , n$ such that \[\sum_{k=1}^{n} k \cdot \cos \frac{2 \pi a_k}{n}=0.\]

Let $ABC$ be an equilateral triangle and $\Gamma$ its incircle. If $D$ and $E$ are points on the segments $AB$ and $AC$ such that $DE$ is tangent to $\Gamma$, show that $\frac{AD}{DB}+\frac{AE}{EC}=1$.

In quadrilateral $ABCD$, $BC=8$, $CD=12$, $AD=10$, and $m\angle A= m\angle B = 60^\circ$. Given that $AB=p + \sqrt{q}$, where $p$ and $q$ are positive integers, find $p+q$.

Evaluate $ \int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{x^2}{(1\plus{}x\tan x)(x\minus{}\tan x)\cos ^ 2 x}\ dx.$

$ x, y, z $ are positive real numbers such that $ x+y+z=1 $. Prove that \[ \sqrt{ \frac{x}{1-x} } + \sqrt{ \frac{y}{1-y} } + \sqrt{ \frac{z}{1-z} } > 2 \]

Let $ ABCD$ be a parellogram with $ AB>AD$. Suposse the ratio between diagonals $ AC$ and $ BD$ is $ \frac {AC} {BD}\equal{}3$. Let $ r$ be the line symmetric to $ AD$ with respect to $ AC$ and $ s$ the line symmetric to $ BC$ with respect to $ BD$. If $ r$ and $ s$ intersect at $ P$ , find the ratio $ \frac {PA} {PB}$ Daniel

Find a real number $c$ and a positive number $L$ for which \[\lim_{r\to\infty}\frac{r^c\int_0^{\pi/2}x^r\sin x\,dx}{\int_0^{\pi/2}x^r\cos x\,dx}=L.\]

Let $s$ be any arc of the unit circle lying entirely in the first quadrant. Let $A$ be the area of the region lying below $s$ and above the $x$-axis and let $B$ be the area of the region lying to the right of the $y$-axis and to the left of $s$. Prove that $A+B$ depends only on the arc length, and not on the position, of $s$.

The sides of a triangle are equal to $\sqrt2, \sqrt3, \sqrt4$ and its angles are $\alpha, \beta, \gamma$, respectively. Prove that the equation $x\sin \alpha + y\sin \beta + z\sin \gamma = 0$ has exactly one solution in integers $x, y, z$.

Assume every side length of a triangle $ABC$ is more than $2$ and two of its angles are given by $\angle ABC = 57^\circ$ and $ACB = 63^\circ$. Point $P$ is chosen on side $BC$ with $BP:PC = 2:1$. Points $M,N$ are chosen on sides $AB$ and $AC$, respectively so that $BM = 2$ and $CN = 1$. Let $Q$ be the point on segment $MN$ for which $MQ:QN = 2:1$. Find the value of $PQ$. Your answer must be in simplest form.

Let $$f(x) =\sum_{k=0}^{n} a_{k} \sin kx +b_{k} \cos kx,$$ where $a_k$ and $b_k$ are constants. Show that if $|f(x)| \leq 1$ for $x \in [0, 2 \pi]$ and there exist $0\leq x_1 < x_2 <\ldots < x_{2n} < 2 \pi$ with $|f(x_i )|=1,$ then $f(x)= \cos(nx +a)$ for some constant $a.$