Determine an equation of third degree with integral coefficients having roots $\sin \frac{\pi}{14}, \sin \frac{5 \pi}{14}$ and $\sin \frac{-3 \pi}{14}.$

Let $ABC$ be a triangle. For every point $M$ belonging to segment $BC$ we denote by $B'$ and $C'$ the orthogonal projections of $M$ on the straight lines $AC$ and $BC$. Find points $M$ for which the length of segment $B'C'$ is a minimum.

Given an acute-angled triangle $ABC$, let points $A' , B' , C'$ be located as follows: $A'$ is the point where altitude from $A$ on $BC$ meets the outwards-facing semicircle on $BC$ as diameter. Points $B', C'$ are located similarly. Prove that $A[BCA']^2 + A[CAB']^2 + A[ABC']^2 = A[ABC]^2$ where $A[ABC]$ is the area of triangle $ABC$.

For $n\in\mathbb{N}$, $n\geq 2$, $a_{i}, b_{i}\in\mathbb{R}$, $1\leq i\leq n$, such that \[\sum_{i=1}^{n}a_{i}^{2}=\sum_{i=1}^{n}b_{i}^{2}=1, \sum_{i=1}^{n}a_{i}b_{i}=0. \] Prove that \[\left(\sum_{i=1}^{n}a_{i}\right)^{2}+\left(\sum_{i=1}^{n}b_{i}\right)^{2}\leq n. \] [i]Cezar Lupu & Tudorel Lupu[/i]

Let $n$ be a natural number. Find all solutions $x$ of the system of equations $$\left\{\begin{matrix} sinx+cosx=\frac{\sqrt{n}}{2}\\tg\frac{x}{2}=\frac{\sqrt{n}-2}{3}\end{matrix}\right.$$ On interval $\left[0,\frac{\pi}{4}\right).$

Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\tfrac12$ is $\tfrac{a-b\pi}c$, where $a$, $b$, and $c$ are positive integers and $\gcd(a,b,c)=1$. What is $a+b+c$? $\textbf{(A) }59\qquad\textbf{(B) }60\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$

Find $ \lim_{n\to\infty} \int_0^1 x^2|\sin n\pi x|\ dx\ (n\equal{}1,\ 2,\cdots)$.

Let $P(t)$ be a nonconstant polynomial with real coefficients. Prove that the system of simultaneous equations $$ \int_{0}^{x} P(t)\sin t \, dt =0, \;\;\;\; \int_{0}^{x} P(t) \cos t \, dt =0 $$ has only finitely many solutions $x.$

A rectangular box has width $12$ inches, length $16$ inches, and height $\tfrac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of $30$ square inches. Find $m+n$.

Let the incircle of the triangle $ABC$ touch $[BC]$ at $D$ and $I$ be the incenter of the triangle. Let $T$ be midpoint of $[ID]$. Let the perpendicular from $I$ to $AD$ meet $AB$ and $AC$ at $K$ and $L$, respectively. Let the perpendicular from $T$ to $AD$ meet $AB$ and $AC$ at $M$ and $N$, respectively. Show that $|KM|\cdot |LN|=|BM|\cdot|CN|$.

What algebraic relationship holds between $ A $, $ B $, and $ C $ if $$ctg A + \frac{\cos B}{\sin A \cos C} = ctg B + \frac{\cos A}{\sin B \cos C}.$$

Three semicircles of radius $ 1$ are constructed on diameter $ AB$ of a semicircle of radius $ 2$. The centers of the small semicircles divide $ \overline{AB}$ into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles? [asy]import graph; unitsize(14mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dashed=linetype("4 4"); dotfactor=3; pair A=(-2,0), B=(2,0); fill(Arc((0,0),2,0,180)--cycle,mediumgray); fill(Arc((-1,0),1,0,180)--cycle,white); fill(Arc((0,0),1,0,180)--cycle,white); fill(Arc((1,0),1,0,180)--cycle,white); draw(Arc((-1,0),1,60,180)); draw(Arc((0,0),1,0,60),dashed); draw(Arc((0,0),1,60,120)); draw(Arc((0,0),1,120,180),dashed); draw(Arc((1,0),1,0,120)); draw(Arc((0,0),2,0,180)--cycle); dot((0,0)); dot((-1,0)); dot((1,0)); draw((-2,-0.1)--(-2,-0.3),gray); draw((-1,-0.1)--(-1,-0.3),gray); draw((1,-0.1)--(1,-0.3),gray); draw((2,-0.1)--(2,-0.3),gray); label("$A$",A,W); label("$B$",B,E); label("1",(-1.5,-0.1),S); label("2",(0,-0.1),S); label("1",(1.5,-0.1),S);[/asy]$ \textbf{(A)}\ \pi\minus{}\sqrt3 \qquad \textbf{(B)}\ \pi\minus{}\sqrt2 \qquad \textbf{(C)}\ \frac{\pi\plus{}\sqrt2}{2} \qquad \textbf{(D)}\ \frac{\pi\plus{}\sqrt3}{2}$ $ \textbf{(E)}\ \frac{7}{6}\pi\minus{}\frac{\sqrt3}{2}$

Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic. [i]Proposed by John Cuya, Peru[/i]

Let $ABC$ be a triangle with $AB=13$, $AC=25$, and $\tan A = \frac{3}{4}$. Denote the reflections of $B,C$ across $\overline{AC},\overline{AB}$ by $D,E$, respectively, and let $O$ be the circumcenter of triangle $ABC$. Let $P$ be a point such that $\triangle DPO\sim\triangle PEO$, and let $X$ and $Y$ be the midpoints of the major and minor arcs $\widehat{BC}$ of the circumcircle of triangle $ABC$. Find $PX \cdot PY$. [i]Proposed by Michael Kural[/i]

Suppose $a,b,c \in \mathbb R^+$and \[\frac1{a^2+1}+\frac1{b^2+1}+\frac1{c^2+1}=2\] Prove that $ab+ac+bc\leq \frac32$

Let $ABC$ be a isosceles triangle with $AB=AC$ an $D$ be the foot of perpendicular of $A$. $P$ be an interior point of triangle $ADC$ such that $m(APB)>90$ and $m(PBD)+m(PAD)=m(PCB)$. $CP$ and $AD$ intersects at $Q$, $BP$ and $AD$ intersects at $R$. Let $T$ be a point on $[AB]$ and $S$ be a point on $[AP$ and not belongs to $[AP]$ satisfying $m(TRB)=m(DQC)$ and $m(PSR)=2m(PAR)$. Show that $RS=RT$

Let $\alpha , \beta ,\gamma$ be the angles of a triangle $ABC$ of perimeter $ 2p $ and $R$ is the radius of its circumscribed circle. $(a)$ Prove that \[\cot^{2}\alpha +\cot^{2}\beta+\cot^{2}\gamma\geq 3\left(9\cdot \frac{R^{2}}{p^{2}} - 1\right).\] $(b)$ When do we have equality?

In a tetrahedral $ABCD$, given that $\angle ADB=\angle BDC =\angle CDA=\frac{\pi}{3}$, $AD=BD=3$, and $CD=2$. Find the radius of the circumsphere of $ABCD$.

Prove that if the circumcircles of the faces of a tetrahedron $ABCD$ have equal radii, then $AB=CD$, $AC=BD$ and $AD=BC$.

Let $ABC$ be an acute triangle and let $k_{1},k_{2},k_{3}$ be the circles with diameters $BC,CA,AB$, respectively. Let $K$ be the radical center of these circles. Segments $AK,CK,BK$ meet $k_{1},k_{2},k_{3}$ again at $D,E,F$, respectively. If the areas of triangles $ABC,DBC,ECA,FAB$ are $u,x,y,z$, respectively, prove that \[u^{2}=x^{2}+y^{2}+z^{2}.\]

Let $n$ be a positive integer, and $a_j$, for $j=1,2,\ldots,n$ are complex numbers. Suppose $I$ is an arbitrary nonempty subset of $\{1,2,\ldots,n\}$, the inequality $\left|-1+ \prod_{j\in I} (1+a_j) \right| \leq \frac 12$ always holds. Prove that $\sum_{j=1}^n |a_j| \leq 3$.

Find the limit \[\lim_{n\to\infty} \frac{1}{n\ln n}\int_{\pi}^{(n+1)\pi} (\sin ^ 2 t)(\ln t)\ dt.\]

Evaluate $\int_0^{\pi} e^{\sin x}\cos ^ 2(\sin x )\cos x\ dx$.

Prove \[\left[\int_\pi^\infty \frac{\cos x}{x}\ dx\right]^2< \frac{1}{{\pi}^2}\]

Let $a,b,c$ be the lengths of the sides of a triangle, and $\alpha, \beta, \gamma$ respectively, the angles opposite these sides. Prove that if \[ a+b=\tan{\frac{\gamma}{2}}(a\tan{\alpha}+b\tan{\beta}) \] the triangle is isosceles.