Let $S_1$ and $S_2$ be two spheres with distinct radii that touch externally. The spheres lie inside a cone $C$, and each sphere touches the cone in a full circle. Inside the cone there are $n$ additional solid spheres arranged in a ring in such a way that each solid sphere touches the cone $C$, both of the spheres $S_1$ and $S_2$ externally, as well as the two neighboring solid spheres. What are the possible values of $n$? [i]Proposed by Iceland.[/i]

Let $ABC$ be and acute triangle. Construct a triangle $PQR$ such that $ AB = 2PQ $, $ BC = 2QR $, $ CA = 2 RP $, and the lines $ PQ, QR,$ and $RP$ pass through the points $ A, B , $ and $ C $, respectively. (All six points $ A, B, C, P, Q, $ and $ R $ are distinct.)

An equilateral triangle $ABC$ is given, together with a point $P$ inside it. [asy] draw((0,0)--(4,0)--(2,3.464)--(0,0)); draw((1.3, 1.2)--(0,0)); draw((1.3, 1.2)--(2,3.464)); draw((1.3, 1.2)--(4,0)); label("$A$",(0,0),SW); label("$B$",(4,0),SE); label("$C$",(2,3.464),N); label("$P$",(1.3,1.2),S); [/asy] Given that $PA = 3$ cm, $PB = 5$ cm, and $PC = 4$ cm, find the side of the equilateral triangle.

Let $f: \mathbb{R}\to\mathbb{R}$ is a function such that $f( \cot x ) = \cos 2x+\sin 2x$ for all $0 < x < \pi$. Define $g(x) = f(x) f(1-x)$ for $-1 \leq x \leq 1$. Find the maximum and minimum values of $g$ on the closed interval $[-1, 1].$

Let $ABC$ be a triangle with $\angle{BAC} > 90$. Let $D$ be a point on the segment $BC$ and $E$ be a point on line $AD$ such that $AB$ is tangent to the circumcircle of triangle $ACD$ at $A$ and $BE$ is perpendicular to $AD$. Given that $CA=CD$ and $AE=CE$. Determine $\angle{BCA}$ in degrees.

Let $O$ be the circumcenter and $H$ the orthocenter of an acute-angled triangle $ABC$ such that $BC>CA$. Let $F$ be the foot of the altitude $CH$ of triangle $ABC$. The perpendicular to the line $OF$ at the point $F$ intersects the line $AC$ at $P$. Prove that $\measuredangle FHP=\measuredangle BAC$.

Evaluate $ \int_0^{\frac {\pi}{2}} \frac {dx}{(1 \plus{} \cos x)^2}$.

For a convex hexagon $ ABCDEF$ with an area $ S$, prove that: \[ AC\cdot(BD\plus{}BF\minus{}DF)\plus{}CE\cdot(BD\plus{}DF\minus{}BF)\plus{}AE\cdot(BF\plus{}DF\minus{}BD)\geq 2\sqrt{3}S \]

Evaluate \[{\int_0^\frac{\pi}{2}} \frac{\sin 2005x}{\sin x}dx\]

Let $R$ and $T$ be points respectively on sides $[BC]$ and $[CD]$ of a square $ABCD$ with side length $6$ such that $|CR|+|RT|+|TC|=12$. What is $\tan (\widehat{RAT})$ $ \textbf{(A)}\ 2\sqrt 3 \qquad\textbf{(B)}\ \sqrt 3 \qquad\textbf{(C)}\ \dfrac 13 \qquad\textbf{(D)}\ \dfrac 12 \qquad\textbf{(E)}\ 1 $

Let $A_1A_2...A_n$ be a regular polygon ($n > 4$), $T$ be the common point of $A_1A_2$ and $A_{n-1}A_n$ and $M$ be a point in the interior of the triangle $A_1A_nT$. Show that the equality $$\sum_{i=1}^{n-1} \frac{\sin^2 \left(\angle A_iMA_{i+1}\right)}{d(M,A_iA_{i+1}}=\frac{\sin^2 \left(\angle A_1MA_n\right)}{d(M,A_1A_n} $$ holds if and only if $M$ belongs to the circumcircle of the polygon.

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?

The $A$-excircle of a triangle $ABC$ touches the side $BC$ at the point $K$ and the extended side $AB$ at the point $L$. The $B$-excircle touches the lines $BA$ and $BC$ at the points $M$ and $N$, respectively. The lines $KL$ and $MN$ meet at the point $X$. Show that the line $CX$ bisects the angle $ACN$.

In plane,let a circle $(O)$ and two fixed points $B,C$ lies in $(O)$ such that $BC$ not is the diameter.Consider a point $A$ varies in $(O)$ such that $A\neq B,C$ and $AB\neq AC$.Call $D$ and $E$ respective is intersect of $BC$ and internal and external bisector of $\widehat{BAC}$,$I$ is midpoint of $DE$.The line that pass through orthocenter of $\triangle ABC$ and perpendicular with $AI$ intersects $AD,AE$ respective at $M,N$. 1/Prove that $MN$ pass through a fixed point 2/Determint the place of $A$ such that $S_{AMN}$ has maxium value

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

Let $ ABC$ be a triangle, let $ AB > AC$. Its incircle touches side $ BC$ at point $ E$. Point $ D$ is the second intersection of the incircle with segment $ AE$ (different from $ E$). Point $ F$ (different from $ E$) is taken on segment $ AE$ such that $ CE \equal{} CF$. The ray $ CF$ meets $ BD$ at point $ G$. Show that $ CF \equal{} FG$.

Let $S_1$ and $S_2$ be two spheres with distinct radii that touch externally. The spheres lie inside a cone $C$, and each sphere touches the cone in a full circle. Inside the cone there are $n$ additional solid spheres arranged in a ring in such a way that each solid sphere touches the cone $C$, both of the spheres $S_1$ and $S_2$ externally, as well as the two neighboring solid spheres. What are the possible values of $n$? [i]Proposed by Iceland.[/i]

Consider the cyclic quadrilateral with side lengths $1$, $4$, $8$, $7$ in that order. What is its circumdiameter? Let the answer be of the form $a\sqrt b+c$, for $b$ squarefree. Find $a+b+c$.

Let $ABC$ be a scalene triangle and $k$ be its circumcircle. Let $t_A,t_B,t_C$ be the tangents to $k$ at $A, B, C,$ respectively. Prove that points $AB\cap t_C$, $CA\cap t_B$, and $BC\cap t_A$ exist, and that they are collinear.

Let $ABCDEFGHIJKL$ be a regular dodecagon. Find $\frac{AB}{AF} + \frac{AF}{AB}$.

For $ f(x)\equal{}e^{\frac{x}{2}}\cos \frac{x}{2}$, evaluate $ \sum_{n\equal{}0}^{\infty} \int_{\minus{}\pi}^{\pi}f(x)f(x\minus{}2n\pi)dx\ (n\equal{}0,\ 1,\ 2,\ \cdots)$.

The measures, in degrees, of the angles , $\alpha, \beta$ and $\theta$ are greater than $0$ less than $60$. Find the value of $\theta$ knowing, also, that $\alpha + \beta = 2\theta$ and that $$\sin \alpha \sin \beta \sin \theta = \sin(60 - \alpha ) \sin(60 - \beta) \sin(60 - \theta ).$$

Show that the improper integral \[ \lim_{B \rightarrow \infty} \displaystyle\int_{0}^{B} \sin (x) \sin (x^2) dx \] converges.

In $ \triangle ABC$, $ AB \equal{} BC$, and $ BD$ is an altitude. Point $ E$ is on the extension of $ \overline{AC}$ such that $ BE \equal{} 10$. The values of $ \tan CBE$, $ \tan DBE$, and $ \tan ABE$ form a geometric progression, and the values of $ \cot DBE$, $ \cot CBE$, $ \cot DBC$ form an arithmetic progression. What is the area of $ \triangle ABC$? [asy]unitsize(3mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair D=(0,0), C=(3,0), A=(-3,0), B=(0, 8), Ep=(6,0); draw(A--B--Ep--cycle); draw(D--B--C); label("$A$",A,S); label("$D$",D,S); label("$C$",C,S); label("$E$",Ep,S); label("$B$",B,N);[/asy]$ \textbf{(A)}\ 16 \qquad \textbf{(B)}\ \frac {50}{3} \qquad \textbf{(C)}\ 10\sqrt3 \qquad \textbf{(D)}\ 8\sqrt5 \qquad \textbf{(E)}\ 18$

Find the smallest natural number $n$ for which $sin \Big(\frac{1}{n+1934}\Big)<\frac{1}{1994}$ .