- Determine the sum of $2005$ terms $$S=\frac{2^{2}}{1.3}+\frac{3^{2}}{2.4}+\frac{4^{2}}{3.5}+\ldots+\frac{2006^{2}}{2005.2007}$$
- Let $ABC$ be an isosceles right triangle with right angle at $A .$ Pick an arbitrary point $M$ on $A C$. The foot of the altitude with $B C$ through $M$ is $H .$ Let $I$ be the midpoint of $BM$. Find the value of the angle $\angle H A I .$
- Prove that there exist infinitely many positive integers $n$ such that $n !$ is a multiple of $n^{2}+1$
- Let $a, b, c$ be positive numbers such that $a+b+c=a b c .$ Prove that $$\frac{a}{b^{3}}+\frac{b}{c^{3}}+\frac{c}{a^{3}} \geq 1.$$
- Solve the equation $$\frac{4}{x}+\sqrt{x-\frac{1}{x}}=x+\sqrt{2 x-\frac{5}{x}}$$
- Let $A B C$ be a right triangle at $A$ with $A B<A C$, $B C=2+2 \sqrt{3}$ and its incircle has radius is $1 .$ Determine the value of the angles $\angle B$ and $\angle C$.
- Let $A B C$ be an acute triangle with orthocenter $H .$ Let $A G$ be the altitude through $A$ and let $D$ be the midpoint of $B C .$ The circles, whose diameters are $B C$ and $A D$ respectively, meet at $E$ and $F$. Prove that a) The circumcircles of $H, E$ and $G$ and of $H$, $F$ and $G$ are both tangent to the circle whose diameter is the side $B C$. b) $H, E$, and $F$ lie on the same line.
- Find the maximum value of the following expression $$x_{1}^{3} x_{2}^{2}+x_{2}^{3} x_{3}^{2}+\ldots+x_{n}^{3} x_{1}^{2}+n^{2(n-1)} x_{1}^{3} x_{2}^{3} \ldots x_{n}^{3}$$ where $x_{1}, x_{2}, \ldots, x_{n}$ are non-negative numbers whose sum is $1$ $(n \geq 2)$.
- Solve the system of equations $$\begin{cases}20\left(x+\dfrac{1}{x}\right) &=11\left(y+\dfrac{1}{y}\right) &=2007\left(z+\dfrac{1}{z}\right) \\ x y+y z+z x &=1 \end{cases}$$
- Find the limit at $x=0$ of the following function $$\frac{\sqrt{\frac{\cos 2 x+\sqrt[3]{1+3 x}}{2}}-\sqrt[3]{\frac{\cos 3 x+3 \cos x-\ln (1+x)^{4}}{4}}}{x}$$
- Let $\left(O_{1}\right)$ be a circle inside a triangle $A B C$ and touches both sides $A B$ and $A C$. Construct a second circle $\left(O_{2}\right)$ through $B$, $C$ and touches outside of $\left(O_{1}\right)$ at $T .$ Prove that the angle bisector of $\widehat{B T C}$ passes through the incenter of the triangle $A B C$.
- Let $A_{1} A_{2} A_{3} A_{4}$ be a tetrahedron with perpendicular opposite edges. Denote by $S_{i}$ the area of the opposing faces of the vertices $A_{I}$ $(i=1,2,3,4) .$ Find all possible points $M$ where the sum $$S_{1} \cdot M A_{1}+S_{2} \cdot M A_{2}+S_{3} \cdot M A_{3}+S_{4} \cdot M A_{4}$$ is smallest possible.