- Find positive integers $x,y,z$ such that $$x^{y}+y^{z}+z^{x}=105.$$
- Given a right triangle $ABC$ with the right angle $A$, $AB=AC$ and $BC=a\sqrt{2}$. Let $M,D$ and $E$ be arbitrary points on the sides $BC,AB$ and $AC$ respectively. Find the minimum value of $MD+ME$
- Solve the system of equation \begin{align*} \sqrt{x}-\frac{2}{\sqrt{y}}+\frac{3}{\sqrt{z}} & =1\\ \sqrt{y}-\frac{2}{\sqrt{z}}+\frac{3}{\sqrt{x}} & =2\\ \sqrt{z}-\frac{2}{\sqrt{x}}+\frac{3}{\sqrt{y}} & =3 \end{align*}
- Two circles $(O_{1},R_{1})$ and $(O_{2,}R_{2})$ intersect at $A$ and $B$. A line $d$ is tangent to $(O_{1},R_{1})$ and $(O_{2},R_{2})$ at $C$ and $D$ respectively. Let $R_{3},R_{4}$ respectively be the circumradii of $ACD$ and $BCD$. Prove that $R_{1}\cdot R_{2}=R_{3}\cdot R_{4}$
- Suppose that $a,b,c$ are three positive numbers and given $\alpha,\beta,\lambda\geq2$. Prove that \[\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\] \[\geq\frac{3}{2}+\frac{(a-b)^{2}}{\alpha(b+c)(c+a)}+\frac{(b-c)^{2}}{\beta(c+a)(a+b)}+\frac{(c-a)^{2}}{\lambda(a+b)(b+c)}.\]
- Solve the equation \[ 4x^{3}-24x^{2}+45x-26=\sqrt{-x^{2}+4x-3}.\]
- Let $x_{1},x_{2}$ be two solutions of the equation $x^{2}-2ax-1=0$ where a is an integer. Prove that for every natural number $n$, $\frac{1}{8}(x_{1}^{2n}-x_{2}^{2n})(x_{1}^{4n}-x_{2}^{4n})$ is always a product of three consecutive natural numbers.
- The incircle $I$ of the triangle $ABC$ is tangent to $BC$, $CA$ and $AB$ at $D,E$ and $F$ respectivly. Suppose that $X$ is the intersection of the lines $DE$ and $AB$, and $Y$ is the intersection of the lines $CF$ and $DE$. Prove that $IY\perp CX$.
- Given three positive numbers $a,b,c$ such that $24ab+44bc+33ca\leq1$. Find the minimum value of the expression $P=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$.
- Find all sets of $19$ distinct positive integers such that for each set the sum of the numbers is 2017 and the sums of all the digits of the numbers are all equal.
- The sequence $(a_{n})$, $n=0,1,2,\ldots$, is determined as follows: $a_{0}=610$, $a_{1}=89$ and $a_{n+2}=7a_{n+1}-a_{n}$. Find all $n$ such that $2a_{n+1}a_{n}-3$ is a fourth power of an integer.
- Given a triangle $ABC$ where $AB\ne AC$ and $\widehat{A}=90^{0}$. Let $BE$ and $CF$ be the altitudes from $B$ and $C$. Let $(O_{1})$, $(O_{2})$ be the circles which pass through $A$ and are tangent to $BC$ at $B,C$ respectively. Assume that $D$ is the second intersection of $(O_{1})$ and $(O_{2})$. Let $M,N$ (resp. $P,Q$) respectively be the second intersections of $BA,BD$ (resp. $CA,CD$) and $(O_{2})$ (resp. $(O_{1})$). Let $R$ be the intersection of $AD$ and $EF$, $S$ the intersection of $MP$ and $NQ$. Show that $RS\perp BC$.