- Determine all triple of prime numbers $a,b,c$ (not necessarily distinct) such that \[abc<ab+bc+ca.\]
- Let $ABC$ be a right triangle, with right angle at $A$ and $AH$ is the altitude from $A$, $\widehat{ACB}=30^{0}$. Construct an equilateral triangle $ACD$ ($D$ and $B$ are in opposite side $AC$). K is the foot of the perpendicular line from $H$ onto $AC$. The line through $H$ and parallel to $AD$ meets $AB$ at $M$. Prove the points $D,K,M$ are colinear.
- Consider a $6\times6$ board of squares with 4 corner squares being deleted. Find the smallest number of squares that can be painted black given that among the 5 squares in any figure, there is at lease one black.
- Let $a,b,c$ be real numbers in the interval $[1,2]$. Prove the inequality \[a^{2}+b^{2}+c^{2}+3\sqrt[3]{(abc)^{2}}\geq2(ab+bc+ca).\]
- Let $ABC$ be a non-right triangle ($AB<AC$) with altitude $AH$. $E,F$ are the orthogonal projection of point $H$ onto $AB$ and $AC$ respectively. $EF$ meets $BC$ at $D$. Draw a semicircle with diameter $CD$ on the half-plane containing $A$ with edge $CD$. The line through $B$ and perpendicular to $CD$ meets the semicircle at $K$. Prove that $DK$ is tangent to the circumcircle of triangle $KEF$.
- Given that the equation \[ax^{3}-x^{2}+ax-b=0\quad(a\ne0,\,b\ne0)\] has three positive real roots. Determine the greatest value of the following expression \[P=\frac{11a^{2}-3\sqrt{3}ab-\frac{1}{3}}{9b-10(\sqrt{3}a-1)}.\]
- Solve the following system of equations \[\begin{cases} \sqrt{x-\frac{1}{4}}+\sqrt{y-\frac{1}{4}} & =\sqrt{3}\\ \sqrt{y-\frac{1}{16}}+\sqrt{z-\frac{1}{16}} & =\sqrt{3}\\ \sqrt{z-\frac{9}{16}}+\sqrt{x-\frac{9}{16}} & =\sqrt{3}\end{cases}.\]
- Let $a,b$ be real constants such that $ab>0$. Let $\{u_{n}\}$ be a sequence where $n=1,2,3,\ldots$ given by \[u_{1}=a,\quad u_{n+1}=u_{n}+bu_{n}^{2},\,\forall n\in\mathbb{N}^{*}.\] Determine the limit \[\lim_{n\to\infty}\left(\frac{u_{1}}{u_{2}}+\frac{u_{2}}{u_{3}}+\ldots+\frac{u_{n}}{u_{n+1}}\right).\]
- Find all positive integers $k$ with the property that there exists a polynomial $f(x)$ with integer coefficients of degree greater than 1 such that for all prime numbers $p$ and natural numbers $a,b$ if $p$ divides $(ab-k)$ then it also divides $(f(a)f(b)-k)$.
- Given $a_{i}\in[0,\alpha]$ ($i=\overline{1,n}$), ($\alpha>0$). Prove the inequality \[\prod_{i=1}^{n}(\alpha-a_{i})\leq\alpha^{n}\left(1-\sum_{i=1}^{n}\frac{a_{i}}{S_{i}+\alpha}\right)\] where ${\displaystyle S_{i}=\sum_{k=1}^{n}a_{k}-a_{i}}$ for all $i=\overline{1,n}$.
- Point $O$ is in the interior of triangle $ABC$. The ray $Ox$ parallel to $AB$ meets $BC$ at $D$, ray $Oy$ parallel to $BC$ meets $CA$ at $E$, ray $Oz$ parallel to $CA$ meets $AB$ at $F$. Prove that

a) $3S_{DEF}\leq S_{ABC}$.

b) $OD\cdot OE\cdot OF\leq27AB\cdot BC\cdot CA$. - The circle $(O)$ and $(O')$ meet at points $A,B$. Point $C$ is fixed on $(O)$ and point $D$ is fixed on $(O')$. A moving point $P$ is on the opposite ray of ray $BA$. The circumcircles of traingles $PBC$, $PDB$ intersect $BD$, $BC$ at seconde points $E,F$ respectively. Prove that the midpoint of line segment $EF$ is always on a fixed segment $EF$ is always on a fixed straight line.