Mathematics and Youth Magazine Problems For 50th Anniversary Contest 2014

Junior

1. Solve the system of equations \[\begin{cases} x^{3}+y^{3} & =x^{2}+xy+y^{2}\\ \sqrt{6x^{2}y^{2}-x^{4}-y^{4}} & =\dfrac{13}{4}(x+y)-2xy-\dfrac{3}{4}\end{cases}.\]
2. In a triangle $ABC$, let $(O)$, $(I)$ and $I_{a}$ denote the circumcircle, incircle and the center of its $A$-excircle. The incircle $(I)$ touches segment $BC$ at $D$, $P$ is the midpoint of arc $BAC$ of the circle $(O)$, and $PI_{a}$ intersects $(O)$ again at $K$. Prove that $\widehat{DAI}=\widehat{KAI}$.
   
3. Find all triples of positive integers $(a,b,c)$ such that $a,b,c$ are the side lengths of a triangle and \[\sqrt{\frac{19}{a+b-c}}+\sqrt{\frac{5}{b+c-a}}+\sqrt{\frac{79}{c+a-b}}\]  is an odd integer, differ from 1.
4. In a triangle $ABC$, segment $CD$ is the altitude from $C$. Points $E,F$ chosen on the side $AB$ such that $\widehat{ACE}=\widehat{BCF}=90^{0}$. $X$ is a point on segment $CD$; point $K$ on segment $FX$ such that $BK=BC$ and point $L$ on segment $EX$ such that $AL=AC$; $AL$ meets $BK$ at $M$. Prove that $ML=MK$. 
5. Find all triple of positive integers $(x,y,p)$ where $p$ is a prime number and \[8x^{3}+y^{3}-6xy=p-1.\]
6. In a triangle $ABC$, let $I$ denote a point on the internal angle-bisector of the angle $BAC$. The line through $B$ and parallel to $CI$ intersects $AC$ at $D$; the line through $C$ and parallel to $BI$ meets $AB$ at $E$. Let $M$, $N$ denote the midpoints of $BD$, $CE$ respectively. Prove that $AI$ is perpendicular to $MN$.

Senior

1. The sequence of positive real numbers $\{a_{n}\}_{n=0}^{\infty}$ satisfies \[a_{n+1}=\frac{2}{a_{n}+a_{n-1}},\quad n=1,2,3,\ldots.\] Prove that there exists a pair of real numbers $s,t$ such that $s\leq a_{n}\leq t$ for all $n=0,1,2,\ldots$.
2. In a triangle $ABC$ ($AC>AB$), the altitudes $BB'$ and $CC'$ intersects at point $H$. Let $M,N$ be the midpoints of $BC'$, $CB'$ respectively. $MH$ meets the circumcircle of triangle $CHB'$ at $I$; $NH$ meets the circumcircle of triangle $BHC'$ at point $J$. If $P$ is the midpoint of segment $BC$, prove that $AP\perp IJ$.
3. Find all triples of positive integers $(a,b,p)$ such that $p$ is a prime number, $a$ and $b$ have no common divisor, and the set of prime divisors of $a+b$ is the same as that of $a^{p}+b^{p}$.
4. Let $ABC$ be a non-isosceles triangle. The incircle $(I)$ touches $BC$, $CA$ and $AB$ at $A_{0}$, $B_{0}$ and $C_{0}$respectively; $AI$, $BI$, $CI$ intersect $BC$, $CA$, $AB$ at $A_{1}$, $B_{1}$, $C_{1}$; $B_{0}C_{0}$, $C_{0}A_{0}$, $A_{0}B_{0}$ meet $B_{1}C_{1}$, $C_{1}A_{1}$, $A_{1}B_{1}$ at $A_{2}$, $B_{2}$, $C_{2}$ respectively. Prove that the lines $A_{0}A_{2}$, $B_{0}B_{2}$, $C_{0}C_{2}$ intersect at a point on the circle $(I)$. 
5. Find all nonempty subsets $A,B$ of the set of positive integers $\mathbb{Z}^{+}$ such that the following conditions are satisfied
• $A\cap B=\emptyset,$
• If $a\in A$, $b\in B$ then $a+b\in A$ and $2a+b\in B$.
6. Let $ABC$ be a triangle, non-isosceles at vertex $A$. Let $O$, $H$ denote its circumcenter and orthocenter respectively. The line through $A$ and perpendicular to $OH$ intersects $BC$ at $K$. Prove that $H$ and the centers of the Euler circles of triangles $ABC$, $ABK$ and $ACK$ lie on a circle.