Mathematics and Youth Magazine Problems - Aug 2015, Issue 458

1. For a given prime number $p$, find positive integers $x,y$ such that  $\frac{1}{x}+\frac{1}{y}=\frac{1}{p}.$
2. Given an acute triangle $ABC$ with the orthocenter $H$. Let $M$ be the midpoint of $BC$. The line through $A$ parallel to $MH$ meets the line through $H$ parallel to $MA$ at $N$. Prove that $AH^{2}+BC^{2}=MN^{2}.$
3. Suppose that $\begin{cases} a^{3}-a^{2}+a-5 & =0\\ b^{3}-2b^{2}+2b+4 & =0 \end{cases}.$ Find $a+b$.
4. From a point $M$ outside the circle $(O)$ draw to tangents $MA,MB$ to $(O)$ ($A,B$ are points of tangency). $C$ is an arbitrary point on the minor arc $AB$ of $(O)$. The rays $AC$ and $BC$ intersect $MB$ and $MA$ at $D$ and $E$ respectively. Prove that the circumcircles of the triangles $ACE$, $BCD$ and $OCM$ meet at another point which is different from $C$.
5. Find all triples of positive integers $(a,b,c)$ such that  $(a^{5}+b)(a+b^{5})=2^{c}.$
6. Solve the following system of equations $\begin{cases} x+y+z+\sqrt{xyz} & =4\\ \sqrt{2x}+\sqrt{3y}+\sqrt{3z} & =\frac{7\sqrt{2}}{2}\\ x & =\min\{x,y,z\}\end{cases}.$
7. Given a diamond $ABCD$ with $\widehat{BAD}=120^{0}$. Let $M$ vary on the side $BD$. Assume that $H$ and $K$ are the orthogonal projections of $M$ on the lines through $AB$ and $AD$ respectively. Let $N$ be the midpoint of $HK$. Prove that the line through $MN$ always passes through a fixed point.
8. Given a triangle $ABC$. Find the maximum value and the minimum value of the expression $P=\cos^{2}2A\cdot\cos^{2}2B\cdot\cos^{2}2C+\frac{\cos4A\cdot\cos4B\cdot\cos4C}{8}.$
9. Find the smallest $k$ such that $S=a^{3}+b^{3}+c^{3}+kabc-\frac{k+3}{6}\left[a^{2}(b+c)+b^{2}(c+a)+c^{2}(a+b)\right]\leq0$ for all triples $(a,b,c)$ which are the lengths of the sides of a triangle.
10. Given a sequence of polynomials $\{P_{n}(x)\}$ satisfying the following conditions $P_{1}(x)=2x$, $P_{2}(x)=2(x^{2}+1)$ and $P_{n}(x)=2xP_{n-1}(x)-(x^{2}-1)P_{n-2}(x),\quad n\in\mathbb{N},n\geq3.$ Prove that $P_{n}(x)$ is divisible by $Q(x)=x^{2}+1$ if and only if $n=4k+2$, $k\in\mathbb{N}$.
11. Consider the funtion $f(n)=1+2n+3n^{2}+\ldots+2016n^{2015}.$ Let $(t_{0},t_{1},\ldots,t_{2016})$ and $(s_{0},s_{1},\ldots,s_{2016})$ be two permutations of $(0,1,\ldots,2016)$. Prove that there exist two different numbers in the following set $$A=\left\{s_{0}f(t_{0}),s_{1}f(t_{1}),\ldots,s_{2016}f(t_{2016})\right\}$$ such that their difference is divisible by $2017$.
12. Given a triangle $ABC$ and an arbitrary point $M$. Prove that $\frac{1}{BC^{2}}+\frac{1}{CA^{2}}+\frac{1}{AB^{2}}\geq\frac{9}{(MA+MB+MC)^{2}}.$