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2016 Issue 466

  1. Let $x,y$ and $z$ be integers satisfying $x\ne y$ and $\frac{x}{y}=\frac{x^{2}+z^{2}}{y^{2}+z^{2}}$. Prove that $x^{2}+y^{2}+z^{2}$ is not a prime number.
  2. Given a triangle $ABC$ with $AB<AC$. Let $AD$ be the angle bisector. Let $E$ and $F$ respectivly be points on the sides $AB$ and $AC$ such that $BE=CF$. If $G$ and $H$ respectively are the midpoints of $BF$ and $CE$. Prove that $GH\perp AD$.
  3. Given $3$ different positive integers $a,b$ and $c$ satisfying $ab+bc+ca\geq674$. Find the minimum values of the expression $$ P=\frac{a^{3}+b^{3}+c^{3}}{3}-abc.$$
  4. Given an acute triangle $ABC$ with the altitude $AH$. Let $E\in AB$ and $F\in AC$ such that $HE$ is perpendicular to $AB$ and $HF$ is perpendicular to $AC$. Assume that $HE$ and $BF$ intersect at $K$ and $HF$ and $CE$ intersect at $G$. Choose $M$ and $N$ respectively on the line segments $HB$ and $HC$ such that $EMHK$ and $FNHK$ are inscribed quadrilaterals. Prove that $KN=GM$.
  5. Solve the system of equations $$x^{5}+y^{3}=2z$$ $$y^{5}+z^{3}=2x$$ $$z^{5}+x^{3}=2y$$
  6. Let $a,b$ and $c$ be nonnegative numbers such that $a+b+c\leq\pi$. Prove that $$ 0\leq a-\sin a-\sin b-\sin c+\sin\left(a+b\right)+\sin\left(a+c\right)\leq\pi.$$
  7. Let $a,b$ and $c$ be three sides of a triangle. Prove that $$\frac{3}{2}<\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}<\frac{4\pi}{5}.$$
  8. Given a triangle $ABC$ with the altitude $AH$ ($H\in BC$). Let $\left(I\right)$ be the circle with the diameter $AH$ and the center $I$. The circle $\left(I\right)$ intersects $AB$ and $AC$ respectively at $T$ and $S$. The tangent of $\left(I\right)$ at $T$ and $S$ intersect at $M$. Let $N$, $K$, $J$ respectively be the intersections of $BC$ and $MA$, $BS$ and $CT$, $TS$ and $IM$. Prove that $NJ$ goes through the midpoint of $AK$.
  9. Given $a_{1},a_{2},\ldots a_{n}\in\mathbb{R}$, $n\in\mathbb{N}$, $n\geq3$. Find $$\max\left(\min\left\{ \frac{a_{1}+a_{2}}{1+a_{3}^{2}};\frac{a_{2}+a_{3}}{1+a_{3}^{2}};\ldots;\frac{a_{n-2}+a_{n-1}}{1+a_{n}^{2}}; \frac{a_{n-1}+a_{n}}{1+a_{1}^{2}};\frac{a_{n}+a_{1}}{1+a_{2}^{2}}\right\} \right).$$
  10. Let $n\geq3$ be a natural number. Prove that $\dfrac{(3n)!}{n!(n+1)!(n+2)!}$ is also a natural number.
  11. Given two sequences $\left(x_{n}\right)$ and $\left(y_{n}\right)$ such that $$\begin{cases} \sqrt{y_{n}-x_{n+1}^{2}+1} & =x_{n}\\ \sqrt{3+x_{n+1}^{2}-y_{n}}+\sqrt{10+y_{n-1}-x_{n-1}^{2}} & =\dfrac{5n-1}{n} \end{cases}\quad\forall n\in\mathbb{N}^{*}.$$ Prove that $\left(x_{n}\right)$ and $\left(y_{n}\right)$ converge. Find the limits.
  12. Given an acute triangle $ABC$ which is not isosceles and is inscribed in a circle $\left(O\right)$. Let $P$ be a point inside the triangle such that $AP\perp BC$. Let $E$ and $F$ respectivly be the circumcircle of the triangle $AEF$ intersects $\left(O\right)$ at another point $G$ besides $A$. Prove that $GP$, $BE$ and $CF$ are concurrent.

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Mathematics & Youth: 2016 Issue 466
2016 Issue 466
Mathematics & Youth
https://www.molympiad.org/2017/06/mathematics-and-youth-magazine-problems_63.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/06/mathematics-and-youth-magazine-problems_63.html
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