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2016 Issue 469

  1. Compare $A$ and $B$ where $$A=\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\ldots+\frac{2015}{2016!}$$ (with $n!$ is the usual notation for the product $1,2\ldots,n$) and $B=1,02015$.
  2. Given 5 line segments where any 3 of them can form a triangle. Prove that there is at least one acute triangle among the ones which can be formed bt any 3 line segments.
  3. Suppose that $a,b$ and $c$ are positive numbers and $n$ is an integer which is greater or equal to 2. Prove that $$\sqrt[n]{\frac{a}{a+nb}}+\sqrt[n]{\frac{b}{b+nc}}+\sqrt[n]{\frac{c}{c+na}}>1.$$
  4. Given a rhombus $ABCD$ and let the length of the side $AB$ be $2a$. Let $R_{1},R_{2}$ respectively be the circumradius of the triangles $ABC$ and $ABD$. Prove that $R_{1}R_{2}\geq2a^{2}$. When do we have the equality?.
  5. Find integral solutions of the following equation $$\frac{11x}{5}-\sqrt{2x+1}=3y-\sqrt{4y-1}+2.$$
  6. Given a system of equations $$\begin{cases} x\sqrt{y-m}+y\sqrt{z-m}+z\sqrt{x-m} & =6m\sqrt{m},\\ x^{2}+y^{2}+z^{2} & =12m^{2}, \end{cases}$$ where $m$ is a positive number. Find $x,y$ and $z$ in terms $m$.
  7. Find the maximum and minimum of the function $$y=f\left(x\right)=\cos x+4\cos\frac{x}{2}+7\cos\frac{x}{4}+6\cos\frac{x}{8}.$$
  8. Given a triangle $ABC$. Let $M$ be the midpoint of $BC$. Prove that if $\widehat{BAC}\leq\frac{\pi}{2}$ then $$2AM\leq\sqrt{2\left(AB^{2}+AC^{2}\right)}\cos\frac{\widehat{BAC}}{2}.$$ When do we have the equality?.
  9. Let $x,y$ and $z$ be positive numbers such that $xyz=1$. Prove that $$x+y+z+xy+yz+zx\leq 3+\left(\frac{x}{y}\right)^{n}+\left(\frac{y}{z}\right)^{n}+\left(\frac{z}{x}\right)^{n}$$ for every $n\in\mathbb{N}^{*}$.
  10. Find the largest $n$ for which there exist $n$ different positive numbers $x_{1},x_{2},\ldots,x_{n}$ such that $$\frac{x_{i}}{x_{j}}+\frac{x_{j}}{x_{i}}+8\left(\sqrt{3}-2\right)\geq\left(7-4\sqrt{3}\right)\left(\frac{1}{x_{i}x_{i}}+x_{i}x_{j}\right) $$ for any two different indices $i,j$.
  11. Find all functions $f:\mathbb{R}^{+}\to\mathbb{R}$ satisfying $$f\left(\frac{x+y}{2016}\right)=\frac{f\left(x\right)+f\left(y\right)}{2015}$$ for all $x,y\in\mathbb{R}^{+}$.
  12. Two circles $\left(O\right)$ and $\left(O_{1}\right)$ intersect at $B$ and $C$, Let $M$ be the midpoint of $BC$. Let $A$ be a point which varies on $\left(O\right)$ but is different from $B$ and $C$. The lines $AB$ and $CA$ respectively intersect $\left(O_{1}\right)$ at $F$ and $E$. Assume that $P$ and $Q$ respectively be the perpendicular projections of $M$ on $BE$ and $CF$. Construct the parallelogram $MPKQ$. Prove that $AK$ always goes through a fixed point.

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Mathematics & Youth: 2016 Issue 469
2016 Issue 469
Mathematics & Youth
https://www.molympiad.org/2017/06/mathematics-and-youth-magazine-problems_51.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/06/mathematics-and-youth-magazine-problems_51.html
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