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2012 Issue 416

  1. Find all natural numbers $x,y,z$ such that \[2010^{x}+2011^{y}=2012^{z}.\]
  2. The natural numbers $a_{1},a_{2},\ldots,a_{100}$ satisfy the equation \[\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{100}}=\frac{101}{2}.\]Prove that there are at least two equal numbers.
  3. Let $a,b,c$ be positive real numbers. Prove the inequality \[\frac{(a+b)^{2}}{ab}+\frac{(b+c)^{2}}{bc}+\frac{(c+a)^{2}}{ca}\geq9+2\left(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\right).\]
  4. Solve the equation \[4x^{2}+14x+11=4\sqrt{6x+10}.\]
  5. In a triange $ABC$, te incircle $(I)$ meets $BC$, $CA$ at $D$, $E$ respectively. Let $K$ be the point of reglection of $D$ through the midpoint of $BC$, the line through $K$ and perpendicular to $BC$ meets $DE$ at $L$, $N$ is the midpoint of $KL$. Prove that $BN$ and $AK$ are orthogonal.
  6. Determine the maximum value of the expression \[A=\frac{mn}{(m+1)(n+1)(m+n+1)}\] where $m,n$ are natural numbers.
  7. Triangle $ABC$ ($AB>AC$) is inscribed in circle $(O)$. The exterior angle bisector of $BAC$ meets $(O)$ at another point $E$; $M,N$ are the midpoints of $BC$, $CA$ respectively; $F$ os the perpendicular foot of $E$ on $AB$, $K$ is the intersection of $MN$ and $AE$. Prove that $KF$ and $BC$ are parallel.
  8. Solve the equation \[\sin^{2n+1}x+\sin^{n}2x+(\sin^{n}x-\cos^{n}x)^{2}-2=0\] where $n$ is a given positive integer.
  9. Find all polynomials $P(x)$ such that \[P(2)=12,\quad P(x^{2})=x^{2}(x^{2}+1)P(x),\:\forall x\in\mathbb{R}.\]
  10. Let $r_{1},r_{2},\ldots,r_{n}$ be $n$ rational numbers such that $0<r_{i}\leq\dfrac{1}{2}$, ${\displaystyle \sum_{i=1}^{n}r_{i}=1}$ ($n>1$), and let $f(x)=[x]+\left[x+\dfrac{1}{2}\right]$. Find the greatest value of the expression ${\displaystyle P(k)=2k-\sum_{i=1}^{n}f(kr_{i})}$ where $k$ runs over the integers $\mathbb{Z}$ (the notation $[x]$ means the greatest integer not exceeding $x$).
  11. Suppose that $f:\mathbb{R}\to\mathbb{R}$ is a continuous funtion such that $f(x)+f(x+1006)$ is a rational number if and only if $x\in\mathbb{R}$, \[f(x+20)+f(x+12)+f(x+2012)\] is itrational. Prove that $f(x)=f(x+2012)$ for all $x\in\mathbb{R}$.
  12. Prove the following inequality \[\frac{m_{a}}{h_{a}}+\frac{m_{b}}{h_{b}}=\frac{m_{c}}{h_{c}}\leq1+\frac{R}{r},\] where $m_{a},b_{b},m_{c}$ are medians; $h_{a},h_{b},h_{c}$ are the altitudes from $A,B,C$ and $R,r$ are the circumradius and inradius, respectively. 

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Mathematics & Youth: 2012 Issue 416
2012 Issue 416
Mathematics & Youth
https://www.molympiad.org/2017/10/mathematics-and-youth-magazine-problems_8.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/10/mathematics-and-youth-magazine-problems_8.html
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