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2012 Issue 418

  1. Given \[A=1^{5}+2^{5}+3^{5}+\ldots+2011^{5}.\] Find the last digit of $A$.
  2. Let $ABC$ be an isosceles right triangle with right angle at $A$. On the half-plane defined by $AB$ containing $C$ draw an isosceles right triangle $ABD$ with right angle at $B$. Let $E$ be the midpoint of segment $BD$. Draw $CM$ perpendicular to $AE$ at $M$. Let $N$ be the midpoint of segment $CM$, $K$ is the intersection of $BM$ and $DN$. Find the measure of the angle $BKD$.
  3. Find all positive integer solutions of the equation \[3^{x}-32=y^{2}.\]
  4. Find all minimal value of the expression \[A=\frac{1}{x^{3}+xy+y^{3}}+\frac{4x^{2}y^{2}+2}{xy}\] where $x$ and $y$ are positive real numbers satisfying $x+y=1$.
  5. Let $ABC$ be an acute triangle with orthocenter $H$. Prove that $ABC$ is an equilateral triangle if and only if \[\frac{AH}{BC}=\frac{BH}{CA}=\frac{CH}{AB}.\]
  6. Let $ABC$ be a triangle with circumcenter $O$, and incenter $I$. $BC$ touches the circle $(I)$ at $D$. The circle whose diameter is $AI$ meets $(O)$ at $M$ ($M\ne A$) and cuts the line passing through $A$ parallel to $BC$ at $N$. Prove that $MO$ passes through the midpoint of $DN$.
  7. Solve the system of equations \[\begin{cases}\sqrt{xy+(x-y)(\sqrt{xy}+2)}+\sqrt{x} & =y+\sqrt{y}\\(x+1)(y+\sqrt{xy}+x(1-x)) & =4\end{cases}.\]
  8. Let $ABC$ be an acute triangle. Prove the inequaltiy \[\cos^{3}A+\cos^{3}B+\cos^{3}C+\cos A.\cos B.\cos C\geq\frac{1}{2}.\]
  9. For each natural number $n$, let $(S_{n})$ be the sum of all digits of $n$ (in the decimal system). Put $S_{k}(n)=S(S(\ldots(S(n))\ldots))$ ($k$ times). Find all natural numbers $n$ such that \[S_{1}(n)+S_{2}(n)+\ldots S_{k}(n)+\ldots+S_{223}(n)=n.\]
  10. Does there exist a set $X$ satisfying the following two conditions
    • $X$ contains $2012$ natural numbers.
    • The sum of any arbitrary elements in $X$ is the $k$-th power of a positive integer ($k\geq2$).
  11. Find all functions $f:\mathbb{R}\to\mathbb{R}$ satisfing \[f\left(\frac{xf(y)}{2}\right)+f\left(\frac{yf(x)}{2}\right)=4xy,\:\forall x,y\in\mathbb{R}.\]
  12. Fix two circles $(K)$ and $(O)$, where $(K)$ is inside $(O)$. Two circles $(O_{1})$, $(O_{2})$ are moving so that they always externally touch each other at $M$. Both also internally touch $(O)$, and externally touch $(K)$. Prove that $M$ belongs to a fixed circle.

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Mathematics & Youth: 2012 Issue 418
2012 Issue 418
Mathematics & Youth
https://www.molympiad.org/2017/10/mathematics-and-youth-magazine-problems_28.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/10/mathematics-and-youth-magazine-problems_28.html
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