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2012 Issue 417

  1. Which number is bigger, $2^{3100}$ or $3^{2100}$?.
  2. Let $ABC$ be an isosceles triangle with $AB=AC$. $BM$ is the median from $B$. $N$ is a point on $BC$ such that $\widehat{CAN}=\widehat{ABM}$. Prove that $CM\geq CN$.
  3. Let $a,b,c$ be positive numbers such that \[|a+b+c|\leq1,\,|a-b+c|\leq1,\,|4a+2b+c|\leq8,\,|4a-2b+c|\leq8.\] Prove the inequality \[|a|+3|b|+|c|\leq7.\]
  4. Solve the equation \[(x-2)(x^{2}+6x-11)^{2}=(5x^{2}-10x+1)^{2}.\]
  5. Let $ABC$ be a right triangle, with right angle at$A$, $AH$ is the altitude from $A$ and $I,J$ ae the incenters of triangles $HAB$ and $HAC$, respectively. $IJ$ cuts $AB$ at $M$ and meets $AC$ at $N$. Let $X$ and $Y$ be the intersections of $HI$ with $AB$ and $HJ$ with $AC$; $BY$, $CX$ cuts $MN$ at $P$ and $Q$ respectively. Prove that \[\frac{AI}{AJ}=\frac{HP}{HQ}.\]
  6. Let $x,y,z$ be real numbers such that $x^{2}+y^{2}+z^{2}=3$. Find the minimum and maximum value of the expression \[P=(x+2)(y+2)(z+2).\]
  7. In a triangle $ABC$, let $m_{a},m_{b},m_{c}$ be its median lengths, and $l_{a},l_{b},l_{c}$ be the lengths of its inner bisectors, $p$ is half of its perimeter. Prove the inequality \[m_{a}+m_{b}+m_{c}+l_{a}+l_{b}+l_{c}\leq2\sqrt{3}p.\]
  8. Let $S.ABC$ be a pyramid where surface $SAB$ is a isosceles triangle at $S$ and $\widehat{BSA}=120^{0}$, the plane $(SAB)$ is perpendicular to $(ABC)$. Prove that $\dfrac{S_{ABC}}{S_{SAC}}\leq\sqrt{3}$, when does the equality occur?. (Denote by $S_{DEF}$ the area of triangle $DEF$)
  9. A natural number $n$ is a good number if it is possible to partition any square into $n$ smaller squares such that at least two of them are not equal.
    a) Prove that if $n$ is a good number, then $n\geq4$.
    b) Prove that both $4$ and $5$ are not good.
    c) Find all good numbers.
  10. A sequence $a_{0},a_{1},\ldots,a_{n}$ ($n\geq2$) is defined by \[a_{0}=0,\quad a_{k}=\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{n+k},\,k=1,2,\ldots,n.\] Prove the inequality \[\sum_{k=0}^{n-1}\frac{e^{a_{k}}}{n+k+1}+(\ln2-a_{n})e^{a_{n}}<1\] where ${\displaystyle e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}}$.
  11. Find all functions $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ satisfying \[f(x)f(yf(x))=f(y+f(x)),\quad x,y\in\mathbb{R}^{+}.\]
  12. Given a triangle $ABC$ inscribed in a circle $(O,R)$, with center $G$ and area $S$. Prove that \[a^{2}+b^{2}+c^{2}\geq\left(4\sqrt{3}+\frac{OG^{2}}{R^{2}}\right)S+(a-b)^{2}+(b-c)^{2}+(c-a)^{2}.\]

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Mathematics & Youth: 2012 Issue 417
2012 Issue 417
Mathematics & Youth
https://www.molympiad.org/2017/10/mathematics-and-youth-magazine-problems_22.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/10/mathematics-and-youth-magazine-problems_22.html
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