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 \[a^{2}+b^{2}+c^{2}\geq\left(4\sqrt{3}+\frac{OG^{2}}{R^{2}}\right)S+(a-b)^{2}+(b-c)^{2}+(c-a)^{2}.\]




Mathematics & Youth: 2012 Issue 417
2012 Issue 417
Mathematics & Youth
Loaded All Posts Not found any posts VIEW ALL Readmore Reply Cancel reply Delete By Home PAGES POSTS View All RECOMMENDED FOR YOU LABEL ARCHIVE SEARCH ALL POSTS Not found any post match with your request Back Home Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sun Mon Tue Wed Thu Fri Sat January February March April May June July August September October November December Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec just now 1 minute ago $$1$$ minutes ago 1 hour ago $$1$$ hours ago Yesterday $$1$$ days ago $$1$$ weeks ago more than 5 weeks ago Followers Follow THIS CONTENT IS PREMIUM Please share to unlock Copy All Code Select All Code All codes were copied to your clipboard Can not copy the codes / texts, please press [CTRL]+[C] (or CMD+C with Mac) to copy