$show=home

2010 Issue 393

  1. Let $a_{1}, a_{2}, \ldots, a_{2010}$ be natural numbers such that $$\frac{1}{a_{1}^{11}}+\frac{1}{a_{2}^{11}}+\frac{1}{a_{3}^{11}}+\ldots+\frac{1}{a_{2010}^{11}}=\frac{1005}{1024}.$$ Determine the value of the following expression $$A=\frac{a_{2010}^{6}}{a_{1}^{5}}+\frac{a_{2009}^{6}}{a_{2}^{5}}+\frac{a_{2008}^{6}}{a_{3}^{5}}+\ldots+\frac{a_{1}^{6}}{a_{2010}^{5}}$$
  2. In a triangle $A B C$ where $B$ and $C$ are acute angles, let $B D$ and $A H$ be respectively the angle bisector and the altitude. Given that $\widehat{A D B}=\widehat{A H D}=\alpha,$ find the measure of $\alpha$.
  3. Find all integer solutions of the equation $$y^{3}=x^{6}+2 x^{4}-1000$$
  4. Find the minimum value of the expression $$P=\frac{a}{b+c+d-a}+\frac{b}{c+d+a-b}+\frac{c}{d+a+b-c}+\frac{d}{a+b+c-d}$$ where $a, b, c, d$ are the length of 4 sides of a convex quadrilateral.
  5. Let $A B C$ be an isosceles triangle with vertex at $C$. Let $O$, $I$ be its circumcenter and incenter respctively. $D$ is a point chosen on the side $B C$ so that $D O$ is perpendicular to $B I$. Prove that $D I$ is parallel to $A C$.
  6. Let $a, b, c$ be positive real numbers satisfying the condition $$\frac{1}{a+b+1}+\frac{1}{b+c+1}+\frac{1}{c+a+1} \geq 1.$$ Prove that $$a+b+c=a b+b c+c a$$
  7. Let $A B C D E$ be a cyclic pentagon with $A C \parallel D E$ and $\widehat{A M B}=\widehat{B M C}$ where $M$ is the midpoint of $B D$. Prove that the line $B E$ passes through the midpoint of $A C$.
  8. Let $O A B C$ be a tetrahedron with right trihedral angle at vertex $O$. Prove that $$\cot A B \cdot \cot B C+\cot B C \cdot \cot C A+\cot C A \cdot \cot A B \leq \frac{3}{2}$$ where cot $A B$ is the cotangent of the dihedral angle of side $A B$.
  9. Let $A B C$ be an acute triangle. The altitudes $B K$ and $C L$ meet at $H$. The line passing through $H$ meets $A B$, $A C$ at $P$, $Q$ respectively. Prove that $H P=H Q$ if and only if $M P=M Q$ where $M$ is the midpoint of $B C$
  10. Find all real numbers $k$ and $m$ such that $$k\left(x^{3}+y^{3}+z^{3}\right)+m x y z \geq(x+y+z)^{3}$$ for any non-negative numbers $x, y, z$
  11. Let $\left(x_{n}\right)$ be a sequence of real numbers, $n=1,2, \ldots$ satisfying $$\ln \left(1+x_{n}^{2}\right)+n x_{n}=1$$ for any positive integers $n$. Find $$\lim _{n \rightarrow+\infty} \frac{n\left(1+n x_{n}\right)}{x_{n}}$$
  12. Find all continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x+f(y))=2 y+f(x),\, \forall x, y \in \mathbb{R}$$

$type=three$c=3$source=random$title=oot$p=1$h=1$m=hide$rm=hide

Anniversary_$type=three$c=12$title=oot$h=1$m=hide$rm=hide

Name

2006,1,2007,12,2008,12,2009,12,2010,12,2011,12,2012,12,2013,12,2014,12,2015,12,2016,12,2017,12,2018,11,2019,12,2020,12,2021,6,Anniversary,4,
ltr
item
Mathematics & Youth: 2010 Issue 393
2010 Issue 393
Mathematics & Youth
https://www.molympiad.org/2020/09/2010-issue-393.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2010-issue-393.html
true
8958236740350800740
UTF-8
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