$show=home

2010 Issue 397

  1. Compare the following two fractions $$A=\frac{1010^{1010}}{2010^{2010}} \text { and } B=\frac{2010^{2010}}{3010^{3010}}$$
  2. Let $A B C$ be a triangle with $\widehat{B A C} \geq 60^{\circ}$. Prove that $A B+A C \leq 2 B C$.
  3. Find the remainder when dividing $3^{2^{n}}$ by $2^{n+3}$ where $n$ is a positive integer.
  4. Let $A B C$ be a triangle. Construct a parallelogram $AMPN$ so that the points $M$, $N$ are in $A B$, $A C$ respectively; $P$ lies inside the triangle $A B C$. Let $Q$ be the intersection of the line $A P$ and $B C$. Prove that $$\frac{A M \cdot A N \cdot P Q}{A B \cdot A C \cdot A Q} \leq \frac{1}{27}.$$ Find the position of $P$ when the equality occurs.
  5. Solve the equation $$\left(x+\frac{5-x}{\sqrt{x}+1}\right)^{2}=\frac{-192(\sqrt{x}+1)}{5 \sqrt{x}-x \sqrt{x}}$$
  6. Two circles $\left(O_{1}\right)$ and $\left(O_{2}\right)$ meet at points $K$ and $L$ such that their centers $O_{1}$ and $O_{2}$ lie on the same side of the line $K L .$ The tangent line to $\left(O_{1}\right)$ at $K$ meets $\left(O_{2}\right)$ at $A .$ The tangent line to $\left(O_{2}\right)$ at $K$ meets $\left(O_{1}\right)$ at $B$. Find the area of the triangle $A K B,$ given that $A L=3$, $B L=6$ and $\tan \widehat{A K B}=-\dfrac{1}{2}$.
  7. Let $a, b, c$ be positive real numbers satisfying $a+b+c=3$. Prove the following inequality $$\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}+5 \geq(a+b)(b+c)(c+a).$$ When does equality occur?
  8. Solve the system of equations $$\begin{cases}x&=3 z^{3}+2 z^{2} \\ y&=3 x^{3}+2 x^{2} \\ z&=3 y^{3}+2 y^{2}\end{cases}$$
  9. Let $A B C$ be an acute triangle. Let $a$, $b$, $c$ be the side lengths of the triangle and $h_{a}$, $h_{b}$, $h_{c}$ be the length of the corresponding altitudes. Let $r$, $R$ be respectively the inradius and circumradius of this triangle. Prove the inequality $$\frac{9 R}{a^{2}+b^{2}+c^{2}} \leq \frac{1}{h_{a}+\sqrt{h_{b} h_{c}}}+\frac{1}{h_{b}+\sqrt{h_{c} h_{a}}}+\frac{1}{h_{c}+\sqrt{h_{a} h_{b}}} \leq \frac{1}{2 r}$$
  10. Let $A$ be the set of $n$ distinct points on the plane $(n \geq 2)$ and $T(A)$ be the set of vectors whose endpoints are in $A$. Find the maximum and minimum value of $|T(A)|$. (where $|T(A)|$ denotes the cardinality of $T(A)$.)
  11. Let $f$ be a continuous function on $\mathbb{R}$ satisfying the following two conditions $$f(2012)=2011,\quad f(x)f_{4}(x)=1,\, \forall x \in \mathbb{R}.$$ Denote $f_{n}(x)=\underbrace{f(f \ldots f(x)))}_{n \text { times } f}$. Find $f(2010)$
  12. Let $\left(x_{n}\right)$ $(n=1,2, \ldots)$ be a sequence given by $$x_{1}=2,1;\quad x_{n+1}=\frac{x_{n}-2+\sqrt{x_{n}^{2}+8 x_{n}-4}}{2}.$$ For each positive integer $n$, let $\displaystyle y_{n}=\sum_{i=1}^{n} \frac{1}{x_{i+1}^{2}-4}$. Find $\displaystyle\lim_{n \rightarrow+\infty} y_{n}$.

$type=three$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 397
2010 Issue 397
Mathematics & Youth
https://www.molympiad.org/2020/09/2010-issue-397.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2010-issue-397.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