$show=home

2010 Issue 395

  1. Compare the following two numbers $$A=\frac{2^{2009}+1}{2^{2010}+1} \text { and } B=\frac{2^{2010}+1}{2^{2011}+1}.$$
  2. Let $A B C$ be a triangle with $\widehat{B A C}=45^{\circ}$, $A M$ is its median, $A D$ is the angle bisector of the triangle $M A C$, draw $D K$ perpendicular to $A B$ ($K$ lies on $A B$). $A M$ cuts $D K$ at $I$. Prove that if $A M$ is the angle bisector of $\widehat{B A D}$ then $B I$ is the angle bisector of $\widehat{A B D}$.
  3. Find all positive numbers $a$ and $b$ such that $\dfrac{a^{2}+b}{b^{2}-a}$ and $\dfrac{b^{2}+a}{a^{2}-b}$ are both integers.
  4. Find the minimum value of the expression $P=a+b+c$ given that $3 \leq a, b,c \leq 5$ and $a^{2}+b^{2}+c^{2}=50$.
  5. Let $A B C$ be a triangle. The angle bisector of $\widehat{B A C}$ cuts the angle bisector of $\widehat{A B C}$ at $I$ and meets $B C$ at $E$. The line perpendicular to $A E$ at $E$ meets the arc $\widehat{B I C}$ of the circumcircle of the triangle $B I C$ at $H$. Prove that $A H$ touches the arc $\widehat{B I C}$.
  6. Let $I$ be the incenter of the triangle $A B C$ with $B C=a$, $C A=b$ and $A B=c$. The lines $A I$, $B I$, $C I$ cut the circumcircle of $A B C$ at $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ respectively. Prove that $$(p-a) I A^{\prime 2}+(p-b) I B^{\prime 2}+(p-c) I C^{\prime 2}=\frac{1}{2} a b c$$ where $p=\dfrac{1}{2}(a+b+c)$.
  7. Let $A B C$ ($B C=a$, $A C=b$, $A B=c$) be a triangle where $A$, $B$, $C$ satisfying the condition $$\cos A+\cos B=2 \cos C.$$ Prove the inequality $$c \geq \frac{8}{9} \max \{a, b\}.$$ When does the equality occur?
  8. Solve the equation $$x^{\log _{7} 11}+3^{\log _{7} x}=2 x.$$
  9. Solve the equation $$\sqrt[3]{3 x+4}=x^{3}+3 x^{2}+x-2$$
  10. Let $X$ be the subset of the set $\{1,2,\ldots, 2010\}$ satisfying conditions $|X|=62$ and for every $x \in X$, there exist $a, b \in X \cup\{0 ; 2011\}$ ($a$ and $b$ differ from $x$) such that $x=\dfrac{a+b}{2}$. Prove that there exist two elements $x$, $y$ in $X$ such that $|x-y| \geq 11$ and $\dfrac{x+y}{2}$ is not in $X$.
  11. Make a torus-shaped chessboard by first identifying a pair of opposite edges of an $n \times n$ chessboard to get a cylinder and then identifying the opposite bases of the resulting cylinder. Prove that it is possible to place $n$ queens on this torus chessboard so that none of them are able to capture any other using the standard chess queen's moves if and only if $(n, 6)=1$. (A queen can capture another if they share the same row, column or diagonal.)
  12. Let ABCDEF be an inscribed hexagon, $A C$ is parallel to $D F$ and $B E$ is the circumdiameter. $A B$ cuts $E F$ at $M$ and $B C$ cuts $D E$ at $N$; $I$ is the intersection point of $A N$ and $CM$. Prove that $E I$ is perpendicular to $A C$.

$type=three$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 395
2010 Issue 395
Mathematics & Youth
https://www.molympiad.org/2020/09/2010-issue-395.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2010-issue-395.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