Mathematical Olympic Revenge 2014

Let $ABC$ an acute triangle and $\Gamma$ its circumcircle. The bisector of $BAC$ intersects $\Gamma$ at $M\neq A$. A line $r$ parallel to $...

  1. Let $ABC$ an acute triangle and $\Gamma$ its circumcircle. The bisector of $BAC$ intersects $\Gamma$ at $M\neq A$. A line $r$ parallel to $BC$ intersects $AC$ at $X$ and $AB$ at $Y$. Also, $MX$ and $MY$ intersect $\Gamma$ again at $S$ and $T$, respectively. If $XY$ and $ST$ intersect at $P$, prove that $PA$ is tangent to $\Gamma$.
  2. a) Let $n$ a positive integer. Prove that $$\gcd(n, \lfloor n\sqrt{2} \rfloor)<\sqrt[4]{8}\sqrt{n}.$$ b) Prove that there are infinitely many positive integers $n$ such that $$\gcd(n, \lfloor n\sqrt{2} \rfloor)>\sqrt[4]{7.99}\sqrt{n}.$$
  3. Let $n$ a positive integer. In a $2n\times 2n$ board, $1\times n$ and $n\times 1$ pieces are arranged without overlap. Call an arrangement maximal if it is impossible to put a new piece in the board without overlapping the previous ones. Find the least $k$ such that there is a maximal arrangement that uses $k$ pieces.
  4. Let $a>1$ be a positive integer and $f\in \mathbb{Z}[x]$ with positive leading coefficient. Let $S$ be the set of integers $n$ such that $n \mid a^{f(n)}-1$. Prove that $S$ has density $0$; that is, prove that $$\lim_{n\rightarrow \infty} \frac{|S\cap \{1,...,n\}|}{n}=0.$$

COMMENTS

Name

Balkan,1,Bosonia,1,Brazil,1,Bulgary,1,Canada,1,CentroAmerican,1,CGMO,1,China,5,Cono Sur,2,France,1,Germany,2,Greece,2,IberoAmerican,1,IMO,1,India,2,Indonedia,1,International,40,Italy,1,Itan,1,Japan,1,JBMO,2,Kazakhstan,1,Korea,2,Macedonia,1,Mediterrane,7,Mediterranean,3,MEMO,1,Mexico,1,Miklós Schweitzer,1,Moldova,1,National,29,Olympic Revenge,1,Paenza,1,Paraguayan,1,Rusia,1,TST,9,Turkey,1,Tuymaada,19,Undergraduate,1,Zhautykov,1,
ltr
item
MATHEMATICAL OLYMPIAD PROBLEMS: Mathematical Olympic Revenge 2014
Mathematical Olympic Revenge 2014
MATHEMATICAL OLYMPIAD PROBLEMS
http://www.molympiad.ml/2017/11/mathematical-olympic-revenge-2014.html
http://www.molympiad.ml/
http://www.molympiad.ml/
http://www.molympiad.ml/2017/11/mathematical-olympic-revenge-2014.html
true
3289146460604631361
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