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 $$\lim_{n\rightarrow \infty} \frac{|S\cap \{1,...,n\}|}{n}=0.$$



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MATHEMATICAL OLYMPIAD PROBLEMS: Mathematical Olympic Revenge 2014
Mathematical Olympic Revenge 2014
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