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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