OBM 2015

Let $\triangle ABC$ be an acute-scalene triangle, and let $N$ be the center of the circle wich pass trough the feet of altitudes. Let $D$ be the intersection of tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$ . Prove that $A$ , $D$ and $N$ are collinear iff $\measuredangle BAC = 45º$ .

Consider $S=\{1, 2, 3, \cdots, 6n\}$ , $n>1$ . Find the largest $k$ such that the following statement is true: every subset $A$ of $S$ with $4n$ elements has at least $k$ pairs $(a,b)$ , $a<b$ and $b$ is divisible by $a$ .

Given a natural $n>1$ and its prime fatorization $n=p_1^{\alpha 1}p_2^{\alpha_2} \cdots p_k^{\alpha_k}$ , its false derived is defined by $$f(n)=\alpha_1p_1^{\alpha_1-1}\alpha_2p_2^{\alpha_2-1}...\alpha_kp_k^{\alpha_k-1}.$$ Prove that there exist infinitely many naturals $n$ such that $f(n)=f(n-1)+1$ .

Let $n$ be a integer and let $n=d_1>d_2>\cdots>d_k=1$ its positive divisors. a) Prove that $$d_1-d_2+d_3-\cdots+(-1)^{k-1}d_k=n-1$$ iff $n$ is prime or $n=4$ . b) Determine the three positive integers such that $$d_1-d_2+d_3-...+(-1)^{k-1}d_k=n-4.$$

Is that true that there exist a polynomial $f(x)$ with rational coefficients, not all integers, with degree $n>0$ , a polynomial $g(x)$ , with integer coefficients, and a set $S$ with $n+1$ integers such that $f(t)=g(t)$ for all $t \in S$ ?

Let $\triangle ABC$ be a scalene triangle and $X$ , $Y$ and $Z$ be points on the lines $BC$ , $AC$ and $AB$ , respectively, such that $\measuredangle AXB = \measuredangle BYC = \measuredangle CZA$ . The circumcircles of $BXZ$ and $CXY$ intersect at $P$ . Prove that $P$ is on the circumference which diameter has ends in the ortocenter $H$ and in the baricenter $G$ of $\triangle ABC$ .