CR-MO 2018

Consider $f (n, m)$ the number of finite sequences of $1$ 's and $0$ 's such that each sequence that starts at $0$ , has exactly n $0$ 's and $m$ $1$ 's, and there are not three consecutive $0$ 's or three $1$ 's. Show that if $m, n> 1$ , then $$f (n, m) = f (n-1, m-1) + f (n-1, m-2) + f (n-2, m-1) + f (n-2, m-2)$$

Let $O$ be the center of the circle circumscribed to $\vartriangle ABC$ , and let $P$ be any point on $BC$ ( $P \ne B$ and $P \ne C$ ). Suppose that the circle circumscribed to $\vartriangle BPO$ intersects $AB$ at $R$ ( $R \ne A$ and $R \ne B$ ) and that the circle circumscribed to $\vartriangle COP$ intersects $CA$ at point $Q$ ( $Q \ne C$ and $Q \ne A$ ).

1. Show that $\vartriangle PQR \sim \vartriangle ABC$ and that $O$ is orthocenter of $\vartriangle PQR$ .
2. Show that the circles circumscribed to the triangles $\vartriangle BPO$ , $\vartriangle COP$ , and $\vartriangle PQR$ all have the same radius.

Consider $\vartriangle ABC$ , with $AD$ bisecting $\angle BAC$ , $D$ on segment $BC$ . Let $E$ be a point on $BC$ , such that $BD = EC$ . Through $E$ we draw the line $\ell$ parallel to $AD$ and consider a point $P$ on it and inside the $\vartriangle ABC$ . Let $G$ be the point where line $BP$ cuts side $AC$ and let F be the point where line $CP$ to side $AB$ . Show that $BF = CG$ .

In the accompanying figure, semicircles with centers $A$ and $B$ have radii $4$ and $2$ , respectively. Furthermore, they are internally tangent to the circle of diameter $PQ$ . Also the semicircles with centers $A$ and $B$ are externally tangent to each other. The circle with center $C$ is internally tangent to the semicircle with diameter $PQ$ and externally tangent to the others two semicircles. Determine the value of the radius of the circle with center $C$ .

Prove that there are only two sets of consecutive positive integers that satisfy that the sum of its elements is equal to $100$ .

Determine all triples $(a, b, c)$ of nonnegative integers that satisfy: $$(c-1) (ab- b -a) = a + b-2$$

Let $p$ be a prime number such that $p = 10^{d -1} + 10^{d-2} + ...+ 10 + 1$ . Show that $d$ is a prime.