CR-MO 2017

Let $P (x)$ be a polynomial of degree $2n$ , such that $P (k) =\frac{k}{k + 1}$ for $k = 0,...,2n$ . Determine $P (2n + 1)$ .

Let $f: Z ^+ \to R$ , such that $f (1) = 2018$ and $f (1) + f (2) + ...+ f (n) = n^2f (n)$ , for all $n> 1$ . Find the value $f (2017)$ .

Consider the right triangle $\vartriangle ABC$ right at $A$ and let $D$ be a point on the hypotenuse $BC$ . Consider the line that passes through the incenters of $\vartriangle ABD$ and $\vartriangle ACD$ , and let $K$ and $L$ the intersections of said line with $AB$ and $AC$ respectively. Show that if $AK = AL$ then $D$ is the foot of the altitude on the hypotenuse.

In triangle $ABC$ with incenter $I$ and circumcircle $\omega$ , the tangent through $C$ to $\omega$ intersects $AB$ at point $D$ . The angle bisector of $\angle CDB$ intersects $AI$ and $BI$ at $E$ and $F$ , respectively. Let $M$ be the midpoint of $[EF]$ . Prove that line $MI$ passes through the midpoint of arc $ACB$ of $w$ .

A positive integer is said to be "nefelibata" if, upon taking its last digit and placing it as the first digit, keeping the order of all the remaining digits intact (for example, 312 -> 231), the resulting number is exactly double the original number. Find the smallest possible nefelibata number.