OME 2001

Prove that the graph of the polynomial $P(x)$ is symmetric in respect to point $A(a,b)$ if and only if there exists a polynomial $Q(x)$ such that: $P(x) = b + (x-a)Q((x-a)^2)).$

Let $P$ be a point on the interior of triangle $ABC$ , such that the triangle $ABP$ satisfies $AP = BP$ . On each of the other sides of $ABC$ , build triangles $BQC$ and $CRA$ exteriorly, both similar to triangle $ABP$ satisfying: $$BQ = QC$$ and $$CR = RA.$$ Prove that the point $P,Q,C,$ and $R$ are collinear or are the vertices of a parallelogram.

You have five segments of lengths $a_1, a_2, a_3, a_4,$ and $a_5$ such that it is possible to form a triangle with any three of them. Demonstrate that at least one of those triangles has angles that are all acute.

The integers between $1$ and $9$ inclusive are distributed in the units of a $3$ x $3$ table. You sum six numbers of three digits: three that are read in the rows from left to right, and three that are read in the columns from top to bottom. Is there any such distribution for which the value of this sum is equal to $2001$ ?

A quadrilateral $ABCD$ is inscribed in a circle of radius 1 whose diameter is $AB$ . If the quadrilateral $ABCD$ has an incircle, prove that $CD \leq 2 \sqrt{5} - 2$ .

Define the function $f: \mathbb{N} \rightarrow \mathbb{N}$ which satisfies, for any $s, n \in \mathbb{N}$ , the following conditions: $f(1) = f(2^s)$ and if $n < 2^s$ , then $f(2^s + n) = f(n) + 1.$ Calculate the maximum value of $f(n)$ when $n \leq 2001$ and find the smallest natural number $n$ such that $f(n) = 2001.$