DE-BUNDESWETTBEWERB 2001

Let $ABC$ an acute triangle with circumcircle center $O.$ The line $(BO)$ intersects the circumcircle again in $D,$ and the extension of the altitude from $A$ intersects the circle in $E.$ Prove that the quadrilateral $BECD$ and the triangle $ABC$ have the same area.

Prove: For each positive integer is the number of divisors whose decimal representations ends with a 1 or 9 not less than the number of divisors whose decimal representations ends with 3 or 7.

10 vertices of a regular 100-gon are coloured red and ten other (distinct) vertices are coloured blue. Prove that there is at least one connection edge (segment) of two red which is as long as the connection edge of two blue points.

For each $n \in \mathbb{N}$ we have two numbers $p_n, q_n$ with the following property: For exactly $n$ distinct integer numbers $x$ the number $$x^2 + p_n \cdot x + q_n$$ is the square of a natural number. (Note the definition of natural numbers includes the zero here.)

Let $ABC$ be a triangle. Points $A',B',C'$ are on the sides $BC, CA, AB,$ respectively such that we have $$\overline{A'B'} = \overline{B'C'} = \overline{C'A'}$$ and $$\overline{AB'} = \overline{BC'} = \overline{CA'}.$$ Prove that triangle $ABC$ is equilateral.

A square $R$ of sidelength $250$ lies inside a square $Q$ of sidelength $500$ . Prove that: One can always find two points $A$ and $B$ on the perimeter of $Q$ such that the segment $AB$ has no common point with the square $R$ , and the length of this segment $AB$ is greater than $521$ .