IT-CESENATICO 2000

A possitive integer is called special if all its decimal digits are equal and it can be represented as the sum of squares of three consecutive odd integers. (a) Find all $4$ -digit special numbers (b) Are there $2000$ -digit special numbers?

Let $ABCD$ be a convex quadrilateral, and write $\alpha=\angle DAB$ , $\beta=\angle ADB$ , $\gamma=\angle ACB$ , $\delta= \angle DBC$ and $\epsilon=\angle DBA$ . Assuming that $\alpha<\pi/2$ , $\beta+\gamma=\pi /2$ , and $\delta+2\epsilon=\pi$ , prove that $(DB+BC)^2=AD^2+AC^2$ .

A pyramid with the base $ABCD$ and the top $V$ is inscribed in a sphere. Let $AD = 2BC$ and let the rays $AB$ and $DC$ intersect in point $E$ . Compute the ratio of the volume of the pyramid $VAED$ to the volume of the pyramid $VABCD$ .

Let $n > 1$ be a fixed integer. Alberto and Barbara play the following game: (i) Alberto chooses a positive integer, (ii) Barbara chooses an integer greater than $1$ which is a multiple or submultiple of the number Alberto chose (including itself), (iii) Alberto increases or decreases the Barbara's number by $1$ . Steps (ii) and (iii) are alternatively repeated. Barbara wins if she succeeds to reach the number $n$ in at most $50$ moves. For which values of $n$ can she win, no matter how Alberto plays?

A man disposes of sufficiently many metal bars of length $2$ and wants to construct a grill of the shape of an $n \times n$ unit net. He is allowed to fold up two bars at an endpoint or to cut a bar into two equal pieces, but two bars may not overlap or intersect. What is the minimum number of pieces he must use?

Let $p(x)$ be a polynomial with integer coefficients such that $p(0) = 0$ and $0 \le p(1) \le 10^7$ . Suppose that there exist positive integers $a,b$ such that $p(a) = 1999$ and $p(b) = 2001$ . Determine all possible values of $p(1)$ .

(Note: $1999$ is a prime number.)