IT-CESENATICO 1998

Calculate the sum $\sum_{n=1}^{1.000.000}[ \sqrt{n} ]$ .

You may use the formula $\sum_{i=1}^{k} i^2=\frac{k(k +1)(2k +1)}{6}$ without a proof.

Alberto wants to organize a poker game with his friends this evening. Bruno and Barbara together go to gym once in three evenings, whereas Carla, Corrado, Dario and Davide are busy once in two evenings (not necessarily the same day). Moreover, Dario is not willing to play with Davide, since they have a quarrel over a girl. A poker game requires at least four persons (including Alberto). What is the probability that the game will be played?

Let $ABCD$ be a trapezoid with the longer base $AB$ such that its diagonals $AC$ and $BD$ are perpendicular. Let $O$ be the circumcenter of the triangle $ABC$ and $E$ be the intersection of the lines $OB$ and $CD$ . Prove that $BC^2 = CD \cdot CE$ .

Suppose $a_1,a_2,a_3,a_4$ are distinct integers and $P(x)$ is a polynomial with integer coefficients satisfying $P(a_1) = P(a_2) = P(a_3) = P(a_4) = 1$ . (a) Prove that there is no integer $n$ such that $P(n) = 12$ . (b) Do there exist such a polynomial and $a_n$ integer $n$ such that $P(n) = 1998$ ?

We say that a function $f : N \to N$ is increasing if $f(n) < f(m)$ whenever $n < m$ , multiplicative if $f(nm) = f(n)f(m)$ whenever $n$ and $m$ are coprime, and completely multiplicative if $f(nm) = f(n)f(m)$ for all $n,m$ . (a) Prove that if $f$ is increasing then $f(n) \ge n$ for each $n$ . (b) Prove that if $f$ is increasing and completely multiplicative and $f(2) = 2$ , then $f(n) = n$ for all $n$ . (c) Does (b) remain true if the word "completely" is omitted?