IT-CESENATICO 1990

A cube of edge length $3$ consists of $27$ unit cubes. Find the number of lines passing through exactly three centers of these $27$ cubes, as well as the number of those passing through exactly two such centers.

In a triangle $ABC$ , the bisectors of the angles at $B$ and $A$ meet the opposite sides at $P$ and $Q$ , respectively. Suppose that the circumcircle of triangle $PQC$ passes through the incenter $R$ of $\vartriangle ABC$ . Given that $PQ = l$ , find all sides of triangle $PQR$ .

Let $a,b,c$ be distinct real numbers and $P(x)$ a polynomial with real coefficients. Suppose that the remainders of $P(x)$ upon division by $(x-a), (x-b)$ and $(x-c)$ are $a,b$ and $c$ , respectively. Find the polynomial that is obtained as the remainder of $P(x)$ upon division by $(x-a)(x-b)(x-c)$ .

Let $a,b,c$ be side lengths of a triangle with $a+b+c = 1$ . Prove that $a^2 +b^2 +c^2 +4abc \le \frac12$ .

Prove that, for any integer $x$ , $x^2 +5x+16$ is not divisible by $169$ .

Some marbles are distributed over $2n + 1$ bags. Suppose that, whichever bag is removed, it is possible to divide the remaining bags into two groups of $n$ bags such that the number of marbles in each group is the same. Prove that all the bags contain the same number of marbles.