CMO 1972

Let $a_1,a_2,\ldots,a_n$ be non-negative real numbers. Define $M$ to be the sum of all products of pairs $a_ia_j$ $(i<j)$ , $\textit{i.e.}$ , $$M = a_1(a_2+a_3+\cdots+a_n)+a_2(a_3+a_4+\cdots+a_n)+\cdots+a_{n-1}a_n.$$ Prove that the square of at least one of the numbers $a_1,a_2,\ldots,a_n$ does not exceed $2M/n(n-1)$ .

a) Prove that $10201$ is composite in all bases greater than 2.

b) Prove that $10101$ is composite in all bases.

Describe a construction of quadrilateral $ABCD$ given:

(i) the lengths of all four sides;

(ii) that $AB$ and $CD$ are parallel;

(iii) that $BC$ and $DA$ do not intersect.

Prove that the equation $x^3+11^3=y^3$ has no solution in positive integers $x$ and $y$ .