CAPS 2001

Let $n\ge2$ be a natural number, and $a_i$ be positive numbers, where $i=1,2,\cdots,n.$ Show that $$\left(a_1^3+1\right)\left(a_2^3+1\right)\cdots\left(a_n^3+1\right) \geq \left(a_1^2a_2+1\right)\left(a_2^2a_3+1\right)\cdots\left(a_n^2a_1+1\right)$$

A triangle $ABC$ has acute angles at $A$ and $B$ . Isosceles triangles $ACD$ and $BCE$ with bases $AC$ and $BC$ are constructed externally to triangle $ABC$ such that $\angle ADC = \angle ABC$ and $\angle BEC = \angle BAC$ . Let $S$ be the circumcenter of $\triangle ABC$ . Prove that the length of the polygonal line $DSE$ equals the perimeter of triangle $ABC$ if and only if $\angle ACB$ is right.

Distinct points $A$ and $B$ are given on the plane. Consider all triangles $ABC$ in this plane on whose sides $BC,CA$ points $D,E$ respectively can be taken so that (i) $\frac{BD}{BC}=\frac{CE}{CA}=\frac{1}{3}$ ; (ii) points $A,B,D,E$ lie on a circle in this order.

Find the locus of the intersection points of lines $AD$ and $BE$ .

Find all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy $$f(x^2 + y) + f(f(x) - y) = 2f(f(x)) + 2y^2 \text{ for all }x, y \in \mathbb{R}.$$

Points with integer coordinates in cartesian space are called lattice points. We color $2000$ lattice points blue and $2000$ other lattice points red in such a way that no two blue-red segments have a common interior point (a segment is blue-red if its two endpoints are colored blue and red). Consider the smallest rectangular parallelepiped that covers all the colored points. (a) Prove that this rectangular parallelepiped covers at least $500,000$ lattice points. (b) Give an example of a coloring for which the considered rectangular paralellepiped covers at most $8,000,000$ lattice points.