Show that $ \frac{\frac{1}{1\cdot 2} +\frac{1}{3\cdot 4}+\cdots +\frac1{1997\cdot 1998}}{\frac{2}{1000\cdot 1998} +\frac{1}{1001\cdot 1997}} $ is an integer number. Bogdan Enescu

Consider the rectangle $ ABCD $ and the points $ M,N,P,Q $ on the segments $ AB,BC,CD, $ respectively, $ DA, $ excluding its extremities. Denote with $ p_{\square} , A_{\square} $ the perimeter, respectively, the area of $ \square. $ Prove that: a) $ p_{MNPQ}\ge AC+BD. $ b) $ p_{MNPQ} =AC+BD\implies A_{MNPQ}\le \frac{A_{ABCD}}{2} . $ c) $ p_{MNPQ} =AC+BD\implies MP^2 +NQ^2\ge AC^2. $ Dan Brânzei and Gheorghe Iurea

Let $ n $ be a natural number. Find all integer numbers that can be written as $$ \frac{1}{a_1} +\frac{2}{a_2} +\cdots +\frac{n}{a_n} , $$where $ a_1,a_2,...,a_n $ are natural numbers.

Solve in $ \mathbb{Z}^2 $ the following equation: $$ (x+1)(x+2)(x+3) +x(x+2)(x+3)+x(x+1)(x+3)+x(x+1)(x+2)=y^{2^x} . $$ Adrian Zanoschi

We´re given an inscriptible quadrilateral $ DEFG $ having some vertices on the sides of a triangle $ ABC, $ and some vertices (at least one of them) coinciding with the vertices of the same triangle. Knowing that the lines $ DF $ and $ EG $ aren´t parallel, find the locus of their intersection. Dan Brânzei

Find the smallest natural number for which there exist that many natural numbers such that the sum of the squares of their squares is equal to $ 1998. $ Gheorghe Iurea