DK-GEORGMOHR 1998

For any real number $m$ , the equation $$x^2+(m-2)x- (m+3)=0$$ has two solutions, denoted $x_1$ and $x_2$ . Determine $m$ such that $x_1^2+x_2^2$ is the minimum possible.

Let $a$ and $b$ be positive real numbers with $a + b =1$ . Show that $$\left(a+\frac{1}{a}\right)^2 + \left(b+\frac{1}{b}\right)^2 \ge \frac{25}{2}.$$

A neat fruit arrangement on a large round dish is edged with strawberries. Between $100$ and $200$ berries are used for this border. A deliciously hungry child eats first one of the strawberries and then starts going round and round the dish, she eats strawberries in the following way: When she has eaten a berry, she leaves it next lie, then she eats the next, leaves the next, etc. Thus she continues until there is only one strawberry left. This berry is the one that was lying right after the very first thing she ate. How many berries were there originally?