SE-SKOLORNAS 1988

Let $a > b > c$ be sides of a triangle and $h_a,h_b,h_c$ be the corresponding altitudes. Prove that $a+h_a > b+h_b > c+h_c$ .

Six ducklings swim on the surface of a pond, which is in the shape of a circle with radius $5$ m. Show that at every moment, two of the ducklings swim on the distance of at most $5$ m from each other.

Show that if $x_1+x_2+x_3 = 0$ for real numbers $x_1,x_2,x_3$ , then $x_1x_2+x_2x_3+x_3x_1\le 0$ .

Find all $n \ge 4$ for which $x_1+x_2+...+x_n = 0$ implies $x_1x_2+x_2x_3+...+x_{n-1}x_n+x_nx_1 \le 0$ .

A polynomial $P(x)$ of degree $3$ has three distinct real roots. Find the number of real roots of the equation $P'(x)^2 -2P(x)P''(x) = 0$ .

Show that there exists a constant $a > 1$ such that, for any positive integers $m$ and $n$ , $\frac{m}{n} < \sqrt7$ implies that $$7-\frac{m^2}{n^2} \ge \frac{a}{n^2} .$$

The sequence $(a_n)$ is defined by $a_1 = 1$ and $a_{n+1} = \sqrt{a_n^2 +\frac{1}{a_n}}$ for $n \ge 1$ . Prove that there exists $a$ such that $\frac{1}{2} \le \frac{a_n}{n^a} \le 2$ for $n \ge 1$ .