SE-SKOLORNAS 1965

The feet of the altitudes in the triangle $ABC$ are $A', B', C'$ . Find the angles of $A'B'C'$ in terms of the angles $A, B, C$ . Show that the largest angle in $A'B'C'$ is at least as big as the largest angle in $ABC$ . When is it equal?

Find all positive integers m, n such that $m^3 - n^3 = 999$ .

Show that for every real $x \ge \frac12$ there is an integer $n$ such that $|x - n^2| \le \sqrt{x-\frac{1}{4}}$ .

Find constants $A > B$ such that $\frac{f\left( \frac{1}{1+2x}\right) }{f(x)}$ is independent of $x$ , where $f(x) = \frac{1 + Ax}{1 + Bx}$ for all real $x \ne - \frac{1}{B}$ . Put $a_0 = 1$ , $a_{n+1} = \frac{1}{1 + 2a_n}$ . Find an expression for an by considering $f(a_0), f(a_1), ...$ .

Let $S$ be the set of all real polynomials $f(x) = ax^3 + bx^2 + cx + d$ such that $|f(x)| \le 1$ for all $-1 \le x \le 1$ . Show that the set of possible $|a|$ for $f$ in $S$ is bounded above and find the smallest upper bound.