SE-SKOLORNAS 1962

Find all polynomials $f(x)$ such that $f(2x) = f'(x) f''(x)$ .

$ABCD$ is a square side $1$ . $P$ and $Q$ lie on the side $AB$ and $R$ lies on the side $CD$ . What are the possible values for the circumradius of $PQR$ ?

Find all pairs $(m, n)$ of integers such that $n^2 - 3mn + m - n = 0$ .

Which of the following statements are true? (A) $X$ implies $Y$ , or $Y$ implies $X$ , where $X$ is the statement, the lines $L_1, L_2, L_3$ lie in a plane, and $Y$ is the statement, each pair of the lines $L_1, L_2, L_3$ intersect. (B) Every sufficiently large integer $n$ satisfies $n = a^4 + b^4$ for some integers a, b. (C) There are real numbers $a_1, a_2,... , a_n$ such that $a_1 \cos x + a_2 \cos 2x +... + a_n \cos nx > 0$ for all real $x$ .

Find the largest cube which can be placed inside a regular tetrahedron with side $1$ so that one of its faces lies on the base of the tetrahedron.