SE-SKOLORNAS 1973

$\log_8 2 = 0.2525$ in base $8$ (to $4$ places of decimals). Find $\log_8 4$ in base $8$ (to $4$ places of decimals).

The Fibonacci sequence $f_1,f_2,f_3,\dots$ is defined by $f_1=f_2=1$ , $f_{n+2}=f_{n+1}+f_n$ . Find all $n$ such that $f_n = n^2$ .

$ABC$ is a triangle with $\angle A = 90^\circ$ , $\angle B = 60^\circ$ . The points $A_1$ , $B_1$ , $C_1$ on $BC$ , $CA$ , $AB$ respectively are such that $A_1B_1C_1$ is equilateral and the perpendiculars (to $BC$ at $A_1$ , to $CA$ at $B_1$ and to $AB$ at $C_1$ ) meet at a point $P$ inside the triangle. Find the ratios $PA_1:PB_1:PC_1$ .

$p$ is a prime. Find all relatively prime positive integers $m$ , $n$ such that $$\frac{m}{n}+\frac{1}{p^2}=\frac{m+p}{n+p}$$

$f(x)$ is a polynomial of degree $2n$ . Show that all polynomials $p(x)$ , $q(x)$ of degree at most $n$ such that $f(x)q(x)-p(x)$ has the form $$\sum\limits_{2n<k\leq 3n} (a^k + x^k)$$ have the same $p(x)/q(x)$ .

$f(x)$ is a real valued function defined for $x \geq 0$ such that $f(0) = 0$ , $f(x+1)=f(x)+\sqrt{x}$ for all $x$ , and $$f(x) < \frac{1}{2}f\left(x - \frac{1}{2}\right)+\frac{1}{2}f\left(x + \frac{1}{2}\right) \text{for all} x \geq \frac{1}{2}$$ Show that $f\left(\frac{1}{2}\right)$ is uniquely determined.