SE-SKOLORNAS 1975

$A$ is the point $(1,0)$ , $L$ is the line $y = kx$ (where $k > 0$ ). For which points $P(t,0)$ can we find a point $Q$ on $L$ such that $AQ$ and $QP$ are perpendicular?

Is there a positive integer $n$ such that the fractional part of $$\left(3+\sqrt{5}\right)^n >0.99 ?$$

Show that $$a^n + b^n + c^n \geq ab^{n-1} + bc^{n-1} + ca^{n-1}$$ for real $a,b,c \geq 0$ and $n$ a positive integer.

$P_1$ , $P_2$ , $P_3$ , $Q_1$ , $Q_2$ , $Q_3$ are distinct points in the plane. The distances $P_1Q_1$ , $P_2Q_2$ , $P_3Q_3$ are equal. $P_1P_2$ and $Q_2Q_1$ are parallel (not antiparallel), similarly $P_1P_3$ and $Q_3Q_1$ , and $P_2P_3$ and $Q_3Q_2$ . Show that $P_1Q_1$ , $P_2Q_2$ and $P_3Q_3$ intersect in a point.

Show that $n$ divides $2^n + 1$ for infinitely many positive integers $n$ .

$f(x)$ is defined for $0 \leq x \leq 1$ and has a continuous derivative satisfying $|f'(x)| \leq C|f(x)|$ for some positive constant $C$ . Show that if $f(0) = 0$ , then $f(x)=0$ for the entire interval.