SE-SKOLORNAS 2002

$268$ numbers are written around a circle. The $17$ th number is $3$ , the $83$ rd is $4$ and the $144$ th is $9$ . The sum of every $20$ consecutive numbers is $72$ . Find the $210$ th number.

$A, B, C$ can walk at $5$ km/hr. They have a car that can accomodate any two of them whch travels at $50$ km/hr. Can they reach a point $62$ km away in less than $3$ hrs?

$C$ is the circle center $(0,1)$ , radius $1$ . $P$ is the parabola $y = ax^2$ . They meet at $(0, 0)$ . For what values of $a$ do they meet at another point or points?

For which integers $n \ge 8$ is $n^{\frac{1}{n-7}}$ an integer?

The reals $a, b$ satisfy $$\begin{cases} a^3 - 3a^2 + 5a - 17 = 0 b^3 - 3b^2 + 5b + 11 = 0 .\end{cases}$$ Find $a+b$ .

A tetrahedron has five edges of length $3$ and circumradius $2$ . What is the length of the sixth edge?