SE-SKOLORNAS 1963

How many positive integers have square less than $10^7$ ?

The squares of a chessboard have side $4$ . What is the circumference of the largest circle that can be drawn entirely on the black squares of the board?

What is the remainder on dividing $1234^{567} + 89^{1011}$ by $12$ ?

Given the real number $k$ , find all differentiable real-valued functions $f(x)$ defined on the reals such that $f(x+y) = f(x) + f(y) + f(kxy)$ for all $x, y$ .

A road has constant width. It is made up of finitely many straight segments joined by corners, where the inner corner is a point and the outer side is a circular arc. The direction of the straight sections is always between $NE$ ( $45^o$ ) and $SSE$ ( $157 1/2^o$ ). A person wishes to walk along the side of the road from point $A$ to point $B$ on the same side. He may only cross the street perpendicularly. What is the shortest route?

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The real-valued function $f(x)$ is defined on the reals. It satisfies $|f(x)| \le A$ , $|f''(x)| \le B$ for some positive $A, B$ (and all $x$ ). Show that $|f'(x)| \le C$ , for some fixed $C$ , which depends only on $A$ and $B$ . What is the smallest possible value of $C$ ?