3rd Swedish 1963


1. How many positive integers have square less than 10⁷?
2. 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?
3. What is the remainder on dividing 1234⁵⁶⁷ + 89¹⁰¹¹ by 12?
4. 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.
5. 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 (157½^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?
6. The real-valued function f(x) is defined on the reals. It satisfies \|f(x)\| ≤ A, \|f ''(x)\| ≤ B for some positive A, B (and all x). Show that \|f '(x)\| ≤ C, for some fixed C, which depends only on A and B. What is the smallest possible value of C?

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.