35th Swedish 1995


1. The pages of a booklet are numbered from 1 to 2n. A single sheet (of 2 pages) is removed. The numbers of the remaining pages sum to 963. How many pages did the booklet have originally and which pages were removed?
2. X left home between 4 and 5 and returned between 5 and 6 to find that the hands of his clock had changed places. What time did he leave?
3. a, b, x, y are positive reals such that a + b + x + y < 2. If a + b² = x + y² and a² + b = x² + y, show that a = x and b = y.
4. a ≥ b ≥ c > 0 are reals such that abc = 1 and a + b + c > 1/a + 1/b + 1/c. Show that a > 1 > b.
5. A, B, C, D are points on a circle radius r in that order. AB = BC = CD = s < r and AD = s+r. Find the angles of the quadrilateral.
6. What is the largest number of binary sequences of length 10 such that each pair are different in at least 6 places?

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