12th Swedish 1972


1. Find the largest real number a such that x - 4y = 1, ax + 3y = 1 has an integer solution.
2. A rectangular grid of streets has m north-south streets and n east-west streets. For which m, n > 1 is it possible to start at an intersection and drive through each of the other intersections just once before returning to the start?
3. A steak temperature 5^o is put into an oven. After 15 minutes, it has temperature 45^o. After another 15 minutes it has temperature 77^o. The oven is at a constant temperature. The steak changes temperature at a rate proportional to the difference between its temperature and that of the oven. Find the oven temperature.
4. Put x = log₁₀2, y = log₁₀3. Then 15 < 16 implies 1 - x + y < 4x, so 1 + y < 5x. Derive similar inequalities from 80 < 81 and 243 < 250. Hence show that 0.47 < log₁₀3 < 0.482.
5. Show that ∫ ₀¹ (1/(1 + xⁿ)) dx > 1 - 1/n for all positive integers n.
6. a₁, a₂, a₃, ... and b₁, b₂, b₃, ... are sequences of positive integers. Show that we can find m < n such that a_m ≤ a_n and b_m ≤ b_n.

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