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 5o is put into an oven. After 15 minutes, it has temperature 45o. After another 15 minutes it has temperature 77o. 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 am ≤ an and bm ≤ bn.

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

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