

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 northsouth streets and n eastwest 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 = log_{10}2, y = log_{10}3. 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 < log_{10}3 < 0.482.


5. Show that ∫ _{0}^{1} (1/(1 + x^{n})) dx > 1  1/n for all positive integers n.


6. a_{1}, a_{2}, a_{3}, ... and b_{1}, b_{2}, b_{3}, ... are sequences of positive integers. Show that we can find m < n such that a_{m} ≤ a_{n} and b_{m} ≤ b_{n}.

