The 27th British Mathematical Olympiad
1991年第27届英国奥林匹克数学竞赛 
 ABC is a triangle with ∠B = 90^{o} and M the midpoint of AB. Show that sin ACM ≤ 1/3.
 Twelve dwarfs live in a forest. Some pairs of dwarfs are friends. Each has a black hat and a white hat. Each dwarf consistently wears one of his hats. However, they agree that on the nth day of the New Year, the nth dwarf modulo 12 will visit each of his friends. (For example, the 2nd dwarf visits on days 2, 14, 26 and so on.) If he finds that a majority of his friends are wearing a different color of hat, then he will immediately change color. No other hat changes are made. Show that after a while no one changes hat.
 A triangle has sides a, b, c with sum 2. Show that a^{2} + b^{2} + c^{2} + 2abc < 2 .
 Let N be the smallest positive integer such that at least one of the numbers x, 2x, 3x, ... , Nx has a digit 2 for every real number x. Find N. Failing that, find upper and lower bounds and show that the upper bound does not exceed 20 .

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