(因应版主要求 贴个第二天的英文版题目做纪录) Problem 4. Let n > 0 be an integer. We are given a balance and n weights of weight 2^0, 2^1, ... , 2^{n-1}. We are to place each of the n weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed. Determine the number of ways in which this can be done. Problem 5. Let f be a function from the set of integers to the set of positive integers. Suppose that, for any two integers m and n, the difference f(m) - f(n) is divisible by f(m-n). Prove that, for all integers m and n with f(m) <= f(n), the number f(n) is divisible by f(m). Problem 6. Let ABC be an acute triangle with circumcircle Γ. Let l be a tangent line to Γ, and let l_a, l_b, l_c be the lines obtained by reflecting l in the lines BC, CA and AB, respectively. Show that the circumcircle of the triangle determined by the lines l_a, l_b and l_c is tangent to the circle Γ. -- --

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