(a) Show that the task cannot be done if r is divisible by 2 or 3.
(b) Prove that the task is possible for r = 73.
(c) Can the task be done for r = 97?
RA + RC + RE ≥ p/2.
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A1. We are given a positive integer r and a rectangular board divided into 20 x 12 unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of the two squares is √r. The task is to find a sequence of moves leading between two adjacent corners of the board which lie on the long side.
(a) Show that the task cannot be done if r is divisible by 2 or 3.
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| A2. Let P be a point inside the triangle ABC such that ∠APB - ∠ACB = ∠APC - ∠ABC. Let D, E be the incenters of triangles APB, APC respectively. Show that AP, BD, CE meet at a point. |
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| A3. Let S be the set of non-negative integers. Find all functions f: S→S such that f(m + f(n)) = f(f(m)) + f(n) for all m, n. |
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| B1. The positive integers a, b are such that 15a + 16b and 16a - 15b are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? |
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B2. Let ABCDEF be a convex hexagon such that AB is parallel to DE, BC is parallel to EF, and CD is parallel to FA. Let RA, RC, RE denote the circumradii of triangles FAB, BCD, DEF respectively, and let p denote the perimeter of the hexagon. Prove that:
RA + RC + RE ≥ p/2. |
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| B3. Let p, q, n be three positive integers with p + q < n. Let x0, x1, ... , xn be integers such that x0 = xn = 0, and for each 1 ≤ i ≤ n, xi - xi-1 = p or -q. Show that there exist indices i < j with (i, j) not (0, n) such that xi = xj. |
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