| A1. Let f(x) = xn + 5xn-1 + 3, where n > 1 is an integer. Prove that f(x) cannot be expressed as the product of two non-constant polynomials with integer coefficients. |
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A2. Let D be a point inside the acute-angled triangle ABC such that ∠ADB = ∠ACB + 90o, and AC·BD = AD·BC.
(a) Calculate the ratio AB·CD/(AC·BD). (b) Prove that the tangents at C to the circumcircles of ACD and BCD are perpendicular. |
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| A3. On an infinite chessboard a game is played as follows. At the start n2 pieces are arranged in an n x n block of adjoining squares, one piece on each square. A move in the game is a jump in a horizontal or vertical direction over an adjacent occupied square to an unoccupied square immediately beyond. The piece which has been jumped over is removed. Find those values of n for which the game can end with only one piece remaining on the board. |
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B1. For three points P, Q, R in the plane define m(PQR) as the minimum length of the three altitudes of the triangle PQR (or zero if the points are collinear). Prove that for any points A, B, C, X:
m(ABC) ≤ m(ABX) + m(AXC) + m(XBC). |
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| B2. Does there exist a function f from the positive integers to the positive integers such that f(1) = 2, f(f(n)) = f(n) + n for all n, and f(n) < f(n+1) for all n? |
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B3. There are n > 1 lamps L0, L1, ... , Ln-1 in a circle. We use Ln+k to mean Lk. A lamp is at all times either on or off. Initially they are all on. Perform steps s0, s1, ... as follows: at step si, if Li-1 is lit, then switch Li from on to off or vice versa, otherwise do nothing. Show that:
(a) There is a positive integer M(n) such that after M(n) steps all the lamps are on again; (b) If n = 2k, then we can take M(n) = n2 - 1. (c) If n = 2k + 1, then we can take M(n) = n2 - n + 1. |
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