Mathematics Olympiads: Good Problems Appeal
8:50 PM
Muhammad Yusuf
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Question 1 : Question 2 of the 2008 AMO contest
Let ABC be an acute-angled triangle, and let D be the point on AB (extended if necessary) such that AB and CD are perpendicular. Further, let tA and tB be the tangents to the circumcircle of ABC through A and B, respectively, and let E and F be the points on tA and tB, respectively, such that CE is perpendicular to tA and CF is perpendicular to tB. Prove that CD/CE = CF/CD
Question 2 : Question 7 of the 2008 AMO contest, proposed by Bolis Basit
Let A1A2A3 and B1B2B3 be triangles. If p = A1A2 + A2A3 + A3A1 + B1B2 + B2B3 + B3B1 and q = A1B1 + A1B2 + A1B3 + A2B1 + A2B2 + A2B3 + A3B1 + A3B2 + A3B3, prove that 3p ≤ 4q.
Question 3 : Question 7 of the 2009 AMO contest, proposed by Angelo Di Pasquale
Let I be the incentre of a triangle ABC in which AC ≠ BC. Let Ø be the circle passing through A, I and B. Suppose Ø intersects the line AC at A and X and intersects the line BC at B and Y . Show that AX = BY
Question 4 : proposed by Joseph Kupka, Melbourne:
Prove that
Question 5 : proposed by Ian Wanless, Melbourne
Let a, b, c, d be integers satisfying 0 < a < b < c < d < 2010. Prove that there exists an integer e satisfying: (i) 0 < e < 2010, (ii) e is a divisor of 2010, and (iii) no two of a, b, c, d give the same remainder when divided by e.
Question 6 : this problem submitted by Ivan Guo
Larry and Rob are two robots travelling in a car from Argovia to Zillis. Both robots have control over the steering and steer according to the following algorithm. Larry makes a 900 left turn after every l kilometres and Rob makes a 900 right turn after every r kilometres driving from the start, where l and r are relatively prime positive integers. In the event of both turns occurring simultaneously, the car will keep going without changing direction. Assume that the ground is flat and that the car can move in any direction. Let the car start from Argovia facing towards Zillis. For which choices of the pair (l, r) is the car guaranteed to reach Zillis, regardless of how far it is from Argovia?
Question 7 : the AMOC Senior Problems Committee by Ross Atkins
Let n be a positive integer and let a1, . . . , ak (k ≥ 2) be distinct integers in the set {1, . . . , n} such that n divides ai(ai+1 − 1) for i = 1, . . . , k − 1. Prove that n does not divide ak(a1 − 1).
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