Maths Olympiad Exam

Let ABC be a triangle. Let D, E, F be points respectively on the segments BC, CA, AB such that AD, BE, CF concur at the point K. Suppose BD/DC = BF/FA and ADB = AFC. Prove that ABE = CAD

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Maths Olympiad Other Question
Find all real numbers a for which the equation
x^2 + (a + 2)x + 1 = 3|x| has exactly three distinct real solutions in x. Let (a1, a2, a3, . . . , a2011) be a permutation (that is a rearrangement) of the numbers
1, 2, 3, . . . , 2011. Show that there exist two numbers j, k such that 1  j < k  2011 and aj + j = ak + k .