Let O be the centre of a parallelogram ABCD and P be any point in the plane. Let M,N be the midpoints of AP,BP, respectively and Q be the intersection of MC and ND. Prove that O,P and Q are collinear.
2012 Singapore Junior Math Olympiad
2nd Round
Does there exist an integer A such that each of the ten digits 0,1,...,9 appears exactly once as a digit in exactly one of the numbers A,A2,A3 ?
In △ABC, the external bisectors of ∠A and ∠B meet at a point D. Prove that the circumcentre of △ABD and the points C,D lie on the same straight line.
Determine the values of the positive integer n for which the following system of equations has a solution in positive integers x1,x2,...,,xn. Find all solutions for each such n. {x1+x2+...+xn=161x1+1x2+...+1xn=1
Suppose S={a1,a2,...,a15} is a set of 15 distinct positive integers chosen from 2,3,...,2012 such that every two of them are coprime. Prove that S contains a prime number. (Note: Two positive integers m,n are coprime if their only common factor is 1)