Indian National Mathematics Olympiad 2004 Problems1.  ABCD is a convex quadrilateral. K, L, M, N are the midpoints of the sides AB, BC, CD, DA. BD bisects KM at Q. QA = QB = QC = QD, and LK/LM = CD/CB. Prove that ABCD is a square.

Indian National Mathematics Olympiad 2003 Problems1.  ABC is acute-angled. P is an interior point. The line BP meets AC at E, and the line CP meets AB at F. AP meets EF at D. K is the foot of the perpendicular from D to BC. Show that KD bisects ∠EKF.

Indian National Mathematics Olympiad 2002 Problems1.  ABCDEF is a convex hexagon. Consider the following statements. (1) AB is parallel to DE, (2) BC is parallel to EF, (3) CD is parallel to FA, (4) AE = BD, (5) BF = CE, (6) CA = DF. Show that if any five of these statements are true then the hexagon is cyclic.

Indian National Mathematics Olympiad 2001 Problems1.  ABC is a triangle which is not right-angled. P is a point in the plane. A', B', C' are the reflections of P in BC, CA, AB. Show that [incomplete]. 2.  Show that a2

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Indian National Mathematics Olympiad 2000 Problems1.  The incircle of ABC touches BC, CA, AB at K, L, M respectively. The line through A parallel to LK meets MK at P, and the line through A parallel to MK meets LK at Q. Show that the line PQ bisects AB and bisects AC.

Indian National Mathematics Olympiad 1999 Problems1.  ABC is an acute-angled triangle. AD is an altitude, BE a median, and CF an angle bisector. CF meets AD at M, and DE at N. FM = 2, MN = 1, NC = 3. Find the perimeter of ABC.

Indian National Mathematics Olympiad 1998 Problems1.  C is a circle with center O. AB is a chord not passing through O. M is the midpoint of AB. C' is the circle diameter OM. T is a point on C'. The tangent to C' at T meets C at P. Show that PA2 + PB2 = 4 PT2.

Indian National Mathematics Olympiad 1997 Problems1.  ABCD is a parallelogram. A line through C does not pass through the interior of ABCD and meets the lines AB, AD at E, F respectively. Show that AC2 + CE·CF = AB·AE + AD·AF.

Indian National Mathematics Olympiad 1996 Problems1.  Given any positive integer n, show that there are distinct positive integers a, b such that a + k divides b + k for k = 1, 2, ... , n. If a, b are positive integers such that a + k divides b + k for all positive integers k, show that a = b.

Indian National Mathematics Olympiad 1995 Problems1.  ABC is an acute-angled triangle with ∠A = 30o. H is the orthocenter and M is the midpoint of BC. T is a point on HM such that HM = MT. Show that AT = 2 BC. 2. 

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