Indian National Mathematics Olympiad 2000 Problems

Indian National Mathematics Olympiad 2000 Problems

1.  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.

2.  Find the integer solutions to a + b = 1 - c, a³ + b³ = 1 - c².

3.  a, b, c are non-zero reals, and x is real and satisfies [bx + c(1-x)]/a = [cx + a(1-x)]/b = [ax + b(1-x)]/b. Show that a = b = c.

4.  In a convex quadrilateral PQRS, PQ = RS, SP = (√3 + 1)QR, and ∠RSP - ∠SQP = 30^o. Show that ∠PQR - ∠QRS = 90^o.

5.  a, b, c are reals such that 0 ≤ c ≤ b ≤ a ≤ 1. Show that if α is a root of z³ + az² + bz + c = 0, then |α| ≤ 1.

6.  Let f(n) be the number of incongruent triangles with integral sides and perimeter n, eg f(3) = 1, f(4) = 0, f(7) = 2. Show that f(1999) > f(1996) and f(2000) = f(1997).