Olympiad problems

2015 India Regional MathematicaI Olympiad, 2

Let $P(x) = x^2 + ax + b$ be a quadratic polynomial with real coefficients. Suppose there are real numbers $ s \neq t$ such that $P(s) = t$ and $P(t) = s$. Prove that $b-st$ is a root of $x^2 + ax + b - st$.

2015 India Regional MathematicaI Olympiad, 2

Tags: RMO 2015

2.Let $P(x) = x^2 + ax + b$ be a quadratic polynomial where a, b are real numbers. Suppose $P(-1)^2$ , $P(0)^2$, $P(1)^2$ is an Arithmetic progression of positive integers. Prove that a, b are integers.

2015 India Regional MathematicaI Olympiad, 5

Tags: RMO 2015

Two circles X and Y in the plane intersect at two distinct points A and B such that the centre of Y lies on X. Let points C and D be on X and Y respectively, so that C, B and D are collinear. Let point E on Y be such that DE is parallel to AC. Show that AE = AB.

2015 India Regional MathematicaI Olympiad, 5

Tags: RMO 2015 , geometry , incenter , circumcircle , ratio

Let ABC be a right triangle with $\angle B = 90^{\circ}$.Let E and F be respectively the midpoints of AB and AC.Suppose the incentre I of ABC lies on the circumcircle of triangle AEF,find the ratio BC/AB.

2015 India Regional MathematicaI Olympiad, 6

Tags: RMO 2015

Find all real numbers $a$ such that $4 < a < 5$ and $a(a-3\{a\})$ is an integer. ({x} represents the fractional part of x)

2015 India Regional MathematicaI Olympiad, 4

Tags: RMO 2015

Suppose 32 objects are placed along a circle at equal distances. In how many ways can 3 objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?