Q1) A, B, C are three points on a circle. The distance of C from the tangents at A and B to the circle are a and b respectively. If the distance of C from the chord AB is c, show that c is the geometric mean of a and b.
Q2)Find all integer solutions to the equation x³+(x+4)²=y².
Q3)Two right angled triangles are such that the incircle of one triangle is equal in size to the circumcircle of the other. If P is the area of the first triangle and Q is the area of the second triangle, then show that P/Q≥3+2√2.
Q4) (a)Find the maximum value k for which one can choose k integers from 1,2,.....,2n so that none of the chosen integers is divisible by any other chosen integer.
(b) F(x) is a polynomial of degree 2016 such that all the coefficients are non negative and non exceed F(0).Show that the coefficient of x^2017 in (F(x))² is at most F(1)²/2.
Q5) (a) n>=3 and a1, a2, a3,....., an are different positive integers. Given that, except the first and the last, each one is a harmonic mean of its immediate neighbors. Show that none of the given integers is less than n-1. (b)Show that the shortest side of a cyclic quadrilateral with circumradius 1 is at most √2.
Q6)C1,C2,C3 are circles with radii 1,2,3 respectively, touching each other as shown in Figure. Two circles can be drawn touching all these circles. Find the radii of these two circles.