Let \(ABC\) be triangle in which \(AB=AC\). Suppose the orthocentre of the triangle lies on the incircle. Find the ratio \(AB/BC\).

For positive real numbers \(a, b, c,\) which of the following statements necessarily implics \(a= b= c:\) (I) \(a (b^3+c^3)\) = \(b {(c^3 +a3)}\) = \(c {(a^3 +b^3)}\),
(II) \(a {(a^3 + b^3)}\) = \(b {(b^3 + c^3)}\) = \(c {(c^3 + a^3)}\) ? Justify your answer.

Let \(N\) denote the set of all natural numbers. Define a function \(T : N \rightarrow {N}\) by \(T(2k) = k and T (2k +1) = 2k + 2\). We write \(T^2 {(n)}\) = \( T (T(n))\) and in general \(T^k{(n)}\) = \(T ^{k-1} {(T(n))}\) for any k>1.(i) Show that for each \(n \(\in\) N\), there exists \(k\) such thst \( {T^k} {(n)} =1\).(ii) For \(k\in\) N , let \( c_k \) denote the number of elements in the set { n : \(T^k {(n)}\) = 1}. Prove that \( c_{k +2} = c_{k +1} +{c_k}\), for \(k\ge 1\).

Suppose 2016 points of the circumference of a circle are coloured red and the remaining points are coloured blue. Given any natural number \( n \ge 3\), prove that there is a regular \(n\)-sided polygon all of whose vertices are blue.

Let \(ABC\) be a right-angled triangle with \({\angle B} = 90^{\circ}\) . Let \(D\) be a point on \(AC\) such that the inradii of the triangles \(ABD\) and \(C B D\) are equal. If this common value is r’ and if ris the inradius of triangle \(ABC\), prove that \(\frac {1} {r’}\)= \(\frac {1} {r}\) +\(\frac {1} {BD}\).

Consider a nonconstant arithmetic progression \({a_1}, {a_2},….{ a_n}\),….. Suppose there exist relatively prime positive integers \(p > 1\;{\textbf{and}}\; q > 1\)1 such that \({a^2_1}\), \(a^2_{p+1}\) and \(a^2_{q+1}\) are also the terms of the same arithmetic progression. Prove that the terms of the arithmetic progression are all integers.