1. Let $$ABC$$ be triangle in which $$AB=AC$$. Suppose the orthocentre of the triangle lies on the incircle. Find the ratio $$AB/BC$$.
2. 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.
3. 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 $$k\ge 1$$.
4. 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.
5. 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 r is the inradius of triangle $$ABC$$, prove that
$$\frac {1} {r’}$$= $$\frac {1} {r}$$ +$$\frac {1} {BD}$$.
6. 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.

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