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January 29, 2018

Indian National Math Olympiad, INMO 2016 Problems

This post contains problems from Indian National Mathematics Olympiad, INMO 2016. Try them and share your solution in the comments.

INMO 2016, Problem 1

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

INMO 2016, Problem 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.

INMO 2016, Problem 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 \(k\ge 1\).

INMO 2016, Problem 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.

INMO 2016, Problem 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
 \(\frac {1} {r'}\)= \(\frac {1} {r}\) +\(\frac {1} {BD}\).

INMO 2016, Problem 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.

Also Visit: Math Olympiad Program.

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