This post contains problems from Indian National Mathematics Olympiad, INMO 2017. Try them and share your solution in the comments.

INMO 2017, Problem 1

In the given figure, \(ABCD\) is a square sheet of paper. It is folded along \(E F\) such that \(A\) goes to a point \(A'\) different from Band \(C\), on the side \(BC\) and \(D\) goes to \(D'\) . The line \(A' D'\) cuts \(C D\) in \(G\). Show that the inradius of the triangle \(GC A'\) is the sum of the inradii of the triangles \(GD'F\) and \(A' BE\).

INMO 2017, Problem 2

Suppose \(n \ge 0\) is an integer and all the roots of \( x^3 + ax + 4 -(2 \times {2016^n})\) = 0 are integers. Find all possible values of \(\alpha\).

INMO 2017, Problem 3

Find the number of triples \((x, a, b)\) where \(x\) is a real number and a, b belong to the set \({{1,2,3,4,5,6,7,8,9}}\) such that
\(x^2 - a \{x\} + b = 0\)
where \(\{x\}\) denotes the fractional part of the real number \(x\).    (For example \(\{1.1\}\) = 0.1 =\(\{-0.9\}\) ).

INMO 2017, Problem 4

Let \(ABCDE\) be a convex pentagon in which \({\angle A} ={\angle B} ={\angle C} ={\angle D}\) =\(120^{\circ}\) and side lengths are five consecutive integers in some order. Find all possible values of  \(AB + BC + CD\).

INMO 2017, Problem 5

Let \(ABC\) be a triangle with \(\angle A =90^{\circ}\) and \(AB < AC\). Let \(AD\) be the altitude from \(A\) on to BC. Let \(P, Q\) and I denote respectively the incentres of triangles \(ABD, ACD\) and \(ABC\). Prove that \(AI\) is prependicular to \(PQ\) and \(AI = PQ\).

INMO 2017, Problem 6

Let $n \geq 1$ be an integer and consider the sum
$$
x=\sum_{k \geq 0}\left(\begin{array}{c}
n \\
2 k
\end{array}\right) 2^{n-2 k} 3^{k}=\left(\begin{array}{l}
n \\
0
\end{array}\right) 2^{n}+\left(\begin{array}{l}
n \\
2
\end{array}\right) 2^{n-2} \cdot 3+\left(\begin{array}{l}
n \\
4
\end{array}\right) 2^{n-4} \cdot 3^{2}+\cdots
$$
Show that $2 x-1,2 x, 2 x+1$ form the sides of a triangle whose area and inradius are also integers..

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.

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This post contains problems from Indian National Mathematics Olympiad, INMO 2018. Try them and share your solution in the comments.

INMO 2018, Problem 1

Let ABC be a non-equilateral triangle with integer sides. Let D and E be respectively the mid-points BC and C A, let G be the centroid of triangle ABC. Suppose D, C, E, G are concyclic. Find the least possible perimeter of triangle ABC.

INMO 2018, Problem 2

For any natural number n, consider a \(1 \times n\) rectangular board made up of n unit squares. This is covered by three types of tiles; \(1\times 1\) red tile, \(1\times 1\) green tile and \(1\times 2\) blue domino. (For example, we can have 5 types of tiling when n = 2; red-red; red-green; green-red; green-green; and  blue.) Let \(t_n\) denote the number of ways of covering \(1\times n\) rectangular board by these three types of tiles. Prove that \(t_n\) divides \(t_{2n+1}\).

INMO 2018, Problem 3

Let \(\Gamma_1\) and \(\Gamma_2\) be two circles with respective centres \(O_1\) and \(O_2\) intersecting in two distinct points A and B such that \({\angle O_1}A{O_2}\) is an obtuse angle. Let the circumcircle of triangle \({O_1}A{O_2}\) intersect \(\Gamma_1\) \(T_2\) respectively in points \(C{(\not= A)}\) and \(D{(\not= A)}\). Let the line C B intersect \(\Gamma_2\) in E; let the line D B intersect \(\Gamma_1\) in F . Prove that the points C, D, E, F are concyclic.

INMO 2018, Problem 4

Find all polynomials with real coefficients P(x) such that \(P{(x^2+x+1)}\) divides \(P{(x^3-1)}\).

INMO 2018, Problem 5

There are \(n\ge 3\) girls in a class sitting around a circular table, each having some apples with her, Every time the teacher notices a girl having more apples than both of her neighnors combined, the teacher takes away one apple from that girl and gives one apple each to her neighbors. Prove that this process stops after a finite numberof steps. (Assume that the teacher has an abundant supply of apples.)

INMO 2018, Problem 6

Let N denote the set of all natural numbers and let \(f : N\rightarrow N\) be a function such that
(a) \(f{(mn)} = f {(m)} f{(n)}\) for all m,n in N ;
(b) m+n divides \(f {(m)} + f {(n)} \) for all m, n in N
Prove that there exists an odd natural number \(k\) such that \(f {(n)} = n^k\) for all n in N.

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