
08 Mar Indian National Mathematical Olympiad 2018 Question Paper
The Indian National Mathematical Olympiad (INMO) is a high school mathematical competition held annually in India since 1989. It is the second tier in the Indian team selection procedure for the International Mathematical Olympiad and is conducted by the Homi Bhabha Centre for Science Education (HBCSE) under the aegis of the National Board of Higher Mathematics (NBHM). The exam has six questions which have to be solved in four hours. The 33rd INMO was held on 21st January.
- Let
be a non-equilateral triangle with integer sides. Let
and
be the respectively the midpoints of
and
; let
be the centroid of triangle
. Suppose
,
,
,
are concyclic. Find the least possible perimeter of triangle
.
-
For any natural number
consider a
rectangular board made up of
unit squares. This is covered by three types of tiles:
red tile,
green tile and
blue domino. Let
denote the number of ways of covering
rectangular board by these three types of tiles. Prove that
divides
.
-
Let
and
be two circles with respective centres
and
intersecting in two distinct points
and
such that
is an obtuse angle. Let the circumcircle of triangle
intersect
and
respectively in points
and
. Let the line
intersect
in
; let the line
intersect
in
. Prove that the points
,
,
,
are concyclic.
-
Find all polynomials with real coefficients
such that
divides
.
-
There are
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 neighbors 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 number of steps. (Assume that the teacher has an abundant supply of apples.)
-
Let
denote the set of all natural numbers and let
be a function such that:
for all
,
in
; and
divides
for all
,
in
. Prove that there exists an odd natural number
such that
for all
in
.