
11 Sep Assam Academy of Mathematics Olympiad 2017 Questions
The Assam Academy of Mathematics (AAM) organizes a mathematics Olympiad for school students every year. This year the Olympiad was held on 10th September, 2017 in three categories. We post below the questions for Category III (for students from classes 9, 10 and 11).
- Prove that
is a multiple of
for every positive integer
.
- Prove that the diagonals of a quadrilateral are perpendicular if and only if the sum of the squares of one pair of opposite sides equal that of the other.
- Prove that for every integer
,
.
- Prove that
is divisible by
for every natural number
.
- Suppose
are natural numbers. Prove that
.
- Let
be a function such that (a)
whenever
, (b)
for all
and (c)
is a prime number whenever
is a prime number. Find
.
is a set of positive integers. None of the elements of
is divisible by
. Prove that there exists a subset of
such that the sum of its elements is divisible by
.
- Consider a row of
seats. A child sits on each. Each child may move at most by one seat. Find the number of ways that they can rearrange.
- Let
be a polynomial over
. If
with integers
, then prove that
has no integral zeroes.
- Solve the equation
.
- Show that the number
with
zeroes is composite.
- In the polynomial
, one zero is the sum of the other two zeroes. Find the relation between
and
.Questions 2 and 3 are of 8 marks each, question 4 is of 9 marks, questions 10, 11 and 12 are of 5 marks each, and the rest of the questions are of 10 marks each.

Managing Editor of the English Section, Gonit Sora and Research Associate, Cardiff University, UK.