## 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.

## Chaman Guwala

Posted at 13:38h, 10 DecemberVery nice

## Parag Dey

Posted at 10:02h, 13 MarchEasy

## Parag Dey

Posted at 10:05h, 13 MarchSolution

1) $A_1 =8$ ,$8|A_1$

Let $p(k) be true for all n=k\ge 1 \in\mathbb{N}$

$A_k=8m,m\in\mathbb{Z}$

And in same way

$A_{k+1}=8p$.

## Parag Dey

Posted at 10:28h, 13 March2) an easy question(vector solution is good)

This question is probably from 1988 Dutch competition.

3) question(unavailable)

4) simple induction

5)2^k =a, 2^m=b,2^l =c

Then ab+bc+ac= 2

1/a +1/b +1/c <= 3/2

This on expanding we get it

6) BJV (functional equations question) .easy

7) pigeonhole principle will suffice

9)|3x+2|=±(3x+2) and case take

10) (10^3)^654 +1=(10^n)^3+1

Clearly composite

11) α,β,£ as roots and apply viete

8) Arthur engel