# Indian Regional Mathematical Olympiad 2013(problem 2)

If $$a$$,$$b$$ and $$c$$ are positive integers such that $$a | b^{5}$$ , $$b|c^{5}$$ and $$c|a^{5}$$.Prove that $abc|(a+b+c)^{31}$.

4 years, 5 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

in general if $$a | b^n$$ , $$b | c^n$$ , $$c | a^n$$ , $$abc | (a+b+c)^{n^2 + n + 1}$$

- 4 years, 5 months ago

Here a , b, c are becoming equal no?

- 4 years, 5 months ago

$$a,b,c$$ are not necessarily equal, if that's what you mean.

- 4 years, 5 months ago

You re correct ...at first glance it looked like that....:p I sometimes suffer from brain malfunctions like these.....

- 4 years, 5 months ago

This was my favorite question of the set. It looks so elegant, and is simple to approach if you are careful with the details.

Staff - 4 years, 5 months ago

Outline:

Consider each term of the expansion of $$(a+b+c)^{31},$$ which is of the form $$m a^{p} b^{q} c^{r},$$ where $$m, a, b, r \in \mathbb{Z^+}$$ and $$p+q+r=31.$$ If $$p,q,r>0,$$ $$abc \mid a^p b^q c^r.$$ You just need to handle the exceptional cases when atleast one of $$p,q,r$$ is zero, which is not hard. I'll post my full solution if necessary.

I got this problem when I sat for RMO last time. Heaven knows why I couldn't solve it in the examination hall... it was pretty easy.

- 4 years, 5 months ago

I solved this problem orally with you while coming down the staircase :P

- 4 years, 5 months ago

Yep, I remember! :P I couldn't solve it in the hall, though.

- 4 years, 5 months ago

This seems fairly easy for an Olympiad problem

- 3 years, 11 months ago