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 $abc|(a+b+c)^{31}$ .

#NumberTheory #CosinesGroup #Goldbach'sConjurersGroup #TorqueGroup

Easy Math Editor

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Math	Appears 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}`	$\boxed{123}$

Comments

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

Sagnik Saha - 7 years, 2 months ago

Here a , b, c are becoming equal no?

Eddie The Head - 7 years, 2 months ago

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

Daniel Liu - 7 years, 2 months ago

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

Eddie The Head - 7 years, 2 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.

Calvin Lin Staff - 7 years, 2 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.

Sreejato Bhattacharya - 7 years, 2 months ago

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

Sagnik Saha - 7 years, 2 months ago

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

Sreejato Bhattacharya - 7 years, 2 months ago

This seems fairly easy for an Olympiad problem

Aritra Jana - 6 years, 8 months ago

Math	Appears 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}`	$\boxed{123}$