# Difference between revisions of "2005 IMO Shortlist Problems/N3"

(→Problem: double comma fixed) |
|||

Line 2: | Line 2: | ||

(''Mongolia'') | (''Mongolia'') | ||

− | Let <math> | + | Let <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math>, <math>e</math>, and <math>f </math> be positive integers. Suppose that the sum <math>S = a+b+c+d+e+f </math> divides both <math>abc + def </math> and <math>ab+bc+ca - de-ef-fd </math>. Prove that <math>S </math> is composite. |

''This was also Problem 1 of the 2nd 2006 German TST, and a problem at the 2006 Indian IMO Training Camp.'' | ''This was also Problem 1 of the 2nd 2006 German TST, and a problem at the 2006 Indian IMO Training Camp.'' |

## Latest revision as of 09:13, 29 August 2011

## Problem

(*Mongolia*)
Let , , , , , and be positive integers. Suppose that the sum divides both and . Prove that is composite.

*This was also Problem 1 of the 2nd 2006 German TST, and a problem at the 2006 Indian IMO Training Camp.*

## Solution

For all integers we have

,

since each coefficient of the first two polynomials is congruent to the corresponding coefficient of the second two polynomials, mod . Now, suppose is prime. Since

,

one of is divisible by , say . Since , this means . But since are positive integers, we then have

,

a contradiction. ∎

*Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.*