American Math Olympiad Problem

American Math Olympiad Problem | AIMO

Find the largest value of n for which (n^3+100)/(n+10) is an Integer.

american math olympiad
math olympiad

[tharagleb]

Or you can divide the two polynomials and then you get a remainder of 900/(n+10) so n is 890.

[timc]

Perhaps also: let m = n + 10 ; then n^3 + 100 = (m - 10)^3 + 100 = m^3 - 30m^2 + 300m - 900, thence max m= 900, max n=890.

[jarikosonen]

Then there could be also smallest n when n=-910.
Basically it looks checking that "remainder" is one or zero, once if it can be extracted first.
Infinity probably is not an integer (then try n->floor(infinity)).

[kabirsethi]

Hello there. Solved this problem recently and my solution is similar to yours.
n³+1000-900|n+10
N+10=900
N=890
Hope this solution is correct.

[arijitchakraborty]

The question should be largest integer value of n, not any n. Bcoz summation and subtraction of fractions also give integer.

[pedrojose]

Good morning, from Brasil.
For this expression in particular is easy to to what you did. But for others may be not. Só it is better teach to find the rest of ter division p(x) ÷ q(x) that for polynomials with Integer coefficients is a Integer.