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Can These Cubics Have Integer Roots Only? (Indian Math Olympiad)
Here is a really nice algebra problem from the 2021 Indian National Math Olympiad.
Find all pairs of integers (a,b) such that each of the polynomials
x³ + ax + b and x³ + bx + a
has all the roots to be integers.
There is no cut and paste formula to solve something like this — otherwise it wouldn’t be an olympiad problem! A good way to start is try to simplify the problem and perhaps find some easy solutions. For example, let a = 0 and then look for a value of b that gives both polynomials integer solutions.
Well, the first polynomial becomes x³ + b. We want that to be zero, so that would mean x has to be the cube root of −b. That’s going to be an integer as long as b is a perfect cube.