APMO · 2. Polynomials, equations and sequences
Symmetric sums, Vieta's formulas, auxiliary polynomials, floor equations and integer sequences: how to find the hidden structure and write complete proofs, through five fully solved APMO problems.
Evaluate symmetric sums by pairing and telescoping.
Use Vieta's formulas and sums of squares to pin down the roots of a polynomial.
Build an auxiliary polynomial from conditions at many points and determine it.
Solve equations with the floor function by fixing the integer part.
Squeeze integer sequences between consecutive squares and prove 'largest constant' statements in both directions.