SMO Senior · 4. Inequalities, extreme values and the integer part
Four tools for SMO bounds: completing the square and the discriminant, absolute values as distances, the integer part as a staircase, and the AM–GM and Cauchy–Schwarz inequalities, finishing with a Round 2 proof.
Find maxima and minima by completing the square and by the discriminant condition $\Delta\ge0$.
Optimise sums of absolute values using distances, breakpoints and slopes.
Solve equations and counting problems with the integer part by writing $x=n+r$.
Apply AM–GM and Cauchy–Schwarz with a correct, attainable equality case.
Write a complete telescoping proof of an inequality about sums.