Inequality Functional Equations

Olympiad-Style Functional Equations and Inequalities

A focused collection of challenging functional equations and inequalities for curious, ambitious problem-solvers.

Written especially for strong students in Years 10–12 preparing for mathematical olympiads, this booklet is also for anyone who enjoys unfamiliar problems and wants to develop flexible, rigorous problem-solving habits, not just collect answers.

The problems vary in domain, technique, and level of rigidity. Some admit more than one solution; others allow a family of possibilities, or an exceptional value, until the final hypotheses are put to work.

Topics covered

The collection explores functional equations shaped by inequalities across the positive reals, the real numbers, the positive rationals, and the positive integers. Each domain calls for different tools: order and limits on the positive reals; symmetry and regularity on the real line; prime exponents on the positive rationals; and counting or divisibility on the integers.

The booklet is arranged in five themed parts:

• Additive and multiplicative rigidity
• Reciprocal and harmonic transforms
• Iteration, fixed points, and growth
• Mean and norm squeezes
• Real-line and polynomial inequalities

Problems are grouped by theme, with the full solutions collected at the end. For each problem, try to solve it before consulting the hint. Pay close attention to the domain and to what the inequalities imply for special choices of inputs. Use hints as nudges, then compare your proof with the worked solution and consider any alternative approaches provided.

Who it’s for

This booklet is intended for students preparing for olympiad-style competitions, including national mathematical olympiads and the International Mathematical Olympiad (IMO), as well as readers who want to go beyond routine school exercises. It assumes a willingness to work through demanding problems and write careful proofs.