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Curriculum · Olympiad Topics · Inequalities

Olympiad Inequalities

AM-GM, Cauchy-Schwarz, and the strategy of smoothing and normalization.

Objectives

  • Apply AM-GM and Cauchy-Schwarz
  • Normalize cleverly
  • Find equality cases first

The full interactive edition of this lesson ships in v1.5. The topic is live everywhere else — the problems below feed your Inequalities mastery now.

Train the topic

●●●●○What is the minimum value of x+4xx + \tfrac{4}{x} for x>0x > 0?AMC 12 style●●●●○For positive reals a,ba, b, what is the minimum value of (a+b)(1a+1b)(a + b)\left(\tfrac{1}{a} + \tfrac{1}{b}\right)?Olympiad training