The simplest example of power mean inequality is the arithmetic mean - geometric mean inequality. It says the following:
Arithmetic Mean is greater than or equal to Geometric Mean
Caution: All numbers must be non-negative.
Suppose $ a_1, a_2, ... , a_n $ be non-negative numbers. Then the two means are defined as follows:
Arithmetic Mean: $ \displaystyle{\frac{a_1 + a_2 + ... + a_n}{n}} $
Geometric Mean: $ \displaystyle{(a_1 \cdot a_2 \cdots a_n)^{\frac{1}{n}}} $
This problem is from Regional Math Olympiad, India.
Suppose a, b, c, d are positive numbers. Then show that $$ \displaystyle { \frac{a}{b} + \frac {b}{c} + \frac{c}{d} + \frac{d}{a} \geq 4 } $$
Regional Math Olympiad, India
Inequality (AM-GM)
6 out of 10
Secrets in Inequalities.
Notice that product of the fractions is 1. Can you use this fact to compute the geometric mean of the fractions?
The geometric mean of the fractions is $$ \displaystyle{(\frac{a}{b} \cdot \frac{b}{c} \cdot \frac{c}{d} \cdot \frac{d}{a} )^{\frac{1}{4} }}$$
This is equal to $ 1^{\frac{1}{4}} = 1$
Hence the geometric mean of the fractions is 1!
Can you now finish the problem using Arithmetic Mean - Geometric Mean inequality?
Lets use the arithmetic mean - geometric mean inequality on the fractions.
$$ \displaystyle { \frac{\frac{a}{b} + \frac {b}{c} + \frac{c}{d} + \frac{d}{a}}{4} \\ \geq (\frac{a}{b} \cdot \frac{b}{c} \cdot \frac{c}{d} \cdot \frac{d}{a} )^{\frac{1}{4} } } $$
But the geometric mean is 1 (right hand side is 1). Hence by cross multiplying we have the final result.
Publishing original scientific research in a peer-reviewed journal is an achievement many scholars work toward for years. Achieving this feat early in one's academic journey requires exceptional intellectual rigor, creativity, and persistence. Aaditya Punatar and Muskan Sadana have reached this remarkable milestone, successfully publishing their collaborative research paper in the prestigious Journal of Emerging Investigators […]
Her project on quantum-accelerated Steiner trees for 15-minute cities earned a place among the finalists at the world's largest pre-college STEM competition. Prishaa Shrimali of Marymount School of New York, was selected as a Grand Finalist at the 2026 Regeneron International Science and Engineering Fair (ISEF) in Phoenix, Arizona. Her project was Optimizing Urban Accessibility: […]
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]