Try this problem from ISI-MSQMS 2018 which involves the concept of Inequality.
(a) Prove that if $x>0, y>0$ and $x+y=1,$ then $\left(1+\frac{1}{x}\right)\left(1+\frac{1}{y}\right) \geq 9$
Algebra
Inequality
Numbers
Answer: $xy \leq \frac{1}{4}$
ISI - MSQMS - B, 2018, Problem 2A
"INEQUALITIES: AN APPROACH THROUGH PROBLEMS BY BJ VENKATACHALA"
We have to show that ,
$(1+\frac{1}{x})(1+\frac{1}{y}) \geq 9$
i.e $1+ \frac{1}{x} + \frac{1}{y} +\frac{1}{xy} \geq 9$
Since $x+y =1$
Therefore the above equation becomes $\frac{2}{xy} \geq 8$
ie $xy \leq \frac{1}{4}$
Now with this reduced form of the equation why don't you give it a try yourself,I am sure you can do it.
Applying AM $\geq$ GM on $x,y$
So you are just one step away from solving your problem,go on.............
Therefore, $\frac{x+y}{2} \geq (xy)^\frac{1}{2}$
$\Rightarrow \frac{1}{2} \geq (xy)^\frac{1}{2}$
Squaring both sides we get, $xy \leq \frac{1}{4}$
Hence the result follows.
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 […]