Try this beautiful problem from AMC 10A, 2018 based on Number theory.
Let's try this problem number 10 from AMC 10A, 2018 based on Number Theory.
Suppose that the real number $x$ satisfies $\sqrt {49-x^2}$ - $\sqrt {25-x^2}$ = $3$.
What is the value of $\sqrt {49-x^2}$ + $\sqrt {25-x^2}$?
Number Theory
Real number
Square root
Answer: 8
AMC 10 A - 2018 - Problem No.10
Mathematics can be fun by Perelman
As a first hint we can start from here :
In order to get rid of the square roots, we multiply by the conjugate. Its value is the solution. The $x^2$ terms cancel out.
$\left(\sqrt{49-x^2}+\sqrt{25-x^2}\right)\left(\sqrt{49-x^2}\right)-\left(\sqrt{25-x^2}\right)$
$=49-x^2-25+x^2=24$
Given that $\left.\sqrt{49-x^2}\right)-\left(\sqrt{25-x^2}\right)=3$
$\sqrt{49-x^2}+\sqrt{25-x^2}=\frac{24}{3} \quad=8$
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 […]
