Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2000 based on Arithmetic and geometric mean with Algebra.
Find the number of ordered pairs (x,y) of integers is it true that \(0 \lt y \lt 10^{6}\) and that the arithmetic mean of x and y is exactly 2 more than the geometric mean of x and y.
Algebra
Equations
Ordered pair
Answer: is 997.
AIME, 2000, Question 3
Elementary Algebra by Hall and Knight
given that \(\frac{x+y}{2}=2+({xy})^\frac{1}{2}\) then solving we have \(y^\frac{1}{2}\)-\(x^\frac{1}{2}\)=+2 and-2
given that \(y \gt x\) then \(y^\frac{1}{2}\)-\(x^\frac{1}{2}\)=+2 and here maximum integer value of \(y^\frac{1}{2}\)=\(10^{3}-1\)=999 whose corresponding \(x^\frac{1}{2}\)=997 and decreases upto \(y^\frac{1}{2}\)=3 whose corresponding \(x^\frac{1}{2}\)=1
then number of pairs (\(x^\frac{1}{2}\),\(y^\frac{1}{2}\))=number of pairs of (x,y)=997.
.
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 […]