Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1994 based on Right angled triangle.
In \(\Delta ABC\), \(\angle C\) is a right angle and the altitude from C meets AB at D. The lengths of the sides of \(\Delta ABC\) are integers, \(BD={29}^{3}\), and \(cosB=\frac{m}{n}\), where m, n are relatively prime positive integers, find m+n.
Integers
Right angled triangle
Pythagoras Theorem
Answer: is 450.
AIME I, 1994, Question 10
Geometry Vol I to IV by Hall and Stevens
\(\Delta ABC \sim \Delta CBD\)
\(\frac{BC}{AB}=\frac{29^{3}}{BC}\)
\(\Rightarrow {BC}^{2}=29^{3}AB\)
\(\Rightarrow 29^{2}|BC and 29|AB\)
\(\Rightarrow BC and AB are in form 29^{2}x, 29x^{2}\) where x is integer
\(by Pythagoras Theorem, AC^{2}+BC^{2}=AB^{2}\)
\(\Rightarrow (29^{2}x)^{2}+AC^{2}=(29x^{2})^{2}\)
\(\Rightarrow 29x|AC\)
taking y=\(\frac{AC}{29x}\) and dividing by (29x)^{2}\)
\(\Rightarrow 29^{2}=x^{2}-y^{2}=(x-y)(x+y)\)
where x,y are integers, the factors are \((1,29^{2}),(29,29)\)
\(y=\frac{AC}{29x}\) not equals 0 \(\Rightarrow x-y=1, x+y=29\)
\(\Rightarrow x=\frac{1+29^{2}}{2}\)
=421 then\(cosB=\frac{BC}{AB}=\frac{29^{2}x}{29x^{2}}\)=\(\frac{29}{421}\)
m+n=29+421=450.
AMC - AIME Boot Camp... for brilliant students.
Use our exclusive one-on-one plus group class system to prepare for Math Olympiad
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 […]