Try this beautiful problem from the Pre-RMO, 2019 based on Diameter of a circle.
A village has a circular wall around it, and the wall has four gates pointing north, southeast and west. A tree stands outside the village, 16 m north of the north gate, and it can be just seen appearing on the horizon from a point 48 m east of the south gate. Find the diameter in meters of the wall that surrounds the village.
Pythagoras Theorem
Equations
Integer
Answer: is 48.
PRMO, 2019, Question 25
Geometry Vol I to IV by Hall and Stevens
Let radius =r
or,\(AB=\sqrt{AO^{2}-OB^{2}}=\sqrt{(16+r)^{2}-r^{2}}\)
\(=\sqrt{256+32r}\)
or, \(AD^{2}+DC^{2}=CA^{2}\)
or, \(48^{2}+(2r+16)^{2}\)
\(=(48+\sqrt{256+32r})^{2}\)
or, \(r^{2}+8r=24\sqrt{256+32r}\)
or, \(r(r+8)=24(4\sqrt{2})(\sqrt{r+8})\)
or, r=24
or, 2r=diameter=48.
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 […]
In 2025, 8 students from Cheenta Academy cracked the prestigious Regional Math Olympiad. In this post, we will share some of their success stories and learning strategies. The Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) are two most important mathematics contests in India.These two contests are for the students who are […]
Cheenta Academy proudly celebrates the success of 27 current and former students who qualified for the Indian Olympiad Qualifier in Mathematics (IOQM) 2025, advancing to the next stage — RMO. This accomplishment highlights their perseverance and Cheenta’s ongoing mission to nurture mathematical excellence and research-oriented learning.