INTRODUCING 5 - days-a-week problem solving session for Math Olympiad and ISI Entrance. Learn More 

April 22, 2020

Triangle and Integer | PRMO 2019 | Question 28

Try this beautiful problem from the Pre-RMO, 2019 based on Circles,Triangle and largest integer.

Circles, Triangle and largest Integer - PRMO 2019

Let ABC be a triangle with sides 51, 52, 53. Let D denote the incircle of triangle ABC. Draw tangents to D which are parallel to the sides ABC. let \(r_1\). \(r_2\), \(r_3\) be the inradii of the three corner triangles so formed, find the largest integer that does not exceed \(r_1+r_2+r_3\).

Triangle and Integer
  • is 107
  • is 15
  • is 840
  • cannot be determined from the given information

Key Concepts



Largest Integers

Check the Answer

Answer: is 15.

PRMO, 2019, Question 28

Geometry Vol I to IV by Hall and Stevens

Try with Hints

First hint

Let PQ be one of tangents parallel to BC and meet sides AB and AC at P and Q let PQ=x and BC=51

Second Hint

triangle ABC similar with triangle APQ then\(\frac{x}{a}=\frac{r_1}{r}=\frac{s-a}{s}\) which is in same way for \(\frac{y}{b}\) and \(\frac{z}{c}\) then \(\frac{r_1+r_2+r_3}{r}\)=\(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\)=3-2=1

Final Step

then \(r_1+r_2+r_3\)=r and r by given condition of question =\((\frac{s(s-a)(s-b)(s-c)}{s})^\frac{1}{2}\)=\((\frac{78(78-51)(78-52)(78-53)}{78})^\frac{1}{2}\)=15.

Subscribe to Cheenta at Youtube

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.