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

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 \(r_1+r_2+r_3\).

- is 107
- is 15
- is 840
- cannot be determined from the given information

Circles

Triangle

Largest Integers

But try the problem first...

Answer: is 15.

Source

Suggested Reading

PRMO, 2019, Question 28

Geometry Vol I to IV by Hall and Stevens

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.

- https://www.cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA

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