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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 $r_1+r_2+r_3$.

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

### Key Concepts

Circles

Triangle

Largest Integers

But try the problem first…

Source

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.