Try this beautiful problem from the Pre-RMO II, 2019, Question 27 based on Shortest Distance.

## Shortest Distance – Pre-RMO II, Problem 27

A conical glass is in the form of a right circular cone. The slant height is 21 and the radius of the top rim of the glass is 14. An ant at the mid point of a slant line on the outside wall of the glass sees a honey drop diametrically opposite to it on the inside wall of the glass. If d the shortest distance it should crawl to reach the honey drop, what is the integer part of d?

- is 107
- is 36
- is 840
- cannot be determined from the given information

**Key Concepts**

Equation

Algebra

Integers

## Check the Answer

But try the problem first…

Answer: is 36.

PRMO II, 2019, Question 27

Higher Algebra by Hall and Knight

## Try with Hints

First hint

Rotate \(\Delta\)OAP by 120\(^\circ\) in anticlockwise then A will be at B, P will be at P’

Second Hint

or, \(\Delta\)OAP is congruent to \(\Delta\)OBP’

or, PB+PA=P’B+PB \(\geq\) P’P

Minimum PB+PA=P’P equality when P on the angle bisector of \(\angle\)AOB

or, P’P=2(21)sin60\(^\circ\)=21\(\sqrt{3}\)

Final Step

[min(PB+PA)]=[21\(\sqrt{3}\)]=36 (Answer)

## Other useful links

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

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