 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

But try the problem first…

Source

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)