Algebra Arithmetic Geometry Math Olympiad PRMO

Shortest Distance | PRMO II 2019 | Question 27

Try this beautiful problem from the Pre-RMO, 2017, Question 23, based on Solving Equation. You may use sequential hints to solve the problem.

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?

Shortest Distance
  • is 107
  • is 36
  • is 840
  • cannot be determined from the given information

Key Concepts




Check the Answer

But try the problem first…

Answer: is 36.

Suggested Reading

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’

Shortest Distance figure

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)

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