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Try this beautiful problem from the PRMO, 2019 based on Lines and Angles.

On a clock, there are two instants between 12 noon and 1 pm when the hour hand and the minute hand are at right angles. the difference in minutes between these two instants written as a+\(\frac{b}{c}\) where a b c are positive integers with \(b\lt c\) and \(\frac{b}{c} \) in the reduced form. find a+b+c.

- is 107
- is 51
- is 840
- cannot be determined from the given information

Lines

Algebra

Angles

But try the problem first...

Answer: is 51.

Source

Suggested Reading

PRMO, 2019, Question 7

Higher Algebra by Hall and Knight

First hint

Minute hand turns 6 (in degrees) in 1 min and hour hand turns half (in degree) in 1 min

Second Hint

x min after 12 is 6x -\(\frac{x}{2}\)=90 or 270 (in degrees) then x =16\(\frac{4}{11}\) or 49\(\frac{1}{11}\)

Final Step

Difference 32\(\frac{8}{11}\) then a+b+c=32+8+11=51.

- https://www.cheenta.com/smallest-perimeter-of-triangle-aime-2015-question-11/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s

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