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Geometry Math Olympiad PRMO

Lines and Angles | PRMO 2019 | Question 7

Try this beautiful problem from the Pre-RMO, 2019 based on Lines and Angles. You may use sequential hints to solve the problem.

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

Lines and Angles – PRMO 2019


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

Key Concepts


Lines

Algebra

Angles

Check the Answer


But try the problem first…

Answer: is 51.

Source
Suggested Reading

PRMO, 2019, Question 7

Higher Algebra by Hall and Knight

Try with Hints


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.

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