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Try this beautiful problem from the Pre-RMO, 2017, Question 26, based on Area of part of circle.

Let AB and CD be two parallel chords in a circle with radius 6 such that the centre O lies between these chords. Suppose AB=6 and CD=8. Suppose further that the area of the part of the circle lying between the chords AB and CD is \(\frac{m\pi+n}{k}\) where m.n.k are positive integers with gcd(m,n,k)=1. What is the value of m+n+k?

- is 107
- is 75
- is 840
- cannot be determined from the given information

Equation

Algebra

Integers

But try the problem first...

Answer: is 75.

Source

Suggested Reading

PRMO, 2017, Question 26

Higher Algebra by Hall and Knight

First hint

A=2[\(\frac{1}{2} \times 25 \times \theta\)]+\(\frac{1}{2} \times 3 \times 8\)+\(\frac{1}{2} \times 4 \times 6\)

where \(\theta=[\pi-(\theta_1+\theta_2)]=[\pi-(tan^{-1}\frac{4}{3}+tan^{-1}\frac{3}{4})]\)

Second Hint

or, \(\theta=\frac{\pi}{2}\)

or, A=24+\(\frac{25\pi}{2}\)

or, A=\(\frac{48+25\pi}{2}\)

Final Step

(m+n+k)=(48+2+25)=75

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

Content

[hide]

Try this beautiful problem from the Pre-RMO, 2017, Question 26, based on Area of part of circle.

Let AB and CD be two parallel chords in a circle with radius 6 such that the centre O lies between these chords. Suppose AB=6 and CD=8. Suppose further that the area of the part of the circle lying between the chords AB and CD is \(\frac{m\pi+n}{k}\) where m.n.k are positive integers with gcd(m,n,k)=1. What is the value of m+n+k?

- is 107
- is 75
- is 840
- cannot be determined from the given information

Equation

Algebra

Integers

But try the problem first...

Answer: is 75.

Source

Suggested Reading

PRMO, 2017, Question 26

Higher Algebra by Hall and Knight

First hint

A=2[\(\frac{1}{2} \times 25 \times \theta\)]+\(\frac{1}{2} \times 3 \times 8\)+\(\frac{1}{2} \times 4 \times 6\)

where \(\theta=[\pi-(\theta_1+\theta_2)]=[\pi-(tan^{-1}\frac{4}{3}+tan^{-1}\frac{3}{4})]\)

Second Hint

or, \(\theta=\frac{\pi}{2}\)

or, A=24+\(\frac{25\pi}{2}\)

or, A=\(\frac{48+25\pi}{2}\)

Final Step

(m+n+k)=(48+2+25)=75

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

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