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Geometry and Trigonometry | PRMO 2019 | Problem 11

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

Try this beautiful problem from Pre RMO, 2019 based on Geometry and Trigonometry.

Geometry and Trigonometry – PRMO 2019

How many distinct triangles ABC are there, up to similarity, such that the magnitudes of angle A, B and C in degrees are positive integers and satisfy cosAcosB + sinAsinBsinkC=1 for some positive integer k, where kC does not exceed 360 degrees.

• is 13
• is 25
• is 6
• cannot be determined from the given information

Geometry

Trigonometry

Number Theory

Check the Answer

Answer: is 6.

PRMO, 2019

Plane Trigonometry by Loney

Try with Hints

First hint

Here cosAcosB+sinAsinBsinkC=1 then cosAcosB+sinAsinB+sinAsinBsinkC-sinAsinB=1 then sinAsinB(sinkC-1)=1-cos(A-B)

Second Hint

Then sinkC-1=0 and cos(A-B)=1 then kC=90 and A=B

Final Step

Then Number of factors of 90 is 90=(2)($3^{2}$)(5) then number of factors=(2)(3)(2)=12 for 6 factor A,B are integers

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