Geometry and Trigonometry | PRMO 2019 | Problem 11

Content

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

Key Concepts

Geometry

Trigonometry

Number Theory

Check the Answer

But try the problem first...

Answer: is 6.

Source

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

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.