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Algebra Arithmetic Math Olympiad PRMO

Trigonometry Problem | PRMO 2016 | Question 14

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

Try this beautiful problem from the PRMO, 2016 based on trigonometry and nearest integer.

Trigonometry and nearest integer – PRMO 2016


If x=cos1cos2cos3…..cos89 and y=cos2cos6cos10….cos86 then find the integer nearest to \(\frac{2}{7}log_{2}(\frac{y}{x})\)

  • is 107
  • is 19
  • is 840
  • cannot be determined from the given information

Key Concepts


Nearest Integer

Algebra

Trigonometry

Check the Answer


But try the problem first…

Answer: is 19.

Source
Suggested Reading

PRMO, 2016, Question 14

Plane Trigonometry by Loney

Try with Hints


First hint

\(\frac{y}{x}\)=\(\frac{cos2cos6cos10…..cos86}{cos1cos2cos3…..cos89}\)=\(2^{44}{2}^\frac{1}{2}\frac{cos2cos6cos10……cos86}{sin2sin4…….sin88}\)=\(2^\frac{89}{2}\frac{sin4sin8…….sin88}{sin2sin4……sin88}\)=\(\frac{2^\frac{89}{2}}{cos4cos8cos12…..cos88}\)=\(\frac{(2)^\frac{89}{2}}{(\frac{1}{2})^{22}}\) for given condition by question =\(2^\frac{133}{2}\)

Second Hint

\(\frac{2}{7}log_{2}(\frac{y}{x})\)=\(\frac{2}{7}log_{2}{2}^\frac{133}{2}\)

Final Step

=\(\frac{2}{7} \times \frac{133}{2}\)=19.

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