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.
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.
Other useful links
- https://www.cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA
Related Program
Math Olympiad Program… taught by Olympians and researchers