Try this beautiful problem from the PRMO, 2016 based on trigonometry and nearest integer.
If x=cos1cos2cos3.....cos89 and y=cos2cos6cos10....cos86 then find the integer nearest to \(\frac{2}{7}log_{2}(\frac{y}{x})\)
Nearest Integer
Algebra
Trigonometry
But try the problem first...
Answer: is 19.
PRMO, 2016, Question 14
Plane Trigonometry by Loney
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.
Math Olympiad Program... taught by Olympians and researchers