INTRODUCING 5 - days-a-week problem solving session for Math Olympiad and ISI Entrance. Learn More 

August 1, 2020

Nearest value | PRMO 2018 | Question 14

Try this beautiful problem from the PRMO, 2018 based on Nearest value.

Nearest Value - PRMO 2018


If x=cos1cos2cos3.....cos89 and y=cos2cos6cos10....cos86, then what is 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


Algebra

Numbers

Multiples

Check the Answer


Answer: is 19.

PRMO, 2018, Question 14

Higher Algebra by Hall and Knight

Try with Hints


First hint

\(\frac{y}{x}\)=\(\frac{cos2cos6cos10.....cos86}{cos1cos2cos3....cos89}\)

=\(2^{44}\times\sqrt{2}\frac{cos2cos6cos10...cos86}{sin2sin4...sin88}\)

[ since cos\(\theta\)=sin(90-\(\theta\)) from cos 46 upto cos 89 and 2sin\(\theta\)cos\(\theta\)=sin2\(\theta\)]

Second Hint

=\(\frac{2^{\frac{89}{2}}sin4sin8sin12...sin88}{sin2sin4sin6...sin88}\)

[ since sin\(\theta\)=cos(90-\(\theta\))]

=\(\frac{2^{\frac{89}{2}}}{cos4cos8cos12..cos88}\)

[ since cos\(\theta\)=sin(90-\(\theta\))]

Final Step

=\(\frac{2^\frac{89}{2}}{\frac{1}{2}^{22}}\)

[since \(cos4cos8cos12...cos88\)

\(=(cos4cos56cos64)(cos8cos52cos68)(cos12cos48cos72)(cos16cos44cos76)(cos20cos40cos80)(cos24cos36cos84)(cos28cos32cos88)cos60\)

\(=(1/2)^{15}(cos12cos24cos36cos48cos60cos72cos84)\)

\(=(1/2)^{16}(cos12cos48cos72)(cos24cos36cos84)\)

\(=(1/2)^{20}(cos36cos72)\)

\(=(1/2)^{20}(cos36sin18)\)

\(=(1/2)^{22}(4sin18cos18cos36/cos18)\)

\(=(1/2)^{22}(sin72/cos18)\)

\(=(1/2)^{22}\)]

=\(2^\frac{133}{2}\)

\(\frac{2}{7}log_2{\frac{y}{x}}\)=\(\frac{2}{7} \times \frac{133}{2}\)=19.

Subscribe to Cheenta at Youtube


Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com
enter