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# Nearest value | PRMO 2018 | Question 14

Try this beautiful problem from the Pre-RMO, 2018 based on the Nearest value. You may use sequential hints to solve the problem.

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

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.

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