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# 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

PRMO, 2018, Question 14

Higher Algebra by Hall and Knight

## Try with Hints

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

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

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

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

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

=$$\frac{2^{\frac{89}{2}}}{cos4cos8cos12..cos88}$$

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

=$$\frac{2^\frac{89}{2}}{\frac{1}{2}^{22}}$$

[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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