2. Calculate the value of the expression $\left(\sin \frac{38 \pi}{3}\right)^{5}: 4^{\log _{2} 3-\log _{16} 256}$.

## 17th Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools

Preliminary Competition, March 16, 2017

## Solutions to Problems and Scoring

(MARCH 16, 2017, 00:04)

A contestant who arrives at the solution by any correct method (even if the scoring guide does not anticipate it) receives all possible points.

A correct method is considered any procedure that:

- sensibly takes into account the wording of the problem,
- leads to the solution of the problem,
- is mathematically correct and complete.

If an intermediate or final result can be recognized, guessed, read from a diagram, or calculated mentally, the contestant generally receives all the points anticipated. However, if the solution is guessed (and cannot be arrived at without calculation), it is scored with 0 points.

A contestant who has only partially solved the problem, with correct procedures but no clear path to the final solution, cannot receive more than half of the possible points.

The mark ${ }^{\prime * \prime}$ next to the points means that the point or points can be awarded for a correct procedure, even if the calculation may be incorrect.

## First Year
2. We simplify and calculate the first part of the expression:

$$
\left(\sin \frac{38 \pi}{3}\right)^{5}=\sin ^{5}\left(12 \pi+\frac{2 \pi}{3}\right)=\sin ^{5}\left(\frac{2 \pi}{3}\right)=\left(\frac{\sqrt{3}}{2}\right)^{5}=\frac{9 \sqrt{3}}{32}
$$

We simplify and calculate the second part of the expression:

$$
4^{\log _{2} 3-\log _{16} 256}=4^{\log _{2} 3}: 4^{\log _{16} 256}=2^{\log _{2} 9}: 4^{2}=\frac{9}{16}
$$

We calculate the value of the given expression:

$$
\frac{9 \sqrt{3}}{32}: \frac{9}{16}=\frac{\sqrt{3}}{2}
$$

Considering the periodicity $\sin \left(\frac{38 \pi}{3}\right)=\sin \left(12 \pi+\frac{2 \pi}{3}\right)=\sin \left(\frac{2 \pi}{3}\right)$ ..... 1 point
Calculation $\sin \left(\frac{2 \pi}{3}\right)=\frac{\sqrt{3}}{2}$ 1 point
Exponentiation $\left(\frac{\sqrt{3}}{2}\right)^{5}=\frac{9 \sqrt{3}}{32}$ ..... 1 point
Considering the division of powers $4^{\log _{2} 3-\log _{16} 256}=4^{\log _{2} 3}: 4^{\log _{16} 256}$ ..... 1 point
Writing $4^{\log _{2} 3}=2^{\log _{2} 9}$ ..... 1 point
Calculation $2^{\log _{2} 9}=9$ ..... 1 point
Calculation $4^{\log _{16} 256}=4^{2}$ ..... 1 point
Calculation $\frac{9}{16}$ ..... 1 point
Multiplication of fractions $\frac{9 \sqrt{3}}{32} \cdot \frac{16}{9}$ ..... 1 point
Solution $\frac{\sqrt{3}}{2}$ ..... 1 point