## Task B-3.2.

What are the lengths of the clock hands if their tips are $8.5 \mathrm{~cm}$ apart at 9 o'clock and $6.5 \mathrm{~cm}$ apart at 10 o'clock?

## Solution.

In the figure, the positions of the hands at 9 and 10 o'clock are shown. In the first case, the hands form an angle of $90^{\circ}$, and in the second case, an angle of $60^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_30_bbee2d45ecbf3fd03f8eg-13.jpg?height=432&width=837&top_left_y=612&top_left_x=524)

From the drawn triangles, we have

$$
\begin{aligned}
a^{2}+b^{2} & =\frac{289}{4} \\
a^{2}+b^{2}-2 a b \cos 60^{\circ} & =\frac{169}{4}
\end{aligned}
$$

The obtained system is equivalent to

$$
\begin{aligned}
a+b & =\frac{23}{2} \\
a b & =30
\end{aligned}
$$

The solution to this system is $a=4 \mathrm{~cm}, b=7.5 \mathrm{~cm}$, so the shorter hand is $4 \mathrm{~cm}$ long, and the longer hand is $7.5 \mathrm{~cm}$ long.

## Task B-3.3

Prove that the equation $\sin (2016 x)+\sin (x)=2$ has no solutions in the set of real numbers.
## First Solution.

Since the maximum value of the function $\sin x$ is 1, the given equality can only be satisfied if $\sin x=1$ and $\sin 2016 x=1$.

Then we have

$$
x=\frac{\pi}{4032}+\frac{l \pi}{1008}, l \in \mathbb{Z} \quad \text { and } \quad x=\frac{\pi}{2}+2 k \pi, k \in \mathbb{Z}
$$

From this, it follows that

$$
\begin{aligned}
\frac{\pi}{4032}+\frac{l \pi}{1008} & =\frac{\pi}{2}+2 k \pi, \text { so } \\
4 l-8064 k & =2015, \text { that is } \\
4(l-2016 k) & =2015
\end{aligned}
$$

The left side of the last equation is divisible by four, while the right side is not, so we conclude that the given equation has no solutions.

## Second Solution.

$\sin 2016 x=1$ and $\sin x=1$.

2 points

The equation $\sin x=1$ is satisfied by all real numbers of the form $x=\frac{\pi}{2}+2 k \pi, k \in \mathbb{Z}. \quad 1$ point

For all such $x$, $\sin 2016 x=\sin 1008 \pi=0$.

2 points

Therefore, for no real number $x$ for which $\sin x=1$, does $\sin 2016 x$ have the value 1, so the equation has no real solutions.

1 point