B3. Find all real numbers $x$ that satisfy the equation

$$
\cos \left(\pi \sin ^{2} x\right)+\sin \left(\pi \cos ^{2} x\right)=1
$$

## Problems for 4th Grade

Time for solving: 120 minutes. In section A, a correct answer will be awarded two points, while a wrong answer will result in the deduction of one point. Enter your answers for section A in the left table.
![](https://cdn.mathpix.com/cropped/2024_06_07_66bdf7a61d9bfcb32ca4g-08.jpg?height=166&width=802&top_left_y=544&top_left_x=640)
B3. Let $y=\pi \sin ^{2} x$. Since $\sin \left(\pi \cos ^{2} x\right)=\sin \left(\pi\left(1-\sin ^{2} x\right)\right)=\sin \left(\pi-\pi \sin ^{2} x\right)=$ $\sin \left(\pi \sin ^{2} x\right)$, we get the equation $\cos y+\sin y=1$. Since $\cos y+\sin y=\sin y+\sin \left(\frac{\pi}{2}-y\right)=$ $2 \sin \frac{\pi}{4} \cos \frac{2 y-\frac{\pi}{2}}{2}=1$, it follows that $\cos \left(y-\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}$, which gives $y=\frac{\pi}{4} \pm \frac{\pi}{4}+2 k \pi$. Since $y=\frac{\pi}{4} \pm \frac{\pi}{4}+2 k \pi=\pi \sin ^{2} x \in[0, \pi]$, it can only be $k=0$. Therefore, $y=0$ or $y=\frac{\pi}{2}$. If $y=0$, then $x=n \pi$, where $n$ is an integer. For $y=\frac{\pi}{2}$, $\sin ^{2} x=\frac{1}{2}$, which gives $\sin x= \pm \frac{\sqrt{2}}{2}$ or $x=\frac{\pi}{4}+n \frac{\pi}{2}$, where $n$ is an integer.

Introduction of a new unknown and transformation of the equation $\mathbf{v} \cos y+\sin y=1 \ldots \ldots \ldots .1$ point

![](https://cdn.mathpix.com/cropped/2024_06_07_66bdf7a61d9bfcb32ca4g-15.jpg?height=68&width=1576&top_left_y=1662&top_left_x=274)

Conclusion that $k=0$. ..........................................................................................................

Writing both families of solutions for $y=0$ and $y=\frac{\pi}{2} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$ (1+1) point (If the contestant only writes all solutions, award 2 points.)

## Solutions for 4th Year

In section A, a correct answer will be awarded 2 points, while an incorrect answer will result in a deduction of 1 point. To avoid a potential negative final score, each contestant is given 3 initial points.

| A1 | A2 | A3 |
| :---: | :---: | :---: |
| D | E | D |

Justifications: