## Task 1.

Determine all polynomials $P$ with real coefficients such that the expression

$$
P(x+3 y)+P(3 x-y)
$$

has the same value for all real numbers $x, y$ for which $x^{2}+y^{2}=1$.
## Solution.

Notice that the mapping $(x, y) \rightarrow(x+3 y, 3 x-y)$ is a bijection between the sets

$$
\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2}=1\right\} \leftrightarrow\left\{(x, y) \in \mathbb{R}: x^{2}+y^{2}=10\right\}
$$

The inverse function is $(a, b) \rightarrow\left(\frac{a+3 b}{10}, \frac{3 a-b}{10}\right)$.

The problem now reduces to finding all polynomials such that the expression

$$
P(x)+P(y)
$$

has the same value for all real numbers $x$ and $y$ for which $x^{2}+y^{2}=10$.

Let $P$ be such a polynomial. Denote by $A$ the set $\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2}=10\right\}$. Notice that $(x, y) \in A$ if and only if $(x,-y) \in A$, and it holds that

$$
P(x)+P(y)=P(x)+P(-y), \forall(x, y) \in A
$$

or equivalently

$$
P(y)-P(-y)=0, \forall y \in[-\sqrt{10}, \sqrt{10}]
$$

Since the left side of the previous inequality is a polynomial and has infinitely many roots, we conclude that $P$ is an even function. Now we know that there exists a polynomial $R$ such that $P(x)=$ $R\left(x^{2}\right)$. Let $C \in \mathbb{R}$ be such that $P(x)+P(y)=C, \forall(x, y) \in A$. Then it holds that

$$
C=P(x)+P(y)=R\left(x^{2}\right)+R\left(y^{2}\right)=R\left(x^{2}\right)+R\left(10-x^{2}\right), \forall(x, y) \in A
$$

Similarly as before, we conclude that $C=R(x)+R(10-x), \forall x \in \mathbb{R}$, or $C=R(5+x)+$ $R(5-x), \forall x \in \mathbb{R}$. Define the polynomial $T(x)=R(5+x)-\frac{C}{2}$. From the previous identity, we directly obtain that $T$ is an odd function, so there exists a polynomial $Q$ such that $T(x)=x Q\left(x^{2}\right), \forall x \in \mathbb{R}$. We get the following necessary condition on $P$:

$$
P(x)=R\left(x^{2}\right)=T\left(x^{2}-5\right)+\frac{C}{2}=\left(x^{2}-5\right) Q\left(\left(x^{2}-5\right)^{2}\right)+\frac{C}{2}
$$

By direct verification, we see that all polynomials of this form satisfy the conditions of the problem.