Find all polynomials $P(x)$ with real coefficients that satisfy  
$$ P(x \sqrt{2})=P\left(x+\sqrt{1-x^{2}}\right) $$  
for all real numbers $x$ with $|x| \leq 1$.
The answer is any polynomial of the form \( P(x) = f(U(x / \sqrt{2})) \), where \( f \in \mathbb{R}[x] \) and \( U \) is the unique polynomial satisfying \( U(\cos \theta) = \cos (8 \theta) \). Let \( Q(x) = P(x \sqrt{2}) \), then the condition reads 
\[ Q(\cos \theta) = Q\left(\frac{1}{\sqrt{2}}(\cos \theta + \sin \theta)\right) = Q\left(\cos \left(\theta - 45^{\circ}\right)\right) \quad \forall 0 \leq \theta \leq 180^{\circ}. \]
We call a polynomial good if it satisfies this functional equation. ## Lemma
The minimal (by degree) good nonconstant polynomial is \( U \).

\[ \begin{aligned} Q\left(\cos 136^{\circ}\right) & = Q\left(\cos 91^{\circ}\right) = Q\left(\cos 46^{\circ}\right) = Q\left(\cos 1^{\circ}\right) = Q\left(\cos -44^{\circ}\right) \\ & = Q\left(\cos 44^{\circ}\right) = Q\left(\cos 89^{\circ}\right) = Q\left(\cos 134^{\circ}\right) = Q\left(\cos 179^{\circ}\right) \end{aligned} \]
Hence \( Q \) is equal at eight distinct values (not nine since \( \cos -44^{\circ} = \cos 44^{\circ} \) is repeated), so \( \operatorname{deg} Q \geq 8 \) (unless \( Q \) is constant). Now, we claim \( Q(x) \equiv f(U(x)) \) for some \( f \in \mathbb{R}[x] \). Indeed, if \( Q \) is good, then by minimality the quotient \( Q \bmod U \) must be constant, so \( Q(x) = \widetilde{Q}(x) \cdot U(x) + c \) for some constant \( c \), but then \( \widetilde{Q}(x) \) is good too and we finish iteratively.