41. Find the smallest constant $c$, such that for all real numbers $x, y$, we have $1+(x+y)^{2} \leqslant c\left(1+x^{2}\right)$ ( $\left.1+y^{2}\right)$ )(2008 German Mathematical Olympiad problem)
41. Take $x=y=\frac{\sqrt{2}}{}$, substitute into the original inequality to get $3 \leqslant \frac{9}{4} c \Rightarrow c \geqslant \frac{4}{3}$. Now we prove that $c=\frac{4}{3}$ satisfies the condition.

We only need to prove: $1+(x+y)^{2} \leqslant \frac{4}{3}\left(1+x^{2}\right)\left(1+y^{2}\right)$. Simplifying and rearranging gives $(2 x y<1)^{2}+$ $(x-y)^{2} \geqslant 0$ - which is obvious. Therefore, $c_{\text {min }}=\frac{4}{3}$.