1. Do there exist four distinct natural numbers $a, b, c, d$ such that the numbers $a^{2}+2 c d+b^{2}$ and $c^{2}+2 a b+d^{2}$ are squares of natural numbers?
Solution. We want to obtain two squares of sums. To use the formula for abbreviated multiplication, it is enough to make $a b=c d$. We can choose any quartet of numbers, for example, $a b=2 \cdot 12=24=3 \cdot 8=c d$.

The equality $2^{2}+2 \cdot 3 \cdot 8+12^{2}=3^{2}+2 \cdot 2 \cdot 12+8^{2}$, which is the same as $2^{2}+2 \cdot 2 \cdot 12+12^{2}=3^{2}+2 \cdot 3 \cdot 8+8^{2}$, satisfies the condition of the problem.

Answer. Yes, such numbers exist, and an example can be provided.