Problem 1. Determine the distinct non-zero natural numbers $a, b, c, d$, which simultaneously have the properties:

(1) Exactly three of the four numbers are prime;

(2) $a^{2}+b^{2}+c^{2}+d^{2}=2018$.
Solution. $2018=\mathcal{M} 4+2$, two numbers are even and two are odd, one number is 2

![](https://cdn.mathpix.com/cropped/2024_06_07_5985a68f469286ee400bg-1.jpg?height=57&width=1524&top_left_y=801&top_left_x=295)

$2018=\mathcal{M} 3+2$, two numbers are multiples of 3 and two $\mathcal{M} 3 \pm 1$, one number is 3

....................................................................................... 1 point.

If $a<b<c<d$, then $a=2, b=3, c^{2}+d^{2}=2005, c=6 k$ or $d=6 k$

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$k \leq 7$, case analysis .......................................................... 2 points.

$k=3, a=2, b=3, c=18, d=41$ and their circular permutations $\ldots \ldots \ldots . .1$ point.