## Problem 1

Let \( a, b, c \in \mathbb{N} \) satisfy the relation:

\[
a + 2b + 3c + 21 = 4(\sqrt{a+1} - 1) + 6(\sqrt{2b+1} - 1) + 10(\sqrt{3c+1} - 1)
\]

a) Determine the numbers \( a, b \), and \( c \).

b) Prove that \( \overline{abc}^{2016} + \overline{acb}^{2016} \) is not a perfect square.
## Solution:

a) $(\sqrt{a+1}-2)^{2}+(\sqrt{2 b+1}-3)^{2}+(\sqrt{3 c+1}-5)^{2}=0$ $2 p$

$a=3, b=4, c=8$
b) $U\left(348^{2016}+384^{2016}\right)=U\left(8^{2016}+4^{2016}\right)=U(6+6)=2 \neq p p$