10.1. Find all four-digit numbers $\overline{x y z t}$, where all digits $x, y, z, t$ are distinct and not equal to 0, such that the sum of all four-digit numbers obtained from $\overline{x y z t}$ by all possible permutations of the digits is 10 times the number $\overline{x x x x}$.
Answer. The number 9123 and all numbers obtained from it by permuting the last three digits, a total of 6 answers.

Solution. The number of four-digit numbers obtained from $\overline{x y z t}$ by all possible permutations of the digits is 24, in each of which each of the digits $x, y, z, t$ appears exactly 6 times in each place. Therefore, the sum of all such numbers is $6(x+y+z+t)(1000+100+10+1)=6666(x+y+z+t)$, which, by condition, equals $11110 x$. From this, $3(y+z+t)=2 x$. According to the condition, all digits $x, y, z, t$ are distinct and not equal to 0, so the left side of the equation is no less than $3(1+2+3)=18$, from which we get the only possibility $x=9$ and $y, z, t$ are any permutation of the digits $1,2,3$.

Evaluation Criteria. If a complete correct answer with verification is provided: 1 point. Correct finding of the sum on the left side of the equation: 3 points.