Estonian Math Competitions

Answer: (a) Yes; (b) No.

Let $x$ be the smallest of the three numbers initially on the blackboard. We will find the numbers on the blackboard after having made 0, 1, 2, 3, 4, 5, 6, 7 and 8 moves, and their sums:

Number of moves	Numbers on the blackboard	Sum of numbers
0	$x$, $x+1$, $x+2$	$3x+3$
1	$x+1$, $x+2$, $x+3$	$4x+5$
2	$x+2$, $2x+2$, $3x+3$	$6x+7$
3	$2x+2$, $3x+3$, $4x+5$	$9x+10$
4	$3x+3$, $4x+5$, $6x+7$	$13x+15$
5	$4x+5$, $6x+7$, $9x+10$	$19x+22$
6	$6x+7$, $9x+10$, $13x+15$	$28x+32$
7	$9x+10$, $13x+15$, $19x+22$	$41x+47$
8	$13x+15$, $19x+22$, $28x+32$	$60x+69$

(a) The total sum of the numbers on the blackboard after 6 moves will be $28x+32$. Either by direct computation or by considerations modulo 4 and 7, we observe that the equation $28x+32=10^4$ has an integer solution. Therefore it is possible that after the 6th move, the sum of the numbers on the blackboard will be a power of 10.

(b) The total sum of the numbers on the blackboard after 8 moves will be $60x+69$. As $60x$ and $69$ end with 0 and 9, respectively, their sum will also end with the digit 9. But such a number cannot be equal to a power of 10.