Estonian Math Competitions

Answer: (a) Yes; (b) Yes.

a. One possibility is $2 + 0 \cdot 2120 : 2 - 2 = 0$.

b. If the teacher allowed using parentheses, Juku could write the expression $(2 - 0 + 2) : 1 \cdot 2022$ whose value is $8088$. We show that it is impossible to achieve so big value without using parentheses. For that, we show that the value of multiplication must be less than $5000$. Note that placing four operators between eight digits leaves there exactly three pairs of digits lying next to each other without an operator between them. Thus if both factors are numerals then the larger factor contains at most $4$ digits and if the larger factor contains $3$ digits then the smaller factor contains at most $2$ digits. As $2$ is the largest digit in use, the product cannot exceed $2222 \cdot 2$ in the first case and $222 \cdot 22$ in the second case. Both bounds are less than $5000$. If the first factor is the ratio and the second is a numeral then the value cannot be larger, because the ratio cannot be larger than the dividend. Analogously, we see that an addend that does not involve multiplication can contain at most $4$ digits and must be less than $3000$, because division or subtraction inside the addend cannot increase it. Consequently, the value of any expression that can be written without parentheses is less than $8000$. Thus Juku can obtain larger values when parentheses are allowed.

c. Placing operators between digits can be done in $7 \cdot 6 \cdot 5 \cdot 4$ ways in total. The number of different values Juku can obtain cannot exceed this number. As $7 \cdot 6 \cdot 5 \cdot 4 = 840 < 1000$, there exists a positive integer less than $1000$ that cannot be obtained.