MathDB

4

Part of 2022 MMATHS

Problems(2)

2022 MMATHS Team Online p4 - even 3 digit numbrt in base b >=10

Source:

10/1/2023
How many ways are there to choose three digits A,B,CA,B,C with 1≤A≤91 \le A \le 9 and 0≤B,C≤90 \le B,C \le 9 such that ABC‾b\overline{ABC}_b is even for all choices of base bb with b≥10b \ge 10?
number theoryMMATHS
2022 MMATHS Individual p4 - 2-digit favorite number

Source:

10/1/2023
Cat and Claire are having a conversation about Cat’s favorite number. Cat says, “My favorite number is a two-digit perfect square!” Claire asks, “If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I’d know for certain what it is?” Cat says, “Yes! Moreover, if I told you a number and identified it as the sum of the digits of my favorite number, or if I told you a number and identified it as the positive difference of the digits of my favorite number, you wouldn’t know my favorite number.” Claire says, “Now I know your favorite number!” What is Cat’s favorite number?
number theoryMMATHS