Base Systems (AIME / USAJMO / AMO)

Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.

Proof track: treat each problem as a full olympiad writeup on paper. Here you only enter the final answer (integer or short expression).

Base Systems: core idea

Digit DP style.

**(Proof track.)** Give a complete justification. Find $(234)_5+(143)_5$ in base $5$ .

Hint: Expand the numeral as a polynomial in its base and enforce valid-digit constraints. Answer: 432.

No proof submission required — answers only.

1. **(Proof track.)** Give a complete justification. Find base $b$ if $(132)_b=56$ .
2. **(Proof track.)** Give a complete justification. Write $255$ in base $2$ .
3. **(Proof track.)** Give a complete justification. For what base $b$ is $(22)_b$ a square equal to $64$ ?
4. **(Proof track.)** Give a complete justification. Compute $(1000)_7-(1)_7$ in base $7$ .
5. **(Proof track.)** Give a complete justification. Find the sum of digits of $2026$ in base $8$ .
6. **(Proof track.)** Give a complete justification. Find $(345)_6\pmod5$ .
7. **(Proof track.)** Give a complete justification. If $(ab)_9=71$ , find $a+b$ .
8. **(Proof track.)** Give a complete justification. Find largest base in which $(123)_b<200$ .
9. **(Proof track.)** Give a complete justification. Evaluate $(10101)_3$ in base ten.