HMMT 二月 2025 · 冲刺赛 · 第 10 题

HMMT February 2025 — Guts Round — Problem 10

[7] A square of side length 1 is dissected into two congruent pentagons. Compute the least upper bound of the perimeter of one of these pentagons. 2 4

解析

whose digits are all less than 3. This has a natural interpretation in base 3, and converting this interpretation to base 10 gives n 1 0 n 1 0 a . . . a a = a · 3 + · · · + a · 3 + a · 3 ≡ a · 10 + · · · + a · 10 + a · 10 = a . . . a a (mod 7) , n 1 0 n 1 0 n 1 0 n 1 0 3 which is a multiple of 7. This mapping is an injection because base conversion is an injection, and this mapping is also a surjection because for any multiple of 7 in base 10, we can convert it to base 3 and interpret this number as a base 10 decimal. (Reversing the above steps, we see the resulting decimal number is a multiple of 7.) Furthermore, for two positive multiples of 7 whose digits are less than 3, the larger one will have a larger base 3 representation, so this mapping is order-preserving. Thus, this mapping is a order-preserving bijection between multiples of 7 in base 10 with digits less than 3 and multiples of 7 in base 3. Therefore, the answer is the preimage of the 100th smallest positive multiple of 7, which is equal to 700 = 221221 . 10 3