HMMT 二月 2022 · ALGNT 赛 · 第 1 题
HMMT February 2022 — ALGNT Round — Problem 1
题目详情
- Positive integers a , b , and c are all powers of k for some positive integer k . It is known that the 2 equation ax − bx + c = 0 has exactly one real solution r , and this value r is less than 100. Compute the maximum possible value of r .
解析
- Positive integers a , b , and c are all powers of k for some positive integer k . It is known that the 2 equation ax − bx + c = 0 has exactly one real solution r , and this value r is less than 100. Compute the maximum possible value of r . Proposed by: Akash Das Answer: 64 2 Solution: Note that for there to be exactly one solution, the discriminant must be 0, so b − 4 ac = 0. b Thus, b is even, so k = 2. Since r = , then r is also a power of 2, and the largest power of 2 less 2 a 2 2 than 100 is 64. This is achieved by ( x − 64) = x − 128 x + 4096.