1997 AMC 12 Problem 30
Attempt Problem 30 of the 1997 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1997 AMC 12 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
30.
For positive integers denote by the number of pairs of different adjacent digits in the binary (base two) representation of For example, and For how many positive integers less than or equal to does
Answer: C
Small hint:
A valid binary numeral consists of a block of s, then s, then s
Big hint:
Count by bit length through six bits, then handle the seven-bit cutoff separately
Solution:
A -bit numeral with exactly two changes has form with positive giving choices. For the total is Among seven-bit numbers at most the five forms beginning with one all work, and the only form beginning with at least two s is itself. Thus there are and C is correct.
Problem 30 in other years
- 1950 AMC 12
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