1995 AMC 12 Problem 27
Attempt Problem 27 of the 1995 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 1995 AMC 12 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
27.
Consider the triangular array of numbers with along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows through are shown.
Let denote the sum of the numbers in row What is the remainder when is divided by
Answer: E
Small hint:
Relate a row’s sum to the previous row’s sum by counting how often each old entry contributes
Big hint:
Solve the resulting recurrence, then compute the power of modulo
Solution:
Every previous entry contributes to two entries of the next row, and the two new boundary entries contribute an additional Thus with Hence Since and we have Therefore and the correct answer is E.
Problem 27 in other years
- 1950 AMC 12
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- 1952 AMC 12
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- 1954 AMC 12
- 1955 AMC 12
- 1956 AMC 12
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- 1958 AMC 12
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