1954 AMC 12 Problem 44

Attempt Problem 44 of the 1954 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 1954 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

44.

A man born in the first half of the nineteenth century was xx years old in the year x2.x^2. He was born in:
18491849
18251825
18121812
18361836
18061806
Answer: E
Concepts:agesperfect squaredate and time
Small hint:
His birth year is x2−xx^2-x
Big hint:
Use the requirement that x2x^2 is a nineteenth-century year and test the nearby integer xx
Solution:
The year x2x^2 must lie in the nineteenth century, so x=43x=43 because 432=184943^2=1849 while the neighboring squares fall outside the relevant range. His birth year is therefore x2−x=1849−43=1806, x^2-x=1849-43=1806, which is in the first half of the century.
Thus, the correct answer is E.
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