1993 AMC 8 Problem 24

Attempt Problem 24 of the 1993 AMC 8 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 1993 AMC 8 solutions, or check the answer key.

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

24.

What number is directly above 142142 in this array of numbers?
123456789101112⋯\begin{array}{ccccccccc} & & & & 1 & & & & \\ & & & 2 & 3 & 4 & & & \\ & & 5 & 6 & 7 & 8 & 9 & & \\ & 10 & 11 & 12 & \cdots & & & & \end{array}
9999
119119
120120
121121
122122
Answer: C
Concepts:perfect squarepattern recognition
Small hint:
Each row ends at a perfect square: row nn ends at n2n^2
Big hint:
142142 is in the row ending at 144=122;144 = 12^2; the row above ends at 121=112,121 = 11^2, and the rows line up at their right edges
Solution:
Each row ends at a perfect square, so the row containing 142142 ends at 144=122,144 = 12^2, and the row above it ends at 121=112.121 = 11^2.
Since the rows are aligned at their right edges, 121121 sits directly above 143,143, and therefore 120120 sits directly above 142.142.
Thus, the correct answer is C .
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