1998 AMC 8 Problem 24
Attempt Problem 24 of the 1998 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 1998 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
24.
A rectangular board of columns has squares numbered beginning in the upper left corner and moving left to right so row one is numbered through row two is through and so on. A student shades square then skips one square and shades square skips two squares and shades square skips squares and shades square and continues in this way until there is at least one shaded square in each column.
What is the number of the shaded square that first achieves this result?
Answer: E
Small hint:
Columns correspond to residues modulo
Big hint:
The shaded squares are triangular numbers.
Solution:
The shaded squares are the triangular numbers Columns correspond to residues modulo with residue representing the eighth column.
The triangular numbers through have residues and Thus, the eighth-column residue has not appeared yet.
The next triangular number is and This is the first time every column has a shaded square.
Thus, the correct answer is E .
Problem 24 in other years
- 1985 AMC 8
- 1986 AMC 8
- 1987 AMC 8
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- 1989 AMC 8
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