2021 AMC 10B Problem 8

Problem 8 of 25EasierNumber Theory

Mr. Zhou places all the integers from

1

225

into a

15

15

grid. He places

1

in the middle square (eighth row and eighth column) and places other numbers one by one clockwise, as shown in part in the diagram below. What is the sum of the greatest number and the least number that appear in the second row from the top?

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Solution

In the outer

15\times15

ring, the top row contains

211,212,\ldots,225,

so the number just below

211

in the second row is

210.

This is the greatest number in the second row. The inner

13\times13

spiral has

13^2=169

in its upper-right corner. In the second row of the full grid, the inner-ring entries run from

157

169.

Thus the least entry in that row is

157.

The required sum is

210+157=367.

Thus, the answer is A .

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