2016 AMC 12B Problem 11

Below is the professionally curated solution for Problem 11 of the 2016 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2016 AMC 12B solutions, or check the answer key.

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Concepts:counting shapes in figures lattice point casework

Difficulty rating: 1630

11.

How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x,$ the line $y=-0.1,$ and the line $x=5.1?$

$30$

$41$

$45$

$50$

$57$

Solution:

A unit square in the strip $k\le x\le k+1$ fits below $y=\pi x$ up to height $\lfloor\pi k\rfloor.$ Counting $1\times1$ squares in the strips $1\le x\le5$ gives $3+6+9+12=30.$ The $2\times2$ squares give $2+5+8=15,$ and the $3\times3$ squares give $1+4=5.$ There are no larger squares, so the total is $30+15+5=50.$

Thus, the correct answer is D.

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