AMC 10 · 2016 · #11
Grade 8 countingPick an answer.
First sketch the region (tool #1) to see it is a right triangle and to read off what "inside" means: an integer corner (x,y) is inside when y ≤ π x, y ≥ -0.1, and x ≤ 5.1. A square has four corners, but only one of them is the troublemaker against each boundary — that is the Extreme Principle (tool #14): the top-left corner is the one most likely to poke above the slant line, so it is the only corner I have to test. Then the count is a clean Systematic List (tool #2): group the squares by side length, and within each side length count column by column. Tool #7 (Identify Subproblems) just sums the three side-length cases at the end.
Sketch the region
The region is bounded on three sides.
Plotting the three lines turns the wordy boundary into three simple yes/no tests on a corner's coordinates.
6.NS.C.8Draw A DiagramOnly the top-left corner matters
Only one corner can violate the boundary.
Among four corners, the highest-and-leftmost one is the first to break the slant rule, so checking it alone is enough.
6.EE.B.5Extreme PrinciplePi never ties an integer corner
The line never lands on a grid point.
An irrational slope can't pass through a lattice point, so the highest allowed integer height in column a is exactly 3a.
An irrational slope can never pass through a lattice point, so the highest allowed height is a clean whole number.
▸ Why?
A line through two lattice points has a slope that is a ratio of whole numbers, which an irrational number is not.
▸ Why?
Since no tie is possible, the largest whole number strictly below the line is the exact ceiling.
Count the 1x1 squares
The unit squares number 30.
Each step right adds 3 more rows of room under the slant line, so the columns count 3,6,9,12.
8.EE.B.5Make A Systematic ListCount the 2x2 squares
The next size gives 15.
A taller square eats two of the rows, so each column loses one spot compared with the 1×1 count.
6.EE.B.8Make A Systematic ListCount 3x3, rule out 4x4, then add
Adding the rest gives 50, choice (D).
Sorting by size keeps the count organized, and the three sizes that fit add straight to the total.
6.EE.B.5Identify SubproblemsCheck only the corner most likely to break each boundary, then count the squares size by size and add them up.
- Sketch the region
- Only the top-left corner matters
- Pi never ties an integer corner
- Count the 1x1 squares
- Count the 2x2 squares
- Count 3x3, rule out 4x4, then add