AMC 12 Mock Exam 4 Problem 15

Problem 15 of 25

A gardener's triangular plot has corners at $(0,0)$ , $(30,0)$ , and $(0,20)$ in the coordinate plane. She plants a tree at every point with integer coordinates that lies strictly inside the triangle. Points on the boundary of the triangle are reserved for the fence, so no trees are planted there. How many trees does she plant?

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Solution

The area is $A=\frac12\cdot30\cdot20=300$ . Walking around the boundary, consecutive lattice points split the leg on the $x$ -axis into $30$ pieces, the leg on the $y$ -axis into $20$ pieces, and the hypotenuse from $(30,0)$ to $(0,20)$ into $\gcd(30,20)=10$ pieces, so the number of boundary lattice points is $B=30+20+10=60$ . By Pick's theorem $A=I+\frac B2-1$ , so $I=300-30+1=\boxed{271}$ .

(Without Pick: the hypotenuse is $2x+3y=60$ . For $x=1,\dots,29$ the interior points have $1\le y<\frac{60-2x}{3}$ , and summing the counts also gives $271$ .)

Tagged: lattice points in a triangle (Pick's theorem)

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