AMC 10 · 2020 · #16

Grade 8 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Circle-Polygon Hybrid
geometric-probabilityarea-circlesspatial-visualizationestimation easier-related-problemidentify-subproblems ↑ Prerequisites: geometric-probabilityarea-circles
📏 Medium solution 💡 3 insights
Problem
Pick a random point inside a square of side 2020 with a corner at the origin. A lattice point is any point with both coordinates integer. The chance that the random point lands within distance d of some lattice point is exactly one half. Find d rounded to the nearest tenth.

Pick an answer.

(A)
0.3
(B)
0.4
(C)
0.5
(D)
0.6
(E)
0.7
How to solve
Strategy Solve an Easier Related Problem

Tool #9 (Easier Problem): the 2020 × 2020 square is just 2020² copies of a single 1 × 1 tile, each behaving the same way around its four corner lattice points. So we replace the giant square with one unit square — the probability is identical. Tool #7 (Subproblems): inside that unit square, the favorable region is four quarter-circles of radius d at the four corners, which combine into one full circle of area π d². Tool #3 (Eliminate): the resulting equation π d² = 1/2 gives d² = 1/2π ≈ 0.159 — square each answer choice and pick the one closest to 0.159.

1STEP 1

Reduce to one unit square

The big square matches one unit square.

P_big = P_unit
2STEP 2

Combine the four corners

Four quarter-circles make one circle.

A_good = 4 · 1/4π d² = π d²
3STEP 3

Write the chance as area

That area is the probability.

P = (π d²)/1 = 1/2 → d² = 1/2π
4STEP 4

Solve for the distance

Solving and rounding gives 0.4.

d² ≈ 0.159 → d ≈ 0.4 → (B)
Answer
0.4
Sanity-check the magnitude: if d=0.5, the four quarter-disks would just touch and have total area π(0.5)² ≈ 0.785 — way too much. If d=0.3, they cover π(0.3)² ≈ 0.283 — too little. Half-coverage sits between, closer to d=0.4 giving area ≈ 0.503, almost exactly 1/2. The assumption d ≤ 0.5 also checks out, so the four quarter-disks really don't overlap.
💡Key takeaway

This AMC 12 problem only needs Grade 8 number-line estimation you already know — shrink the giant square to one tiny tile, fit four corner pie-slices into one whole pie, set that area to 1/2, and check that d ≈ 0.4 does the trick. The answer is (B).

More like this

Same archetype — closest grade level first.