AMC 10 · 2011 · #22

Grade 7 geometry-2dcounting

Archetype Lattice / Grid Pattern from Small Cases └ Sub-archetype Lattice Constraint Analysis

coordinate-geometryprinciple-of-inclusion-exclusion pattern-recognitioncomplementary-counting ↑ Prerequisites: coordinate-geometry

📏 Long solution 💡 4 insights

Problem

Rays from an inside point cut a square into equal-area triangles. Count the points that work for one number but not another.

Pick an answer.

(A)

1500

(B)

1560

(C)

2320

(D)

2480

(E)

2500

How to solve

Strategy Introduce a Variable

Naming the point X = (x, y) turns a vague geometry question into an equation about areas. A diagram of the four rays hitting the corners shows why each side of the square must carry a whole number of equal triangles, which pins x and y to a grid of fractions. Once the grid is known, counting is a systematic list, spotting the shared 60-grid is a pattern, and the final 'not 60-ray' twist is a count-the-complement subtraction.

1STEP 1

Four rays must hit the corners

Four rays are forced to the corners.

bottom=y/2, top=(1-y)/2, left=x/2, right=(1-x)/2

2STEP 2

Each side holds whole tiles

Each side then holds a whole number of tiles.

y/2=a/n → y=2a/n, x=2c/n

3STEP 3

Count the 100-ray points

The larger count gives 2401 points.

x=c/50, y=a/50, 1 ≤ a,c ≤ 49 → 49²=2401

4STEP 4

Find the shared 60-ray points

The shared points number 81.

c/50=c'/30→ 3c=5c'→ 5 ∣ c, c∈{5,…,45} (9) → 9²=81

5STEP 5

Subtract the overlap

Subtracting gives 2320, choice (C).

2401-81=2320

Answer

2320

The result 2320 sits just below the full 2401 count, which is right because only a thin 9 by 9 layer of points also lands on the coarser 60-ray grid. The construction never let a or c reach 0 or 50, so every counted point is genuinely interior, matching the problem's requirement. The value also matches choice (C) and neatly avoids the trap answers 2500 = 50² and 2401 that come from forgetting the interior cap or the overlap.

💡Key takeaway

A point works for n rays exactly when both its coordinates are fractions over n/2, so count the whole 100-grid (49 by 49 = 2401) and take away the spots that also fit the 60-grid (9 by 9 = 81) to get 2320.

Four rays must hit the corners
Each side holds whole tiles
Count the 100-ray points
Find the shared 60-ray points
Subtract the overlap

Four rays must hit the corners

Each side holds whole tiles

Count the 100-ray points

Find the shared 60-ray points

Subtract the overlap

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