Problem 6 · 2019 AMC 8
Medium
Geometry & Measurement
symmetrycareful-counting
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Answer: C — 2/5.
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Hint 1 of 2
"Line PQ is a line of symmetry" sounds open-ended, but a square has only four symmetry lines, period. So the real question is just: how many grid points sit on those four lines (other than P)?
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Hint 2 of 2
The four axes all pass through center P. Count points per line, multiply by 4 — then watch the double-count: P is shared by all four lines and isn't allowed as Q.
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Approach: Q must land on one of the square's 4 symmetry lines
- A square has exactly 4 axes of symmetry, and every one passes through its center P: the two diagonals and the two midline (perpendicular-bisector) lines. For PQ to be an axis, Q must lie on one of these 4 lines.
- Each axis runs across the 9×9 grid through 9 points (including P). Across 4 lines that's 4 × 9 = 36, but P is on all four and Q can't equal P, so subtract those 4 copies of P: 36 − 4 = 32 valid points for Q.
- Probability = 32 / 80 = 2/5.
- Watch the overlap: the only reason this isn't simply 4×9 is that the lines share the center point — whenever you count points spread over several lines through a common point, subtract the shared point's repeats.
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