Problem 3 · AMC 8 Stretch
Stretch
Counting & Probability
Number TheoryGeometry & Measurement
pigeonholeparity
On graph paper, mark any 5 points that sit exactly on grid corners (their \(x\) and \(y\) coordinates are whole numbers). Show that 2 of your points have a midpoint that also sits exactly on a grid corner. Reminder: the midpoint of \((x_1,y_1)\) and \((x_2,y_2)\) is \(\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)\); it lands on a grid corner exactly when \(x_1+x_2\) is even AND \(y_1+y_2\) is even.
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Answer: 2 points share a parity type
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Hint 1 of 4
When is the average of two whole numbers a whole number? Only when the two numbers are both even or both odd (same 'parity').
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Hint 2 of 4
So for each point, all that matters is whether its \(x\) is even or odd, and whether its \(y\) is even or odd. List the possible (even/odd, even/odd) types.
Still stuck? Show hint 3 →
Hint 3 of 4
There are exactly 4 types: (even,even), (even,odd), (odd,even), (odd,odd). Make these your 4 boxes.
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Approach: Pigeonhole on parity types — 5 points, 4 (even/odd, even/odd) classes
- The midpoint is a grid corner exactly when \(x_1+x_2\) and \(y_1+y_2\) are both even, which happens only when each pair of coordinates has the same parity. So all that matters is the even/odd status of each point's two coordinates.
- Each point falls into one of 4 parity types (our boxes): (even,even), (even,odd), (odd,even), (odd,odd).
- Drop your 5 points into these 4 boxes. Since \(5 > 4\), some box holds 2 points.
- Those two points match in both coordinates' parity, so \(x_1+x_2\) and \(y_1+y_2\) are both even — the midpoint lands right on a grid corner.
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