Pigeonhole Problems.pdf - Math 340 Pigeonhole Problems Jordan Barrett 1 Given(xi yi 1 ≤ i ≤ 5 are integer pairs in R2 prove that the midpoint of(xi

# Pigeonhole Problems.pdf - Math 340 Pigeonhole Problems...

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Math 340 Pigeonhole Problems Jordan Barrett 1. Given { ( x i , y i ) : 1 i 5 } are integer pairs in R 2 , prove that the midpoint of ( x i , y i ) - ( x j , y j ) is again an integer pair for some 1 i < j 5. Proof. For any such i, j , the midpoint will be ( x i + x j 2 , y i + y j 2 ) , so the midpoint is an inte- ger pair if ( x i , y i ) has the same parity as ( x j , y j ). The possible parities are (even , even), (even , odd), (odd , even), and (odd , odd). Since there are 5 points, and only 4 possi- ble parities, 2 points must have the same parity, and hence its midpoint is an integer pair. 2. Suppose 101 people are standing in a line. Prove that, in the order of the line, there are either 11 people that are increasing in height, or 11 people that are decreasing in height. Proof. Suppose the students are ordered x 1 , x 2 , . . . , x
• Fall '09
• KHARLMPOVICH
• Yi, Order theory

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