2

Part of 1999 Polish MO Finals

Problems(2)

Multiple of 5050

Source: Problem 2, Polish NO 1999

101

distinct non-negative integers less than

5050

show that one can choose four

a, b, c, d

a + b - c - d

is a multiple of

5050

combinatoricspigeonhole principle

beautiful estimations with 2n integers [mod edit: reals ;) ]

Source: poland, I think 1999

Prove that for any

2n

a_{1}

a_{2}

a_{n}

b_{1}

b_{2}

b_{n}

, we have \sum_{i < j}{\left|a_{i}\minus{}a_{j}\right|}\plus{}\sum_{i < j}{\left|b_{i}\minus{}b_{j}\right|}\leq\sum_{i,j\in\left[1,n\right]}{\left|a_{i}\minus{}b_{j}\right|}.

inequalitiesintegrationfunctiontriangle inequalityinequalities proposed