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Part of 2015 Tuymaada Olympiad

Problems(2)

Source: Tuymaada 2015, Day 1, Problem 1, Senior League

On the football training there was

n

footballers - forwards and goalkeepers. They made

k

goals. Prove that main trainer can give for every footballer squad number from

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n

such, that for every goal the difference between squad number of forward and squad number of goalkeeper is more than

n-k

combinatoricsalgebra

Table with numbers

Source: Tuymaada 2015, Day 1, Problem 1, Junior League

100

different real numbers. Prove, that we can put it in

10 \times 10

table, such that difference between two numbers in cells with common side are not equals

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