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2020 Taiwan APMO Preliminary Problem 5

Source: 2020 Taiwan APMO Preliminary

July 23, 2020
Setsset theory

Problem Statement

Let SS is the set of permutation of {1,2,3,4,5,6,7,8} (1)For all σ=σ1σ2...σ8∈S\sigma=\sigma_1\sigma_2...\sigma_8\in S Evaluate the sum of S=σ1σ2+σ3σ4+σ5σ6+σ7σ8\sigma_1\sigma_2+\sigma_3\sigma_4+\sigma_5\sigma_6+\sigma_7\sigma_8. Then for all elements in SS,what is the arithmetic mean of S? (Notice SS and S are different.) (2)In SS, how many permutations are there which satisfies "For all k=1,2,...,7k=1,2,...,7,the digit after k is not (k+1)"?
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