MathDB
n-element set

Source: China TST 2006

June 18, 2006
geometrygeometric transformationrotationfunctioncombinatorics unsolvedcombinatorics

Problem Statement

dd and nn are positive integers such that d∣nd \mid n. The n-number sets (x1,x2,⋯xn)(x_1, x_2, \cdots x_n) satisfy the following condition: (1) 0≤x1≤x2≤⋯≤xn≤n0 \leq x_1 \leq x_2 \leq \cdots \leq x_n \leq n (2) d∣(x1+x2+⋯xn)d \mid (x_1+x_2+ \cdots x_n) Prove that in all the n-number sets that meet the conditions, there are exactly half satisfy xn=nx_n=n.
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