MathLabs

Problem 4

Let n>1 n>1 be an odd integer and let k1,k2,…,kn k_1,k_2,\dots,k_n be integers. For each permutation a=(a1,…,an)\mathbf a=(a_1,\dots,a_n) of 1,…,n1,\dots,n , define S(a)=∑i=1nkiai S(\mathbf a)=\sum_{i=1}^n k_i a_i . Prove that there are distinct permutations b,c\mathbf b,\mathbf c such that n! n! divides S(b)S(c) S(\mathbf b)S(\mathbf c).
Step 1 of 4: Sum by positions
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∑aS(a)=(n−1)!n(n+1)2∑iki≡0(modn!).\sum_{\mathbf a}S(\mathbf a)=(n-1)!\frac{n(n+1)}2\sum_i k_i\equiv0\pmod{n!}.
Detailed analysis

Each value occurs (n minus 1) factorial times in each position.