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 3 of 4: Contradict
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∑r=0n!−1r=n!(n!−1)2≢0(modn!).\sum_{r=0}^{n!-1}r=\frac{n!(n!-1)}2\not\equiv0\pmod{n!}.
Detailed analysis

Because n is greater than 1, n factorial is even, so this sum is not divisible by n factorial.