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how many functions are there?

Source: Iran 3rd round 2011-number theory exam-p6

September 5, 2011
functionnumber theory proposednumber theory

Problem Statement

aa is an integer and pp is a prime number and we have p≥17p\ge 17. Suppose that S={1,2,....,p−1}S=\{1,2,....,p-1\} and T={y∣1≤y≤p−1,ordp(y)<p−1}T=\{y|1\le y\le p-1,ord_p(y)<p-1\}. Prove that there are at least 4(p−3)(p−1)p−44(p-3)(p-1)^{p-4} functions f:S⟶Sf:S\longrightarrow S satisfying ∑x∈Txf(x)≡a\sum_{x\in T} x^{f(x)}\equiv a (mod(mod p)p). proposed by Mahyar Sefidgaran
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