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congruent to mod p

Source: SMO 2012 Q4

June 30, 2012
modular arithmeticfloor functionnumber theoryrelatively primenumber theory proposed

Problem Statement

Let pp be an odd prime. Prove that 1p−2+2p−2+⋯+(p−12)p−2≡2−2pp(modp).1^{p-2}+2^{p-2}+\cdots+\left(\frac{p-1}{2}\right)^{p-2}\equiv\frac{2-2^p}{p}\pmod p.
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