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p divides sum of combinations (p-1,k)a^k

Source: 2020 Swedish Mathematical Competition p5

May 1, 2021
number theorycombinatoricsdividesdivisible

Problem Statement

Find all integers aa such that there is a prime number of p≥5p\ge 5 that divides (p−12){p-1 \choose 2} +(p−13)a+ {p-1 \choose 3} a +(p−14)a2+{p-1 \choose 4} a^2+ ...+(p−1p−3)ap−5. {p-1 \choose p-3} a^{p-5} .
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