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NT is dead once again

Source: APMO 2024 P4

July 29, 2024
number theorycombinatoricsAPMO 2024

Problem Statement

Prove that for every positive integer tt there is a unique permutation a0,a1,…,at−1a_0, a_1, \ldots , a_{t-1} of 0,1,…,t−10, 1, \ldots , t-1 such that, for every 0≤i≤t−10 \leq i \leq t-1, the binomial coefficient (t+i2ai)\binom{t+i}{2a_i} is odd and 2ai≠t+i2a_i \neq t+i.
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