Problem

Source: 25 Mar 2013

Tags: modular arithmetic, geometry, geometric transformation, real analysis, number theory proposed, number theory



Let $p$ be a prime number and $a, k$ be positive integers such that $p^a<k<2p^a$. Prove that there exists a positive integer $n$ such that \[n<p^{2a}, C_n^k\equiv n\equiv k\pmod {p^a}.\]