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p|m_km_{\sigma(k)}-m_lm_{\sigma(l)}

Source: Moldova 2007 IMO-BMO TST I problem 2

March 5, 2007
modular arithmeticalgebrapolynomialnumber theorynumber theory proposed

Problem Statement

Consider pp a prime number and pp consecutive positive integers m1,m2,…,mpm_{1}, m_{2}, \ldots, m_{p}. Choose a permutation σ\sigma of 1,2,…,p1, 2, \ldots, p. Show that there exist two different numbers k,l∈{1,2,…,p}k,l \in \{1,2, \ldots, p\} such that mkmσ(k)−mlmσ(l)m_{k}m_{\sigma(k)}-m_{l}m_{\sigma(l)} is divisible by pp.
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