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Sum of floors with primes p,q

Source: Baltic Way 2001

November 17, 2010
floor functioncombinatorics proposedcombinatorics

Problem Statement

Let pp and qq be two different primes. Prove that ⌊pq⌋+⌊2pq⌋+⌊3pq⌋+…+⌊(q−1)pq⌋=12(p−1)(q−1)\left\lfloor\frac{p}{q}\right\rfloor+\left\lfloor\frac{2p}{q}\right\rfloor+\left\lfloor\frac{3p}{q}\right\rfloor+\ldots +\left\lfloor\frac{(q-1)p}{q}\right\rfloor=\frac{1}{2}(p-1)(q-1)
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