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When ax+by+cz+d is divisible by p

Source: Viet Nam TST 2012 Day 1

April 17, 2012
floor functionmodular arithmeticgeometrygeometric transformationsymmetryarithmetic sequencecombinatorial geometry

Problem Statement

Let p≥17p\ge 17 be a prime. Prove that t=3t=3 is the largest positive integer which satisfies the following condition: For any integers a,b,c,da,b,c,d such that abcabc is not divisible by pp and (a+b+c)(a+b+c) is divisible by pp, there exists integers x,y,zx,y,z belonging to the set {0,1,2,…,⌊pt⌋−1}\{0,1,2,\ldots , \left\lfloor \frac{p}{t} \right\rfloor - 1\} such that ax+by+cz+dax+by+cz+d is divisible by pp.
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