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3

Part of 2012 Vietnam Team Selection Test

Problems(2)

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

Source: Viet Nam TST 2012 Day 1

4/17/2012
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.
floor functionmodular arithmeticgeometrygeometric transformationsymmetryarithmetic sequencecombinatorial geometry
42 students split into groups

Source: Vietnamese TST 2012 Problem 6

5/9/2012
There are 4242 students taking part in the Team Selection Test. It is known that every student knows exactly 2020 other students. Show that we can divide the students into 22 groups or 2121 groups such that the number of students in each group is equal and every two students in the same group know each other.
combinatorics unsolvedcombinatorics