MathDB

4

Part of 2018 Korea Winter Program Practice Test

Problems(2)

Sum of squares with prime

Source: 2018 Korea Winter Program Practice Test 1 #4

1/6/2018
Let p=4k+1p=4k+1 be a prime. SS is a set of all possible residues equal or smaller then 2k2k when 12(2kk)nk\frac{1}{2} \binom{2k}{k} n^k is divided by pp. Show that ∑x∈Sx2=p \sum_{x \in S} x^2 =p
number theoryprime numbers
3d graph and hamilton cycle

Source: 2018 Korea Winter Mop Practice Test #8

2/25/2018
Graph GG is defined in 3d space. It has ee edges and every vertex are connected if the distance between them is 1.1. Given that there exists the Hamilton cycle, prove that for e>1,e>1, we have min⁡d(v)≤1+2(e2)0.4.\min d(v)\le 1+2\left(\frac{e}{2}\right)^{0.4}.
combinatoricsgraph theory