MathDB

12.2

Part of VMEO III 2006

Problems(2)

m^2 =\sqrt{n} +\sqrt{2n + 1}

Source: 2006 VMEO III Juniors 12.2 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2463155_vmeo_iii

9/11/2021
Find all positive integers (m,n)(m, n) that satisfy m2=n+2n+1.m^2 =\sqrt{n} +\sqrt{2n + 1}.
number theoryDiophantine EquationsDiophantine equationdiophantine
complete graph of n vertices, 3 colors

Source: 2006 VMEO III Seniors 12.2 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2463155_vmeo_iii

9/17/2021
A complete graph of nn vertices is a set of nn vertices and those vertices are connected in pairs by edges. Suppose the graph has nn vertices A1,A2,...,AnA_1, A_2, ..., A_n, the cycle is a set of edges of the form Ai1Ai2,Ai2Ai3,...,AimAi1A_{i_1}A_{i_2}, A_{i_2}A_{i_3},..., A_{i_m}A_{i_1} with i1,i2,...,im∈1,2,...,ni_1, i_2, ..., i_m \in {1, 2, ..., n} double one different. We call mm the length of this cycle. Find the smallest positive integern n such that for every way of coloring all edges of a complete graph of nn vertices, each edge filled with one of three different colors, there is always a cycle of even length with the same color.
PS. The same problem with another wording [url=https://artofproblemsolving.com/community/c6h151391p852296]here .
Graph coloringgraph theorycombinatorics