MathDB

2

Part of 2021 Iran Team Selection Test

Problems(2)

Connected graphs on a Tst

Source: Iranian TST 2021, first exam day 1, problem 2

5/17/2021
In the simple and connected graph GG let xix_i be the number of vertices with degree ii. Let d>3d>3 be the biggest degree in the graph GG. Prove that if : xd≥xd−1+2xd−2+...+(d−1)x1x_d \ge x_{d-1} + 2x_{d-2}+... +(d-1)x_1 Then there exists a vertex with degree dd such that after removing that vertex the graph GG is still connected.
Proposed by Ali Mirzaie
combinatoricsgraph theory
beautiful functional on Tst

Source: Iranian TST 2021, second exam day 1, problem 2

5/20/2021
Find all functions f:N→Nf : \mathbb{N} \rightarrow \mathbb{N} such that for any two positive integers m,nm,n we have : f(n)+1400m2∣n2+f(f(m))f(n)+1400m^2|n^2+f(f(m))
functionnumber theory