MathDB

1

Part of 2020 Iran Team Selection Test

Problems(2)

Weighted complete graph

Source: Iran TST1 Day1 P1

2/22/2020
A weighted complete graph with distinct positive wights is given such that in every triangle is degenerate that is wight of an edge is equal to sum of two other. Prove that one can assign values to the vertexes of this graph such that the wight of each edge is the difference between two assigned values of the endpoints.
Proposed by Morteza Saghafian
combinatoricsgraph theoryIranian TST
square free mod n

Source: Iran TST2 Day1 P1

3/11/2020
We call a monic polynomial P(x)∈Z[x]P(x) \in \mathbb{Z}[x] square-free mod n if there dose not exist polynomials Q(x),R(x)∈Z[x]Q(x),R(x) \in \mathbb{Z}[x] with QQ being non-constant and P(x)≡Q(x)2R(x)mod  nP(x) \equiv Q(x)^2 R(x) \mod n. Given a prime pp and integer m≥2m \geq 2. Find the number of monic square-free mod p P(x)P(x) with degree mm and coeeficients in {0,1,2,3,...,p−1}\{0,1,2,3,...,p-1\}.
Proposed by Masud Shafaie
polynomialnumber theoryIranian TST