4

Part of ICMC 6

Problems(2)

No two cycles share an edge

Source: ICMC 2023 Round 1 P4

\mathcal{G}

be a simple graph with

n

m

edges such that no two cycles share an edge. Prove that

2m < 3n

.
Note: A simple graph is a graph with at most one edge between any two vertices and no edges from any vertex to itself. A cycle is a sequence of distinct vertices

v_1, \dots, v_n

such that there is an edge between any two consecutive vertices, and between

v_n

v_1

.
Proposed by Ethan Tan

college contestscombinatoricsgraph theoryICMC

Sum of divisors perfect square

Source: ICMC 2023 Round 2 P4

Do there exist infinitely many positive integers

m

such that the sum of the positive divisors of

m

m

itself) is a perfect square?
Proposed by Dylan Toh

ICMCcollege contestsnumber theorysum of divisorsPerfect Square