MathDB
Hexagons

Source: Iran PPCE 2004

January 9, 2009
limitinequalitiescombinatorics proposedcombinatorics

Problem Statement

Let H(n) H(n) be the number of simply connected subsets with n n hexagons in an infinite hexagonal network. Also let P(n) P(n) be the number of paths starting from a fixed vertex (that do not connect itself) with lentgh n n in this hexagonal network. a) Prove that the limits \alpha: \equal{}\lim_{n\rightarrow\infty}H(n)^{\frac1n}, \beta: \equal{}\lim_{n\rightarrow\infty}P(n)^{\frac1n}exist. b) Prove the following inequalities: 2≤β≤2 \sqrt2\leq\beta\leq2 α≤12.5 \alpha\leq 12.5 α≥3.5 \alpha\geq3.5 α≤β4 \alpha\leq\beta^4
Back to ProblemsView on AoPS