MathDB

Problem 2

Part of 2000 VJIMC

Problems(2)

n^(0.5*d(n)) is injective

Source: VJIMC 2000 1.2

7/27/2021
Let f:N→Rf:\mathbb N\to\mathbb R be given by f(n)=n12τ(n)f(n)=n^{\frac12\tau(n)}for n∈N={1,2,…}n\in\mathbb N=\{1,2,\ldots\} where τ(n)\tau(n) is the number of divisors of nn. Show that ff is an injection.
number theoryfunction
writing words around circle, two letters

Source: VJIMC 2000 2.2

7/27/2021
If we write the sequence AAABABBB\text{AAABABBB} along the perimeter of a circle, then every word of the length 33 consisting of letters AA and BB (i.e. AAA\text{AAA}, AAB\text{AAB}, ABA\text{ABA}, BAB\text{BAB}, ABB\text{ABB}, BBB\text{BBB}, BBA\text{BBA}, BAA\text{BAA}) occurs exactly once on the perimeter. Decide whether it is possible to write a sequence of letters from a kk-element alphabet along the perimeter of a circle in such a way that every word of the length ll (i.e. an ordered ll-tuple of letters) occurs exactly once on the perimeter.
combinatorics