tmua.me Tau Paper 2 · Question 20

For a real number $x$ , $\lfloor x\rfloor$ is the greatest integer not exceeding $x$ . Let $0<\alpha<1$ and $-1\le t\le0$ . For each positive integer $q$ , consider the two sequences, both indexed by all integers $n$ :

u_n=\left\lfloor\frac{(2q+1)n}{q}+\alpha\right\rfloor, \qquad v_n=\left\lfloor\frac{(2q+1)n+t-q\alpha}{q+1}\right\rfloor.

Call $(\alpha,t)$ perfect if, for every positive integer $q$ , every integer occurs exactly once in the two sequences combined.

How many values of $\alpha$ make exactly one of

(\alpha,-1/8)\qquad\text{and}\qquad(\alpha,-1/12)

perfect?

AInfinitely many

18

20

24

28

examday

tmua.me Tau Paper 2 · Q20