MathDB

Let

C

be a fixed circle,

u > 0

be a fixed real and let

v_0 , v_1 , v_2 , \ldots

be a sequence of positive real numbers. Two ants

A

and

B

walk around the perimeter of

C

in opposite directions, starting from the same starting point. Ant

A

has a constant speed

u

, while ant

B

has an initial speed

v_0

. For each positive integer

n

, when the two ants collide for the

n

−th time, they change the directions in which they walk around the perimeter of

C

, with ant

A

remaining at speed

u

and ant

B

stops walking at speed

v_{n-1}

to walk at speed

v_n

.
(a) If the sequence

\{v_n\}

is strictly increasing, with

\lim_{n\rightarrow \infty} v_n = +\infty

, prove that there is exactly one point in

C

that ant

A

will pass "infinitely" many times.
(b) Prove that there is a sequence

\{v_n\}

with

\lim_{n\rightarrow\infty} v_n = +\infty

, such that ant

A

will pass "infinitely" many times through all points on the circle

C

Problem Statement