2008 Mock ARML 2 Problems/Problem 5
Problem
Al is thinking of a function, . He reveals to Bob that the function is a polynomial of the form
, where
,
,
,
, and
are complex number coefficients. Bob wishes to determine the value of
. For any complex number
that Bob asks about, Al will tell him the value of
. At least how many values of
must Bob ask about in order to definitively determine the value of
?
Solution
Note that the degree of the term with coefficient ,
, is distinct from the other degrees
. We claim that
values of
are sufficient, namely the 3rd roots of unity.
Let . Consider
. For each of
, note that
. Thus,
This is simply a non-degenerate three-equation linear system in
, which will determine the value of
. It is not difficult to see that
or
values of
will not suffice, so the answer is
.
See also
2008 Mock ARML 2 (Problems, Source) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 |