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 |