2008 Mock ARML 2 Problems/Problem 5

Problem

Al is thinking of a function, $f(x)$ . He reveals to Bob that the function is a polynomial of the form $f(x) = ax^8 + bx^5 + cx^2 + dx + e$ , where $a$ , $b$ , $c$ , $d$ , and $e$ are complex number coefficients. Bob wishes to determine the value of $d$ . For any complex number $x$ that Bob asks about, Al will tell him the value of $f(x)$ . At least how many values of $x$ must Bob ask about in order to definitively determine the value of $d$ ?

Solution

Note that the degree of the term with coefficient $d$ , $1$ , is distinct from the other degrees $\mod{3}$ . We claim that $3$ values of $x$ are sufficient, namely the 3rd roots of unity.

Let $\omega = e^{2\pi/3}$ . Consider $(x_1,x_2,x_3) = (1, \omega, \omega^2)$ . For each of $x_1,x_2,x_3$ , note that $x_i^8 = x_i^5 = x_i^2$ . Thus, $\begin{align*} f(1) &= (a+b+c) + d + e\\ f(\omega) &= \omega^2(a+b+c) + \omega \cdot d + e \\ f(\omega^2) &= \omega (a+b+c) + \omega^2 \cdot d + e\end{align*}$ This is simply a non-degenerate three-equation linear system in $a+b+c,\, d,\, e$ , which will determine the value of $d$ . It is not difficult to see that $1$ or $2$ values of $x$ will not suffice, so the answer is $\boxed{3}$ .

2008 Mock ARML 2 (Problems, Source)
Preceded by Problem 4	Followed by Problem 6
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8

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2008 Mock ARML 2 Problems/Problem 5

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