Difference between revisions of "1976 USAMO Problems/Problem 5"

Latest revision as of 16:14, 16 August 2015

Problem

If $P(x)$ , $Q(x)$ , $R(x)$ , and $S(x)$ are all polynomials such that $P(x^5) + xQ(x^5) + x^2 R(x^5) = (x^4 + x^3 + x^2 + x +1) S(x),$ prove that $x-1$ is a factor of $P(x)$ .

Solutions

Solution 1

In general we will show that if $m$ is an integer less than $n$ and $P_0, \dotsc, P_{m-1}$ and $S$ are polynomials satisfying $P_0(x^n) + x P_1(x^n) + \dotsb + x^{m-1} P_{m-1}(x^n) = \sum_{k=0}^{n-1} x^k S(x),$ then $P_k(1) = 0$ , for all integers $0 \le k \le m-1$ . For the problem, we may set $n=5$ , $m=3$ , and then note that since $P(1)= 0$ , $(x-1)$ is a factor of $P(x)$ .

Indeed, let $\omega_1, \dotsc, \omega_{n-1}$ be the $n$ th roots of unity other than 1. Then for all integers $1 \le i \le n-1$ , $\sum_{k=0}^{m-1} \omega_i^k P_k(\omega_i^n) = \sum_{k=0}^{m-1} \omega_i^k P_k(1) = \sum_{k=0}^{n-1} \omega_i^k S(\omega_i) = \frac{\omega_i^{n-1} - 1}{\omega_i - 1} S(\omega_i) = 0 ,$ for all integers $1 \le i \le n$ . This means that the $(m-1)$ th degree polynomial $\sum_{k=0}^{m-1} x^k P_k(1)$ has $n-1 > m-1$ distinct roots. Therefore all its coefficients must be zero, so $P_k = 0$ for all integers $0 \le k \le m-1$ , as desired. $\blacksquare$

Solution 2

Let $\zeta, \xi, \rho$ be three distinct primitive fifth roots of unity. Setting $x = \zeta, \xi$ , we have $\begin{align*} P(1) + \zeta Q(1) + \zeta^2 R(1) &= \frac{\zeta^5-1}{\zeta-1} S(\zeta) = 0, \\ P(1) + \xi Q(1) + \xi^2 R(1) &= \frac{\xi^5-1}{\xi-1} S(\xi) = 0 . \end{align*}$ These equations imply that $\zeta Q(1) + \zeta^2 R(1) = \xi Q(1) + \xi^2 R(1),$ or $Q(1) = - ( \zeta + \xi)R(1).$ But by symmetry, $Q(1) = - (\zeta + \rho)R(1) .$ Since $\xi \neq \rho$ , it follows that $Q(1) = R(1) = 0$ . Then, as noted above, $P(1) = P(1) + \zeta Q(1) + \zeta^2 R(1) = 0,$ so $(x-1)$ is a factor of $P(x)$ , as desired. $\blacksquare$

Solution 3

Let $z, z^2, z^3$ be three of the 5th roots of unity not equal to one that satisfy $1 + z + z^2 + z^3 + z^4 = 0$ as a result. Plugging them into the equation gives the linear system of equations in $(A(1), B(1), C(1))$ :

$A(1) + zB(1) + z^2C(1) = 0$ $A(1) + z^2B(1) + z^4C(1) = 0$ $A(1) + z^3B(1) + z^6C(1) = 0$

Direct observation gives $(A(1), B(1), C(1)) = (0,0,0)$ as a solution, and there are no others because the system of equations is linear. Hence, $A(1) = 0$ , and so $(x-1)$ is a factor of $A(x)$ , as desired. $\blacksquare$

Note: We can generalize this approach to prove the claim in Solution 1 in an equally fast, concise, and readily understandable fashion.

@@ Line 37: / Line 37: @@
 so <math>(x-1)</math> is a factor of <math>P(x)</math>, as desired.  <math>\blacksquare</math>
+=== Solution 3 ===
+Let <math>z, z^2, z^3</math> be three of the 5th roots of unity not equal to one that satisfy <math>1 + z + z^2 + z^3 + z^4 = 0</math> as a result. Plugging them into the equation gives the linear system of equations in <math>(A(1), B(1), C(1))</math>:
-{{alternate solutions}}
+<cmath>A(1) + zB(1) + z^2C(1) = 0</cmath>
+<cmath>A(1) + z^2B(1) + z^4C(1) = 0</cmath>
+<cmath>A(1) + z^3B(1) + z^6C(1) = 0</cmath>
+Direct observation gives <math>(A(1), B(1), C(1)) = (0,0,0)</math> as a solution, and there are no others because the system of equations is linear. Hence, <math>A(1) = 0</math>, and so <math>(x-1)</math> is a factor of <math>A(x)</math>, as desired. <math>\blacksquare</math>
+''Note: We can generalize this approach to prove the claim in Solution 1 in an equally fast, concise, and readily understandable fashion.''
 == See Also ==
@@ Line 44: / Line 52: @@
 {{USAMO box|year=1976|num-b=4|after=Last Question}}
 * [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=523481#523481 Discussion on AoPS/MathLinks]
+{{MAA Notice}}
 [[Category:Olympiad Algebra Problems]]

1976 USAMO (Problems • Resources)
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