Difference between revisions of "2017 AMC 12A Problems/Problem 21"

Revision as of 20:20, 15 November 2022

Problem

A set $S$ is constructed as follows. To begin, $S = \{0,10\}$ . Repeatedly, as long as possible, if $x$ is an integer root of some polynomial $a_{n}x^n + a_{n-1}x^{n-1} + \dots + a_{1}x + a_0$ for some $n\geq{1}$ , all of whose coefficients $a_i$ are elements of $S$ , then $x$ is put into $S$ . When no more elements can be added to $S$ , how many elements does $S$ have?

$\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ 11$

Solution

At first, $S=\{0,10\}$ .

$\begin{tabular}{r c l c l} $10x+10$ & has root & $x=-1$ & so now & $S=\{-1,0,10\}$ \\ $-x^{10}-x^9-x^8-x^7-x^6-x^5-x^4-x^3-x^2-x+10$ & has root & $x=1$ & so now & $S=\{-1,0,1,10\}$ \\ $x+10$ & has root & $x=-10$ & so now & $S=\{-10,-1,0,1,10\}$ \\ $x^3+x-10$ & has root & $x=2$ & so now & $S=\{-10,-1,0,1,2,10\}$ \\ $x+2$ & has root & $x=-2$ & so now & $S=\{-10,-2,-1,0,1,2,10\}$ \\ $2x-10$ & has root & $x=5$ & so now & $S=\{-10,-2,-1,0,1,2,5,10\}$ \\ $x+5$ & has root & $x=-5$ & so now & $S=\{-10,-5,-2,-1,0,1,2,5,10\}$ \end{tabular}$

At this point, no more elements can be added to $S$ . To see this, let

$\begin{align*} a_{n}x^n + a_{n-1}x^{n-1} + ... + a_{2}x^2 + a_{1}x + a_0 &= 0 \\ x(a_{n}x^{n-1} + a_{n-1}x^{n-2} + ... + a_{2}x + a_{1}) + a_0 &= 0 \\ x(a_{n}x^{n-1} + a_{n-1}x^{n-2} + ... + a_{2}x + a_{1}) &= -a_0 \end{align*}$

with each $a_i$ in $S$ . $x$ is a factor of $a_0$ , and $a_0$ is in $S$ , so $x$ has to be a factor of some element in $S$ . There are no such integers left, so there can be no more additional elements. $\{-10,-5,-2,-1,0,1,2,5,10\}$ has $9$ elements $\to \boxed{\textbf{(D)}}$

Solution 3 (If you are also short on time)

By the Rational Root theorem, notice that we must have $x | a_0$ . Since $a_0 \in S$ , this implies that any $x$ added must be a factor of a certain element in $S$ before. This therefore implies that any $x$ 's added must be a factor of $10$ . Thus, the largest possible set is all the positive and negative factors of $10$ , hence $\boxed{9}$ .

Note: this solution is not a real solution because it does not show that each $x$ actually works (basically we have found the maximum possible elements but we have not shown that there is a polynomial for each of them to work).

Video Solution by Richard Rusczyk

https://www.youtube.com/watch?v=hSYSNBVPLhE&list=PLyhPcpM8aMvLZmuDnM-0vrFniLpo7Orbp&index=1 - AMBRIGGS

@@ Line 32: / Line 32: @@
 with each <math>a_i</math> in <math>S</math>. <math>x</math> is a factor of <math>a_0</math>, and <math>a_0</math> is in <math>S</math>, so <math>x</math> has to be a factor of some element in <math>S</math>. There are no such integers left, so there can be no more additional elements. <math>\{-10,-5,-2,-1,0,1,2,5,10\}</math> has <math>9</math> elements <math>\to \boxed{\textbf{(D)}}</math>
-==Solution 2 (If you are short on time)==
-By Rational Root Theorem, the only rational roots for this function we're dealing with must be in the form <math> \pm \frac p{q} </math>, where <math>p</math> and <math>q</math> are co-prime, <math>p</math> is a factor of <math>a_0</math> and <math>q</math> is a factor of <math>a_n</math>. We can easily see <math>-1</math> is in <math>S</math> because of <math>10x + 10 = 0</math> has root <math>-1</math>. Since we want set <math>S</math> to be as large as possible, we let <math>p=10</math> and <math>q=-1</math>, and quickly see that all possible integer roots are <math>\pm 1</math>, <math>\pm 2</math>, <math>\pm 5</math>, <math>\pm 10</math>, plus the <math>0</math> we started with, we get a total of <math>9</math> elements <math>\to \boxed{\textbf{(D)}}</math>
--BochTheNerd
 == Solution 3 (If you are also short on time) ==

2017 AMC 12A (Problems • Answer Key • Resources)
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