Difference between revisions of "2001 AIME I Problems/Problem 3"

 
m (Solution 2)
 
(9 intermediate revisions by 8 users not shown)
Line 1: Line 1:
 
== Problem ==
 
== Problem ==
 +
Find the sum of the [[root]]s, real and non-real, of the equation <math>x^{2001}+\left(\frac 12-x\right)^{2001}=0</math>, given that there are no multiple roots.
 +
 +
== Solution 1 ==
 +
From [[Vieta's formulas]], in a [[polynomial]] of the form <math>a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0 = 0</math>, then the sum of the roots is <math>\frac{-a_{n-1}}{a_n}</math>.
 +
 +
From the [[Binomial Theorem]], the first term of <math>\left(\frac 12-x\right)^{2001}</math> is <math>-x^{2001}</math>, but <math>x^{2001}+-x^{2001}=0</math>, so the term with the largest degree is <math>x^{2000}</math>. So we need the coefficient of that term, as well as the coefficient of <math>x^{1999}</math>.
 +
 +
<cmath>\begin{align*}\binom{2001}{1} \cdot (-x)^{2000} \cdot \left(\frac{1}{2}\right)^1&=\frac{2001x^{2000}}{2}\\
 +
\binom{2001}{2} \cdot (-x)^{1999} \cdot \left(\frac{1}{2}\right)^2 &=\frac{-x^{1999}*2001*2000}{8}=-2001 \cdot 250x^{1999}
 +
\end{align*}</cmath>
 +
 +
Applying Vieta's formulas, we find that the sum of the roots is <math>-\frac{-2001 \cdot 250}{\frac{2001}{2}}=250 \cdot 2=\boxed{500}</math>.
 +
 +
== Solution 2 ==
 +
 +
We find that the given equation has a <math>2000^{\text{th}}</math> degree polynomial. Note that there are no multiple roots. Thus, if <math>\frac{1}{2} - x</math> is a root, <math>x</math> is also a root. Thus, we pair up <math>1000</math> pairs of roots that sum to <math>\frac{1}{2}</math> to get a sum of <math>\boxed{500}</math>.
 +
 +
==Solution 3==
 +
Note that if <math>r</math> is a root, then <math>\frac{1}{2}-r</math> is a root and they sum up to <math>\frac{1}{2}.</math>  We make the substitution <math>y=x-\frac{1}{4}</math> so <cmath>(\frac{1}{4}+y)^{2001}+(\frac{1}{4}-y)^{2001}=0.</cmath> Expanding gives <cmath>2\cdot\frac{1}{4}\cdot\binom{2001}{1}y^{2000}-0y^{1999}+\cdots</cmath> so by Vieta, the sum of the roots of <math>y</math> is 0.  Since <math>x</math> has a degree of 2000, then <math>x</math> has 2000 roots so the sum of the roots is <cmath>2000(\sum_{n=1}^{2000} y+\frac{1}{4})=2000(0+\frac{1}{4})=\boxed{500}.</cmath>
  
== Solution ==
 
  
 
== See also ==
 
== See also ==
* [[2001 AIME I Problems]]
+
{{AIME box|year=2001|n=I|num-b=2|num-a=4}}
 +
 
 +
[[Category:Intermediate Algebra Problems]]
 +
{{MAA Notice}}

Latest revision as of 21:28, 18 September 2018

Problem

Find the sum of the roots, real and non-real, of the equation $x^{2001}+\left(\frac 12-x\right)^{2001}=0$, given that there are no multiple roots.

Solution 1

From Vieta's formulas, in a polynomial of the form $a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0 = 0$, then the sum of the roots is $\frac{-a_{n-1}}{a_n}$.

From the Binomial Theorem, the first term of $\left(\frac 12-x\right)^{2001}$ is $-x^{2001}$, but $x^{2001}+-x^{2001}=0$, so the term with the largest degree is $x^{2000}$. So we need the coefficient of that term, as well as the coefficient of $x^{1999}$.

\begin{align*}\binom{2001}{1} \cdot (-x)^{2000} \cdot \left(\frac{1}{2}\right)^1&=\frac{2001x^{2000}}{2}\\ \binom{2001}{2} \cdot (-x)^{1999} \cdot \left(\frac{1}{2}\right)^2 &=\frac{-x^{1999}*2001*2000}{8}=-2001 \cdot 250x^{1999} \end{align*}

Applying Vieta's formulas, we find that the sum of the roots is $-\frac{-2001 \cdot 250}{\frac{2001}{2}}=250 \cdot 2=\boxed{500}$.

Solution 2

We find that the given equation has a $2000^{\text{th}}$ degree polynomial. Note that there are no multiple roots. Thus, if $\frac{1}{2} - x$ is a root, $x$ is also a root. Thus, we pair up $1000$ pairs of roots that sum to $\frac{1}{2}$ to get a sum of $\boxed{500}$.

Solution 3

Note that if $r$ is a root, then $\frac{1}{2}-r$ is a root and they sum up to $\frac{1}{2}.$ We make the substitution $y=x-\frac{1}{4}$ so \[(\frac{1}{4}+y)^{2001}+(\frac{1}{4}-y)^{2001}=0.\] Expanding gives \[2\cdot\frac{1}{4}\cdot\binom{2001}{1}y^{2000}-0y^{1999}+\cdots\] so by Vieta, the sum of the roots of $y$ is 0. Since $x$ has a degree of 2000, then $x$ has 2000 roots so the sum of the roots is \[2000(\sum_{n=1}^{2000} y+\frac{1}{4})=2000(0+\frac{1}{4})=\boxed{500}.\]


See also

2001 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 2
Followed by
Problem 4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Invalid username
Login to AoPS