Difference between revisions of "2023 AMC 10B Problems/Problem 12"

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When the roots of the polynomial
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#redirect[[2023 AMC 12B Problems/Problem 6]]
 
 
<math>P(x)  = (x-1)^1 (x-2)^2 (x-3)^3 \cdot \cdot \cdot (x-10)^{10}</math>
 
 
 
are removed from the number line, what remains is the union of 11 disjoint open intervals. On how many of these intervals is <math>P(x)</math> positive?
 
 
 
==Solution 1==
 
 
 
<math>P(x)</math> is a product of <math>(x-r_n)</math> or 10 terms. When <math>x < 1</math>, all terms are <math>< 0</math>, but <math>P(x) > 0</math> because there is an even number of terms. The sign keeps alternating <math>+,-,+,-,....,+</math>.  There are 11 intervals, so there are <math>\boxed{\textbf{ 6}}</math> positives and 5 negatives. <math>\boxed{\textbf{(C) 6}}</math>
 
 
 
~<math>\textbf{Techno}\textcolor{red}{doggo}</math>
 
 
 
==Solution==
 
 
 
Denote by <math>I_k</math> the interval <math>\left( k - 1 , k \right)</math> for <math>k \in \left\{ 2, 3, \cdots , 10 \right\}</math> and <math>I_1</math> the interval <math>\left( - \infty, 1 \right)</math>.
 
 
 
Therefore, the number of intervals that <math>P(x)</math> is positive is
 
<cmath>
 
\begin{align*}
 
1 + \sum_{i=1}^{10} \Bbb I \left\{
 
\sum_{j=i}^{10} j \mbox{ is even}
 
\right\}
 
& = 1 + \sum_{i=1}^{10} \Bbb I \left\{
 
\frac{\left( i + 10 \right) \left( 11 - i \right)}{2} \mbox{ is even}
 
\right\} \\
 
& = 1 + \sum_{i=1}^{10} \Bbb I \left\{
 
\frac{- i^2 + i + 110}{2} \mbox{ is even}
 
\right\} \\
 
& = 1 + \sum_{i=1}^{10} \Bbb I \left\{
 
\frac{i^2 - i}{2} \mbox{ is odd}
 
\right\} \\
 
& = \boxed{\textbf{(C) 6}} .
 
\end{align*}
 
</cmath>
 
 
 
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
 

Latest revision as of 19:33, 15 November 2023