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Difference between revisions of "2005 AMC 12B Problems/Problem 23"

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<cmath>\log_{10}(x+y) = z \text{ and } \log_{10}(x^{2}+y^{2}) = z+1.</cmath>
 
<cmath>\log_{10}(x+y) = z \text{ and } \log_{10}(x^{2}+y^{2}) = z+1.</cmath>
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There are real numbers <math>a</math> and <math>b</math> such that for all ordered triples <math>(x,y.z)</math> in <math>S</math> we have <math>x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.</math> What is the value of <math>a+b?</math>
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There are real numbers <math>a</math> and <math>b</math> such that for all ordered triples <math>(x,y,z)</math> in <math>S</math> we have <math>x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.</math> What is the value of <math>a+b?</math>
  
 
<math>
 
<math>
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</math>
 
</math>
  
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== Solution ==
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== Solution 1 ==
 
Let <math>x + y = s</math> and <math>x^2 + y^2 = t</math>.   
 
Let <math>x + y = s</math> and <math>x^2 + y^2 = t</math>.   
 
Then, <math>\log(s)=z</math> implies <math>\log(10s) = z+1= \log(t)</math>,so <math>t=10s</math>.   
 
Then, <math>\log(s)=z</math> implies <math>\log(10s) = z+1= \log(t)</math>,so <math>t=10s</math>.   
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Therefore, <math>x^3 + y^3 = s*\frac{3t-s^2}{2} = s(15s-\frac{s^2}{2})</math>.   
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Therefore, <math>x^3 + y^3 = s\cdot\dfrac{3t-s^2}{2} = s(15s-\dfrac{s^2}{2})</math>.   
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Since <math>s = 10^z</math>, we find that <math>x^3 + y^3 = 15\times10^2 - (1/2)\times10^3</math>.   
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Since <math>s = 10^z</math>, we find that <math>x^3 + y^3 = 15\cdot10^{2z} - (1/2)\cdot10^{3z}</math>.   
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Thus, <math>a+b = \boxed{\frac{29}{2}}</math>.
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Thus, <math>a+b = \frac{29}{2}</math> <math>\Rightarrow</math> <math>\boxed{B}</math>
  
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==Alternate Solution==
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== Solution 2 ==
 
First, remember that <math>x^3 + y^3</math> factors to <math>(x + y) (x^2 - xy + y^2)</math>. By the givens, <math>x + y = 10^z</math> and <math>x^2 + y^2 = 10^{z + 1}</math>. These can be used to find <math>xy</math>:
 
First, remember that <math>x^3 + y^3</math> factors to <math>(x + y) (x^2 - xy + y^2)</math>. By the givens, <math>x + y = 10^z</math> and <math>x^2 + y^2 = 10^{z + 1}</math>. These can be used to find <math>xy</math>:
 
<cmath>(x + y)^2 = 10^{2z}</cmath>
 
<cmath>(x + y)^2 = 10^{2z}</cmath>
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It follows that <math>a = -\frac{1}{2}</math> and <math>b = 15</math>, thus <math>a + b = \frac{29}{2}.</math>
 
It follows that <math>a = -\frac{1}{2}</math> and <math>b = 15</math>, thus <math>a + b = \frac{29}{2}.</math>
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 +
== Solution 3 ==
 +
 +
We can rearrange <math>\log_{10}(x+y)=z</math> into <math>x+y=10^{z}</math> and <math>\log_{10}(x^2+y^2)=z+1</math> into <math>x^2+y^2=10^{z+1}</math>
 +
 +
We can then put <math>x+y</math> to the third power or <math>(x+y)^{3}=10^{3z}</math>. Basic polynomial multiplication shows us that <math>(x+y)^{3}=x^3+3x^{2}y+3xy^{2}+y^3=10^{3z}</math> Thus, <math>x^3+y^3=10^{3z}-3x^{2}y-3xy^{2}</math> or  <math>x^3+y^3=10^{3z}-3xy(x+y)</math>. We know that <math>x+y=10^{z}</math> so we have <math>x^3+y^3=10^{3z}-3xy(10^{z})</math>.
 +
 +
Now we need to find out what <math>xy</math> is equal to in terms of <math>z</math>. We will find <math>xy</math> by squaring <math>x+y</math>. It is basic polynomial multiplication to figure out that <math>(x+y)^{2}=x^2+2xy+y^2</math>. We are also given that <math>x^2+y^2=10^{z+1}</math> and <math>x+y=10^{z}</math>. Thus <math>(10^{z})^{2}=10^{z+1}+2xy</math> or <math>10^{2z}=10^{z+1}+2xy</math>. Rearranging the terms of this equation we obtain that <math>xy=\frac{10^{2z}-10^{z+1}}{2}</math> or <math>xy=\frac{10^z(10^z-10)}{2}</math>. Now plugging this equation into our original equation  <math>x^3+y^3=10^{3z}-3xy(10^{z})</math>, we obtain the equation <math>x^3+y^3=10^{3z}-3(\frac{10^z(10^z-10)}{2})(10^{z})</math>
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Simple rearranging of this equation yields the result that <math>x^3+y^3=10^{3z}-3(\frac{10^{3z}}{2})+30(\frac{10^{2z}}{2})</math>.
 +
Combining like terms we obtain the equation <math>x^3+y^3= -(\frac{10^{3z}}{2})+30(\frac{10^{2z}}{2})</math>.
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Now we know the coefficients of <math>10^{3z}</math> and <math>10^{2z}</math> which are <math>\frac{-1}{2}</math> and <math>\frac{30}{2}</math> respectively. Adding the two coefficients we obtain an answer of <math>\frac{29}{2}.</math> <math>\Rightarrow</math> <math>\boxed{B}</math> (Author: David Camacho)
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 +
== Solution 4 (Alcumus)==
 +
From the given conditions it follows that
 +
<cmath>\[
 +
x+y=10^z, \quad x^2+y^2=10\cdot 10^z\quad\text{and}\quad 10^{2z}=(x+y)^2=x^2+2xy+y^2.
 +
\]</cmath>
 +
Thus<cmath>\[
 +
xy=\frac{1}{2}(10^{2z}-10\cdot 10^z).
 +
\]</cmath>Also<cmath>\[
 +
(x+y)^3=10^{3z}\quad\text{and}\quad x^3+y^3=(x+y)^3-3xy(x+y),
 +
\]</cmath>which yields<cmath>\begin{align*}
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x^3+y^3&=10^{3z}-\frac{3}{2}(10^{2z}-10\cdot 10^z)(10^z) \\
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&=10^{3z}-\frac{3}{2}(10^{3z}-10\cdot 10^{2z})
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=-\frac{1}{2}10^{3z}+15\cdot 10^{2z},
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\end{align*}</cmath>and <math>a+b=-\frac{1}{2}+15=\boxed{29/2}</math>.
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 +
No other value of <math>a + b</math> is possible for all members of <math>S</math>, because the triple <math>\left(\frac{1}{2}(1 + \sqrt{19}), \frac{1}{2}(1 - \sqrt{19}), 0\right)</math> is in <math>S</math>, and for this ordered triple, the equation <math>x^3 + y^3 = a\cdot 10^{3z} + b\cdot 10^{2z}</math> reduces to <math>a + b = 29/2</math>.
  
 
== See Also ==
 
== See Also ==

Latest revision as of 10:36, 19 May 2024

Problem

Let $S$ be the set of ordered triples $(x,y,z)$ of real numbers for which

\[\log_{10}(x+y) = z \text{ and } \log_{10}(x^{2}+y^{2}) = z+1.\] There are real numbers $a$ and $b$ such that for all ordered triples $(x,y,z)$ in $S$ we have $x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.$ What is the value of $a+b?$

$\textbf{(A)}\ \frac {15}{2} \qquad \textbf{(B)}\ \frac {29}{2} \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ \frac {39}{2} \qquad \textbf{(E)}\ 24$

Solution 1

Let $x + y = s$ and $x^2 + y^2 = t$. Then, $\log(s)=z$ implies $\log(10s) = z+1= \log(t)$,so $t=10s$. Therefore, $x^3 + y^3 = s\cdot\dfrac{3t-s^2}{2} = s(15s-\dfrac{s^2}{2})$. Since $s = 10^z$, we find that $x^3 + y^3 = 15\cdot10^{2z} - (1/2)\cdot10^{3z}$. Thus, $a+b = \frac{29}{2}$ $\Rightarrow$ $\boxed{B}$

Solution 2

First, remember that $x^3 + y^3$ factors to $(x + y) (x^2 - xy + y^2)$. By the givens, $x + y = 10^z$ and $x^2 + y^2 = 10^{z + 1}$. These can be used to find $xy$: \[(x + y)^2 = 10^{2z}\] \[x^2 + 2xy + y^2 = 10^{2z}\] \[2xy = 10^{2z} - 10^{z + 1}\] \[xy = \frac{10^{2z} - 10^{z + 1}}{2}\]

Therefore, \[x^3 + y^3 = a \cdot 10^{3z} + b \cdot 10^{2z} = 10^z\left(10^{z + 1} - \frac{10^{2z} - 10^{z + 1}}{2}\right)\] \[= 10^z\left(10^{z + 1} - \frac{10^{2z} - 10^{z + 1}}{2}\right)\] \[= 10^{2z + 1} - \frac{10^{3z} - 10^{2z + 1}}{2}\] \[= -\frac{1}{2} \cdot 10^{3z} + \frac{3}{2} \cdot 10^{2z + 1}\] \[= -\frac{1}{2} \cdot 10^{3z} + 15 \cdot 10^{2z}.\]

It follows that $a = -\frac{1}{2}$ and $b = 15$, thus $a + b = \frac{29}{2}.$

Solution 3

We can rearrange $\log_{10}(x+y)=z$ into $x+y=10^{z}$ and $\log_{10}(x^2+y^2)=z+1$ into $x^2+y^2=10^{z+1}$

We can then put $x+y$ to the third power or $(x+y)^{3}=10^{3z}$. Basic polynomial multiplication shows us that $(x+y)^{3}=x^3+3x^{2}y+3xy^{2}+y^3=10^{3z}$ Thus, $x^3+y^3=10^{3z}-3x^{2}y-3xy^{2}$ or $x^3+y^3=10^{3z}-3xy(x+y)$. We know that $x+y=10^{z}$ so we have $x^3+y^3=10^{3z}-3xy(10^{z})$.

Now we need to find out what $xy$ is equal to in terms of $z$. We will find $xy$ by squaring $x+y$. It is basic polynomial multiplication to figure out that $(x+y)^{2}=x^2+2xy+y^2$. We are also given that $x^2+y^2=10^{z+1}$ and $x+y=10^{z}$. Thus $(10^{z})^{2}=10^{z+1}+2xy$ or $10^{2z}=10^{z+1}+2xy$. Rearranging the terms of this equation we obtain that $xy=\frac{10^{2z}-10^{z+1}}{2}$ or $xy=\frac{10^z(10^z-10)}{2}$. Now plugging this equation into our original equation $x^3+y^3=10^{3z}-3xy(10^{z})$, we obtain the equation $x^3+y^3=10^{3z}-3(\frac{10^z(10^z-10)}{2})(10^{z})$ Simple rearranging of this equation yields the result that $x^3+y^3=10^{3z}-3(\frac{10^{3z}}{2})+30(\frac{10^{2z}}{2})$. Combining like terms we obtain the equation $x^3+y^3= -(\frac{10^{3z}}{2})+30(\frac{10^{2z}}{2})$.

Now we know the coefficients of $10^{3z}$ and $10^{2z}$ which are $\frac{-1}{2}$ and $\frac{30}{2}$ respectively. Adding the two coefficients we obtain an answer of $\frac{29}{2}.$ $\Rightarrow$ $\boxed{B}$ (Author: David Camacho)

Solution 4 (Alcumus)

From the given conditions it follows that \[ x+y=10^z, \quad x^2+y^2=10\cdot 10^z\quad\text{and}\quad 10^{2z}=(x+y)^2=x^2+2xy+y^2. \] Thus\[ xy=\frac{1}{2}(10^{2z}-10\cdot 10^z). \]Also\[ (x+y)^3=10^{3z}\quad\text{and}\quad x^3+y^3=(x+y)^3-3xy(x+y), \]which yields\begin{align*} x^3+y^3&=10^{3z}-\frac{3}{2}(10^{2z}-10\cdot 10^z)(10^z) \\ &=10^{3z}-\frac{3}{2}(10^{3z}-10\cdot 10^{2z}) =-\frac{1}{2}10^{3z}+15\cdot 10^{2z}, \end{align*}and $a+b=-\frac{1}{2}+15=\boxed{29/2}$.

No other value of $a + b$ is possible for all members of $S$, because the triple $\left(\frac{1}{2}(1 + \sqrt{19}), \frac{1}{2}(1 - \sqrt{19}), 0\right)$ is in $S$, and for this ordered triple, the equation $x^3 + y^3 = a\cdot 10^{3z} + b\cdot 10^{2z}$ reduces to $a + b = 29/2$.

See Also

2005 AMC 12B (Problems • Answer Key • Resources)
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Problem 22 		Followed by
Problem 24
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All AMC 12 Problems and Solutions

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