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

Revision as of 19:15, 24 December 2020

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, $x^3 + y^3 = s*\frac{3t-s^2}{2} = s(15s-\frac{s^2}{2})$ . Since $s = 10^z$ , we find that $x^3 + y^3 = 15\times10^{2z} - (1/2)\times10^{3z}$ . 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)

@@ Line 43: / Line 43: @@
 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 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>
+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>
 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>.

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