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Difference between revisions of "2000 AIME I Problems/Problem 7"

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Since <math>x+(1/z)=5, 1=z(5-x)=xyz</math>, so <math>5-x=xy</math>. Also, <math>y=29-(1/x)</math> by the second equation. Substitution gives <math>x=1/5</math>, <math>y=24</math>, and <math>z=5/24</math>, so the answer is 4+1 which is equal to <math>5</math>.
 
Since <math>x+(1/z)=5, 1=z(5-x)=xyz</math>, so <math>5-x=xy</math>. Also, <math>y=29-(1/x)</math> by the second equation. Substitution gives <math>x=1/5</math>, <math>y=24</math>, and <math>z=5/24</math>, so the answer is 4+1 which is equal to <math>5</math>.
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=== Solution 4 ===
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(Hybrid between 1/2)
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Because <math>xyz = 1, \hspace{0.15cm} \frac{1}{x} = yz, \hspace{0.15cm} \frac{1}{y} = xz, </math> and <math>\hspace{0.05cm}\frac{1}{z} = xy</math>. Substituting and factoring, we get <math>x(y+1) = 5</math>, <math>\hspace{0.15cm}y(z+1) = 29</math>, and <math>\hspace{0.05cm}z(x+1) = k</math>. Multiplying them all together, we get, <math>xyz(x+1)(y+1)(z+1) = 145k</math>, but <math>xyz</math> is <math>1</math>, and by the Identity property of multiplication, we can take it out. So, in the end, we get <math>(x+1)(y+1)(z+1) = 145k</math>. And, we can expand this to get <math>xyz+xy+yz+xz+x+y+z+1 = 145k</math>, and if we make a substitution for <math>xyz</math>, and rearrange the terms, we get <math>xy+yz+xz+x+y+z = 145k-2</math> This will be important.
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Now, lets add the 3 equations <math>x(y+1) = 5, \hspace{0.15cm}y(z+1) = 29 </math>, and <math>\hspace{0.05cm}z(x+1) = k</math>. We use the expand the Left hand sides, then, we add the equations to get <math>xy+yz+xz+x+y+z = k+34</math> Notice that the LHS of this equation matches the LHS equation that I said was important. So, the RHS of both equations are equal, and thus <math>145k-2 = k+34</math> We move all constant terms to the right, and all linear terms to the left, to get <math>144k = 36</math>, so <math>k = \frac{1}{4}</math> which gives an answer of <math>1+4 = \boxed{005}</math>
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-AlexLikeMath
  
 
== See also ==
 
== See also ==

Revision as of 15:27, 26 June 2019

Problem

Suppose that $x,$ $y,$ and $z$ are three positive numbers that satisfy the equations $xyz = 1,$ $x + \frac {1}{z} = 5,$ and $y + \frac {1}{x} = 29.$ Then $z + \frac {1}{y} = \frac {m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.


Solution 1

We can rewrite $xyz=1$ as $\frac{1}{z}=xy$.

Substituting into one of the given equations, we have \[x+xy=5\] \[x(1+y)=5\] \[\frac{1}{x}=\frac{1+y}{5}.\]

We can substitute back into $y+\frac{1}{x}=29$ to obtain \[y+\frac{1+y}{5}=29\] \[5y+1+y=145\] \[y=24.\]

We can then substitute once again to get \[x=\frac15\] \[z=\frac{5}{24}.\] Thus, $z+\frac1y=\frac{5}{24}+\frac{1}{24}=\frac{1}{4}$, so $m+n=\boxed{005}$.

Solution 2

Let $r = \frac{m}{n} = z + \frac {1}{y}$.

\begin{align*} (5)(29)(r)&=\left(x + \frac {1}{z}\right)\left(y + \frac {1}{x}\right)\left(z + \frac {1}{y}\right)\\ &=xyz + \frac{xy}{y} + \frac{xz}{x} + \frac{yz}{z} + \frac{x}{xy} + \frac{y}{yz} + \frac{z}{xz} + \frac{1}{xyz}\\ &=1 + x + z + y + \frac{1}{y} + \frac{1}{z} + \frac{1}{x} + \frac{1}{1}\\ &=2 + \left(x + \frac {1}{z}\right) + \left(y + \frac {1}{x}\right) + \left(z + \frac {1}{y}\right)\\ &=2 + 5 + 29 + r\\ &=36 + r \end{align*}

Thus $145r = 36+r \Rightarrow 144r = 36 \Rightarrow r = \frac{36}{144} = \frac{1}{4}$. So $m + n = 1 + 4 = \boxed{5}$.

Solution 3

Since $x+(1/z)=5, 1=z(5-x)=xyz$, so $5-x=xy$. Also, $y=29-(1/x)$ by the second equation. Substitution gives $x=1/5$, $y=24$, and $z=5/24$, so the answer is 4+1 which is equal to $5$.

Solution 4

(Hybrid between 1/2)

Because $xyz = 1, \hspace{0.15cm} \frac{1}{x} = yz, \hspace{0.15cm} \frac{1}{y} = xz,$ and $\hspace{0.05cm}\frac{1}{z} = xy$. Substituting and factoring, we get $x(y+1) = 5$, $\hspace{0.15cm}y(z+1) = 29$, and $\hspace{0.05cm}z(x+1) = k$. Multiplying them all together, we get, $xyz(x+1)(y+1)(z+1) = 145k$, but $xyz$ is $1$, and by the Identity property of multiplication, we can take it out. So, in the end, we get $(x+1)(y+1)(z+1) = 145k$. And, we can expand this to get $xyz+xy+yz+xz+x+y+z+1 = 145k$, and if we make a substitution for $xyz$, and rearrange the terms, we get $xy+yz+xz+x+y+z = 145k-2$ This will be important.


Now, lets add the 3 equations $x(y+1) = 5, \hspace{0.15cm}y(z+1) = 29$, and $\hspace{0.05cm}z(x+1) = k$. We use the expand the Left hand sides, then, we add the equations to get $xy+yz+xz+x+y+z = k+34$ Notice that the LHS of this equation matches the LHS equation that I said was important. So, the RHS of both equations are equal, and thus $145k-2 = k+34$ We move all constant terms to the right, and all linear terms to the left, to get $144k = 36$, so $k = \frac{1}{4}$ which gives an answer of $1+4 = \boxed{005}$

-AlexLikeMath

See also

2000 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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