Difference between revisions of "2015 AIME II Problems/Problem 8"

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==Problem==
 
==Problem==
  
Let <math>a</math> and <math>b</math> be positive integers satisfying <math>\frac{ab+1}{a+b} < \frac{3}{2}</math>. The maximum possible value of <math>\frac{a^3b^3+1}{a^3+b^3}</math> is <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>.
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<!-- don't remove the following tag, for PoTW on the Wiki front page--><onlyinclude>Let <math>a</math> and <math>b</math> be positive integers satisfying <math>\frac{ab+1}{a+b} < \frac{3}{2}</math>. The maximum possible value of <math>\frac{a^3b^3+1}{a^3+b^3}</math> is <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>.<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude>
  
==Solution==
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==Solution 1==
 
Let us call the quantity <math>\frac{a^3b^3+1}{a^3+b^3}</math> as <math>N</math> for convenience. Knowing that <math>a</math> and <math>b</math> are positive integers, we can legitimately rearrange the given inequality so that <math>a</math> is by itself, which makes it easier to determine the pairs of <math>(a, b)</math> that work. Doing so, we have <cmath>\frac{ab+1}{a+b} < \frac{3}{2}</cmath> <cmath>\implies 2ab + 2 < 3a + 3b \implies 2ab - 3a < 3b - 2</cmath> <cmath>\implies a < \frac{3b - 2}{2b - 3}.</cmath> Now, observe that if <math>b = 1</math> we have that <math>N = \frac{a + 1}{a + 1} = 1</math>, regardless of the value of <math>a</math>. If <math>a = 1</math>, we have the same result: that <math>N = \frac{b + 1}{b + 1} = 1</math>, regardless of the value of <math>b</math>. Hence, we want to find pairs of positive integers <math>(a, b)</math> existing such that neither <math>a</math> nor <math>b</math> is equal to <math>1</math>, and that the conditions given in the problem are satisfied in order to check that the maximum value for <math>N</math> is not <math>1</math>.
 
Let us call the quantity <math>\frac{a^3b^3+1}{a^3+b^3}</math> as <math>N</math> for convenience. Knowing that <math>a</math> and <math>b</math> are positive integers, we can legitimately rearrange the given inequality so that <math>a</math> is by itself, which makes it easier to determine the pairs of <math>(a, b)</math> that work. Doing so, we have <cmath>\frac{ab+1}{a+b} < \frac{3}{2}</cmath> <cmath>\implies 2ab + 2 < 3a + 3b \implies 2ab - 3a < 3b - 2</cmath> <cmath>\implies a < \frac{3b - 2}{2b - 3}.</cmath> Now, observe that if <math>b = 1</math> we have that <math>N = \frac{a + 1}{a + 1} = 1</math>, regardless of the value of <math>a</math>. If <math>a = 1</math>, we have the same result: that <math>N = \frac{b + 1}{b + 1} = 1</math>, regardless of the value of <math>b</math>. Hence, we want to find pairs of positive integers <math>(a, b)</math> existing such that neither <math>a</math> nor <math>b</math> is equal to <math>1</math>, and that the conditions given in the problem are satisfied in order to check that the maximum value for <math>N</math> is not <math>1</math>.
  
  
To avoid the possibility that <math>a = 1</math>, we want to find values of <math>b</math> such that <math>\frac{3b - 2}{2b - 3} > 2</math>. If we do this, we will have that <math>a < \frac{3b - 2}{2b - 3} = k</math>, where <math>k</math> is greater than <math>2</math>, and this allows us to choose values of <math>a</math> greater than <math>1</math>. Again, since <math>b</math> is a positive integer, and we want <math>b > 1</math>, we can legitimately multiply both sides of <math>\frac{3b - 2}{2b - 3} > 2</math> by <math>2b - 3</math> to get <math>3b - 2 > 4b - 6 \implies b < 4</math>. For <math>b = 3</math>, we have that <math>a < \frac{7}{3}</math>, so the only possibility for <math>a</math> greater than <math>1</math> is obviously <math>2</math>. Plugging these values into <math>N</math>, we have that <math>N = \frac{8(27) + 1}{8 + 27} = \frac{217}{35} = \frac{31}{5}</math>. For <math>b = 2</math>, we have that <math>a < \frac{4}{1} = 4</math>. Plugging <math>a = 2</math> and <math>b = 3</math> in for <math>N</math> yields the same result of <math>\frac{31}{5}</math>, but plugging <math>a = 2</math> and <math>b = 2</math> into <math>N</math> yields that <math>N = \frac{8(8) + 1}{8 + 8} = \frac{65}{16}</math>. Clearly, <math>\frac{31}{5}</math> is the largest value we can have for <math>N</math>, so our answer is <math>31 + 5 = \boxed{036}</math>.
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To avoid the possibility that <math>a = 1</math>, we want to find values of <math>b</math> such that <math>\frac{3b - 2}{2b - 3} > 2</math>. If we do this, we will have that <math>a < \frac{3b - 2}{2b - 3} = k</math>, where <math>k</math> is greater than <math>2</math>, and this allows us to choose values of <math>a</math> greater than <math>1</math>. Again, since <math>b</math> is a positive integer, and we want <math>b > 1</math>, we can legitimately multiply both sides of <math>\frac{3b - 2}{2b - 3} > 2</math> by <math>2b - 3</math> to get <math>3b - 2 > 4b - 6 \implies b < 4</math>. For <math>b = 3</math>, we have that <math>a < \frac{7}{3}</math>, so the only possibility for <math>a</math> greater than <math>1</math> is obviously <math>2</math>. Plugging these values into <math>N</math>, we have that <math>N = \frac{8(27) + 1}{8 + 27} = \frac{217}{35} = \frac{31}{5}</math>. For <math>b = 2</math>, we have that <math>a < \frac{4}{1} = 4</math>. Plugging <math>a = 3</math> and <math>b = 2</math> in for <math>N</math> yields the same result of <math>\frac{31}{5}</math>, but plugging <math>a = 2</math> and <math>b = 2</math> into <math>N</math> yields that <math>N = \frac{8(8) + 1}{8 + 8} = \frac{65}{16}</math>. Clearly, <math>\frac{31}{5}</math> is the largest value we can have for <math>N</math>, so our answer is <math>31 + 5 = \boxed{036}</math>.
  
==Solution 2 (Proof without words)==
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(Technically, we would have to find that b > 1 before dividing both sides of the inequality by 2b - 3, but otherwise this solution is correct)
<cmath>\frac{ab + 1}{a + b} < \frac{3}{2} \rightarrow 2ab + 2 < 3a + 3b,</cmath>
 
<cmath>\rightarrow 4ab - 6a - 6b + 4 < 0 \rightarrow (2a - 3)(2b - 3) < 5.</cmath>
 
  
<cmath>2a - 3, 2b - 3 \in \{x \neq 2k, k \in Z \}; \rightarrow</cmath>
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==Solution 2 (best solution)==
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<cmath>\frac{ab + 1}{a + b} < \frac{3}{2} \rightarrow 2ab + 2 < 3a + 3b \rightarrow 4ab - 6a - 6b + 4 < 0 \rightarrow (2a - 3)(2b - 3) < 5.</cmath>
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<cmath>2a - 3, 2b - 3 \in \{x \neq 2k, k \in Z \} \rightarrow</cmath> <math>(2a-3)</math> and <math>(2b-3)</math> each cannot be even or else <math>a</math> and <math>b</math> will not be integers
 
<cmath>(2a - 3)(2b - 3) = 1, 3 \rightarrow (2a - 3, 2b - 3) = (1, 1), (1, 3), (3, 1).</cmath>
 
<cmath>(2a - 3)(2b - 3) = 1, 3 \rightarrow (2a - 3, 2b - 3) = (1, 1), (1, 3), (3, 1).</cmath>
 
<cmath>(a, b) = (2, 2), (2, 3), (3, 2).</cmath>
 
<cmath>(a, b) = (2, 2), (2, 3), (3, 2).</cmath>
 
<cmath>\frac{a^3 b^3 + 1}{a^3 + b^3} = \frac{65}{16}, \frac{31}{5}.</cmath>
 
<cmath>\frac{a^3 b^3 + 1}{a^3 + b^3} = \frac{65}{16}, \frac{31}{5}.</cmath>
<cmath>\frac{31}{5} \rightarrow \boxed{36}.</cmath>
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<cmath>\frac{31}{5} \rightarrow \boxed{036}.</cmath>
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(<math>a=1 \rightarrow b=1, 2, 3... \rightarrow \frac{b^3+1}{b^3+1}=1</math>; <math>1<31/5</math>).
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==Solution 3==
  
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Notice that for <math>\frac{a^3b^3+1}{a^3+b^3}</math> to be maximized, <math>\frac{ab+1}{a+b}</math> has to be maximized. We simplify as above to <math>2ab + 2 < 3a + 3b</math>, which is <math>(a-\frac{3}{2})(b-\frac{3}{2}) < \frac{5}{4}</math>. To maximize, <math>a</math> has to be as close to <math>b</math> as possible, making <math>a</math> close to <math>\frac{3+\sqrt{5}}{2}</math>. Because <math>a</math> and <math>b</math> are positive integers, <math>a = 3</math>, and checking back gives <math>b = 2</math> as the maximum or the other way around, which the answer is thus <math>\frac{216+1}{27+8} = \frac{217}{35} = \frac{31}{5} \rightarrow \boxed{036}</math>.
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==Solution 4==
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Guess and check a few values of <math>a</math> and <math>b</math> you will find two things. One, that the highest values of <math>a</math> and <math>b</math> are the closest to <math>\frac{3}{2}</math>. Two, that the pair <math>(2,3)</math> is the highest possible value of <math>a, b</math>. So plugging in <math>a=2</math> and <math>b=3</math> we get <math>((2*3)^3)+1/(8+27)</math> = <math>\frac{217}{35}</math> = <math>\frac{31}{5}</math>, and <math>31+5 = 36</math>.
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Solution by Helloitsaaryan
 
==See also==
 
==See also==
 
{{AIME box|year=2015|n=II|num-b=7|num-a=9}}
 
{{AIME box|year=2015|n=II|num-b=7|num-a=9}}
 +
[[Category: Intermediate Algebra Problems]]
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 13:16, 5 February 2022

Problem

Let $a$ and $b$ be positive integers satisfying $\frac{ab+1}{a+b} < \frac{3}{2}$. The maximum possible value of $\frac{a^3b^3+1}{a^3+b^3}$ is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Solution 1

Let us call the quantity $\frac{a^3b^3+1}{a^3+b^3}$ as $N$ for convenience. Knowing that $a$ and $b$ are positive integers, we can legitimately rearrange the given inequality so that $a$ is by itself, which makes it easier to determine the pairs of $(a, b)$ that work. Doing so, we have \[\frac{ab+1}{a+b} < \frac{3}{2}\] \[\implies 2ab + 2 < 3a + 3b \implies 2ab - 3a < 3b - 2\] \[\implies a < \frac{3b - 2}{2b - 3}.\] Now, observe that if $b = 1$ we have that $N = \frac{a + 1}{a + 1} = 1$, regardless of the value of $a$. If $a = 1$, we have the same result: that $N = \frac{b + 1}{b + 1} = 1$, regardless of the value of $b$. Hence, we want to find pairs of positive integers $(a, b)$ existing such that neither $a$ nor $b$ is equal to $1$, and that the conditions given in the problem are satisfied in order to check that the maximum value for $N$ is not $1$.


To avoid the possibility that $a = 1$, we want to find values of $b$ such that $\frac{3b - 2}{2b - 3} > 2$. If we do this, we will have that $a < \frac{3b - 2}{2b - 3} = k$, where $k$ is greater than $2$, and this allows us to choose values of $a$ greater than $1$. Again, since $b$ is a positive integer, and we want $b > 1$, we can legitimately multiply both sides of $\frac{3b - 2}{2b - 3} > 2$ by $2b - 3$ to get $3b - 2 > 4b - 6 \implies b < 4$. For $b = 3$, we have that $a < \frac{7}{3}$, so the only possibility for $a$ greater than $1$ is obviously $2$. Plugging these values into $N$, we have that $N = \frac{8(27) + 1}{8 + 27} = \frac{217}{35} = \frac{31}{5}$. For $b = 2$, we have that $a < \frac{4}{1} = 4$. Plugging $a = 3$ and $b = 2$ in for $N$ yields the same result of $\frac{31}{5}$, but plugging $a = 2$ and $b = 2$ into $N$ yields that $N = \frac{8(8) + 1}{8 + 8} = \frac{65}{16}$. Clearly, $\frac{31}{5}$ is the largest value we can have for $N$, so our answer is $31 + 5 = \boxed{036}$.

(Technically, we would have to find that b > 1 before dividing both sides of the inequality by 2b - 3, but otherwise this solution is correct)

Solution 2 (best solution)

\[\frac{ab + 1}{a + b} < \frac{3}{2} \rightarrow 2ab + 2 < 3a + 3b \rightarrow 4ab - 6a - 6b + 4 < 0 \rightarrow (2a - 3)(2b - 3) < 5.\]

\[2a - 3, 2b - 3 \in \{x \neq 2k, k \in Z \} \rightarrow\] $(2a-3)$ and $(2b-3)$ each cannot be even or else $a$ and $b$ will not be integers \[(2a - 3)(2b - 3) = 1, 3 \rightarrow (2a - 3, 2b - 3) = (1, 1), (1, 3), (3, 1).\] \[(a, b) = (2, 2), (2, 3), (3, 2).\] \[\frac{a^3 b^3 + 1}{a^3 + b^3} = \frac{65}{16}, \frac{31}{5}.\] \[\frac{31}{5} \rightarrow \boxed{036}.\]

($a=1 \rightarrow b=1, 2, 3... \rightarrow \frac{b^3+1}{b^3+1}=1$; $1<31/5$).

Solution 3

Notice that for $\frac{a^3b^3+1}{a^3+b^3}$ to be maximized, $\frac{ab+1}{a+b}$ has to be maximized. We simplify as above to $2ab + 2 < 3a + 3b$, which is $(a-\frac{3}{2})(b-\frac{3}{2}) < \frac{5}{4}$. To maximize, $a$ has to be as close to $b$ as possible, making $a$ close to $\frac{3+\sqrt{5}}{2}$. Because $a$ and $b$ are positive integers, $a = 3$, and checking back gives $b = 2$ as the maximum or the other way around, which the answer is thus $\frac{216+1}{27+8} = \frac{217}{35} = \frac{31}{5} \rightarrow \boxed{036}$.

Solution 4

Guess and check a few values of $a$ and $b$ you will find two things. One, that the highest values of $a$ and $b$ are the closest to $\frac{3}{2}$. Two, that the pair $(2,3)$ is the highest possible value of $a, b$. So plugging in $a=2$ and $b=3$ we get $((2*3)^3)+1/(8+27)$ = $\frac{217}{35}$ = $\frac{31}{5}$, and $31+5 = 36$. Solution by Helloitsaaryan

See also

2015 AIME II (ProblemsAnswer KeyResources)
Preceded by
Problem 7
Followed by
Problem 9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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