Difference between revisions of "2010 AIME I Problems/Problem 9"

Latest revision as of 07:57, 29 August 2024

Problem

Let $(a,b,c)$ be the real solution of the system of equations $x^3 - xyz = 2$ , $y^3 - xyz = 6$ , $z^3 - xyz = 20$ . The greatest possible value of $a^3 + b^3 + c^3$ can be written in the form $\frac {m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .

Solution

Solution 1

Add the three equations to get $a^3 + b^3 + c^3 = 28 + 3abc$ . Now, let $abc = p$ . $a = \sqrt [3]{p + 2}$ , $b = \sqrt [3]{p + 6}$ and $c = \sqrt [3]{p + 20}$ , so $p = abc = (\sqrt [3]{p + 2})(\sqrt [3]{p + 6})(\sqrt [3]{p + 20})$ . Now cube both sides; the $p^3$ terms cancel out. Solve the remaining quadratic to get $p = - 4, - \frac {15}{7}$ . To maximize $a^3 + b^3 + c^3$ choose $p = - \frac {15}{7}$ and so the sum is $28 - \frac {45}{7} = \frac {196 - 45}{7}$ giving $151 + 7 = \fbox{158}$ .

Solution 2

This is almost the same as solution 1. Note $a^3 + b^3 + c^3 = 28 + 3abc$ . Next, let $k = a^3$ . Note that $b = \sqrt [3]{k + 4}$ and $c = \sqrt [3]{k + 18}$ , so we have $28 + 3\sqrt [3]{k(k+4)(k+18)} = 28+3abc=a^3+b^3+c^3=3k+22$ . Move 28 over, divide both sides by 3, then cube to get $k^3-6k^2+12k-8 = k^3+22k^2+18k$ . The $k^3$ terms cancel out, so solve the quadratic to get $k = -2, -\frac{1}{7}$ . We maximize $abc$ by choosing $k = -\frac{1}{7}$ , which gives us $a^3+b^3+c^3 = 3k + 22 = \frac{151}{7}$ . Thus, our answer is $151+7=\boxed{158}$ .

Solution 3

We have that $x^3 = 2 + xyz$ , $y^3 = 6 + xyz$ , and $z^3 = 20 + xyz$ . Multiplying the three equations, and letting $m = xyz$ , we have that $m^3 = (2+m)(6+m)(20+m)$ , and reducing, that $7m^2 + 43m + 60 = 0$ , which has solutions $m = -\frac{15}{7}, -4$ . Adding the three equations and testing both solutions, we find the answer of $\frac{151}{7}$ , so the desired quantity is $151 + 7 = \fbox{158}$ .

Video Solution by OmegaLearn

https://youtu.be/SpSuqWY01SE?t=1293

~ pi_is_3.14

Remark

It is tempting to add the equations and then use the well-known factorization $x^3+y^3+z^3-3xyz = (x+y+z)(x^2+y^2+z^2-xy-xz-yz)$ . Unfortunately such a factorization is just a red herring: it doesn't give much information on $a^3+b^3+c^3$ .

Another Remark

The real problem with adding the equations is that $x, y, z$ are real numbers based on the problem, but the adding trick only works when $x, y, z$ are integers.

Video Solution

https://youtu.be/LXct4j_rYfw (Video unavailable as of 20240829)

~Shreyas S

@@ Line 3: / Line 3: @@
 == Solution ==
+===Solution 1===
 Add the three equations to get <math>a^3 + b^3 + c^3 = 28 + 3abc</math>.  Now, let <math>abc = p</math>.  <math>a = \sqrt [3]{p + 2}</math>, <math>b = \sqrt [3]{p + 6}</math> and <math>c = \sqrt [3]{p + 20}</math>, so <math>p = abc = (\sqrt [3]{p + 2})(\sqrt [3]{p + 6})(\sqrt [3]{p + 20})</math>.  Now [[cube]] both sides; the <math>p^3</math> terms cancel out.  Solve the remaining [[quadratic]] to get <math>p = - 4, - \frac {15}{7}</math>.  To maximize <math>a^3 + b^3 + c^3</math> choose <math>p = - \frac {15}{7}</math> and so the sum is <math>28 - \frac {45}{7} = \frac {196 - 45}{7}</math> giving <math>151 + 7 = \fbox{158}</math>.
-== See also ==
+===Solution 2===
+This is almost the same as solution 1. Note <math>a^3 + b^3 + c^3 = 28 + 3abc</math>. Next, let <math>k = a^3</math>. Note that <math>b = \sqrt [3]{k + 4}</math> and  <math>c = \sqrt [3]{k + 18}</math>, so we have <math>28 + 3\sqrt [3]{k(k+4)(k+18)} = 28+3abc=a^3+b^3+c^3=3k+22</math>. Move 28 over, divide both sides by 3, then cube to get <math>k^3-6k^2+12k-8 = k^3+22k^2+18k</math>. The <math>k^3</math> terms cancel out, so solve the quadratic to get <math>k = -2, -\frac{1}{7}</math>. We maximize <math>abc</math> by choosing <math>k = -\frac{1}{7}</math>, which gives us <math>a^3+b^3+c^3 = 3k + 22 = \frac{151}{7}</math>. Thus, our answer is <math>151+7=\boxed{158}</math>.
+===Solution 3===
+We have that <math>x^3 = 2 + xyz</math>, <math>y^3 = 6 + xyz</math>, and <math>z^3 = 20 + xyz</math>. Multiplying the three equations, and letting <math>m = xyz</math>, we have that <math>m^3 = (2+m)(6+m)(20+m)</math>, and reducing, that <math>7m^2 + 43m + 60 = 0</math>, which has solutions <math>m = -\frac{15}{7}, -4</math>. Adding the three equations and testing both solutions, we find the answer of <math>\frac{151}{7}</math>, so the desired quantity is <math>151 + 7 = \fbox{158}</math>.
+== Video Solution by OmegaLearn ==
+https://youtu.be/SpSuqWY01SE?t=1293
+~ pi_is_3.14
+== Remark ==
+It is tempting to add the equations and then use the well-known factorization <math>x^3+y^3+z^3-3xyz = (x+y+z)(x^2+y^2+z^2-xy-xz-yz)</math>. Unfortunately such a factorization is just a red herring: it doesn't give much information on <math>a^3+b^3+c^3</math>.
+== Another Remark ==
+The real problem with adding the equations is that <math>x, y, z</math> are real numbers based on the problem, but the adding trick only works when <math>x, y, z</math> are integers.
+==Video Solution==
+https://youtu.be/LXct4j_rYfw (Video unavailable as of 20240829)
+~Shreyas S
+== See Also ==
 {{AIME box|year=2010|num-b=8|num-a=10|n=I}}
 [[Category:Intermediate Algebra Problems]]
+{{MAA Notice}}

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