Difference between revisions of "2014 AIME I Problems/Problem 14"

Latest revision as of 14:09, 13 August 2022

Problem 14

Let $m$ be the largest real solution to the equation

$\frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}=x^2-11x-4$

There are positive integers $a$ , $b$ , and $c$ such that $m=a+\sqrt{b+\sqrt{c}}$ . Find $a+b+c$ .

Solution

The first step is to notice that the 4 on the right hand side can simplify the terms on the left hand side. If we distribute 1 to $\frac{3}{x-3}$ , then the fraction becomes of the form $\frac{x}{x - 3}$ . A similar cancellation happens with the other four terms. If we assume $x = 0$ is not the highest solution (which is true given the answer format) we can cancel the common factor of $x$ from both sides of the equation.

$\frac{1}{x - 3} + \frac{1}{x - 5} + \frac{1}{x - 17} + \frac{1}{x - 19} = x - 11$

Then, if we make the substitution $y = x - 11$ , we can further simplify.

$\frac{1}{y + 8} + \frac{1}{y + 6} + \frac{1}{y - 6} + \frac{1}{y - 8} =y$

If we group and combine the terms of the form $y - n$ and $y + n$ , we get this equation:

$\frac{2y}{y^2 - 64} + \frac{2y}{y^2 - 36} = y$

Then, we can cancel out a $y$ from both sides, knowing that $x = 11$ is not a possible solution given the answer format. After we do that, we can make the final substitution $z = y^2$ .

$\frac{2}{z - 64} + \frac{2}{z - 36} = 1$

$2z - 128 + 2z - 72 = (z - 64)(z - 36)$

$4z - 200 = z^2 - 100z + 64(36)$

$z^2 - 104z + 2504 = 0$

Using the quadratic formula, we get that the largest solution for $z$ is $z = 52 + 10\sqrt{2}$ . Then, repeatedly substituting backwards, we find that the largest value of $x$ is $11 + \sqrt{52 + \sqrt{200}}$ . The answer is thus $11 + 52 + 200 = \boxed{263}$

Video Solution by Punxsutawney Phil

https://youtu.be/pNsmv333SE0

Video Solution by Mathematical Dexterity (Pure magic!)

https://www.youtube.com/watch?v=7b7IPOYZbrk

Video Solution: Math Bear presents... A Fun Algebra Equation Problem

https://www.youtube.com/watch?v=1hdYnbm4Tvo

@@ Line 8: / Line 8: @@
 == Solution ==
-The first step is to notice that the 4 on the right hand side can simplify the terms on the left hand side. If we distribute 1 to <math>3/x-3</math>, then the fraction becomes of the form <math>(x/x - 3)</math>. A similar cancellation happens with the other four terms.
+The first step is to notice that the 4 on the right hand side can simplify the terms on the left hand side. If we distribute 1 to <math>\frac{3}{x-3}</math>, then the fraction becomes of the form <math>\frac{x}{x - 3}</math>. A similar cancellation happens with the other four terms. If we assume <math>x = 0</math> is not the highest solution (which is true given the answer format) we can cancel the common factor of <math>x</math> from both sides of the equation.
-----
+<math>\frac{1}{x - 3} + \frac{1}{x - 5} + \frac{1}{x - 17} + \frac{1}{x - 19} = x - 11</math>
+Then, if we make the substitution <math>y = x - 11</math>, we can further simplify.
+<math>\frac{1}{y + 8} + \frac{1}{y + 6} + \frac{1}{y - 6} + \frac{1}{y - 8} =y </math>
+If we group and combine the terms of the form  <math>y - n</math> and   <math> y + n</math>, we get this equation:
+<math>\frac{2y}{y^2 - 64} + \frac{2y}{y^2 - 36} = y</math>
+Then, we can cancel out a <math>y</math> from both sides, knowing that <math>x = 11</math> is not a possible solution given the answer format.
+After we do that, we can make the final substitution <math>z = y^2</math>.
+<math>\frac{2}{z - 64} + \frac{2}{z - 36} = 1</math>
+<math>2z - 128 + 2z - 72 = (z - 64)(z - 36)</math>
+<math>4z -  200 = z^2 - 100z + 64(36)</math>
+<math>z^2 - 104z + 2504 = 0</math>
+Using the quadratic formula, we get that the largest solution for <math>z</math> is <math>z = 52 + 10\sqrt{2}</math>. Then, repeatedly substituting backwards, we find that the largest value of <math>x</math> is <math>11 + \sqrt{52 + \sqrt{200}}</math>. The answer is thus <math>11 + 52 + 200 = \boxed{263}</math>
+==Video Solution by Punxsutawney Phil==
+https://youtu.be/pNsmv333SE0
+==Video Solution by Mathematical Dexterity (Pure magic!)==
+https://www.youtube.com/watch?v=7b7IPOYZbrk
+==Video Solution: Math Bear presents... A Fun Algebra Equation Problem==
+https://www.youtube.com/watch?v=1hdYnbm4Tvo
 == See also ==
 {{AIME box|year=2014|n=I|num-b=13|num-a=15}}
 {{MAA Notice}}

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