Difference between revisions of "2024 AMC 10B Problems/Problem 13"

Latest revision as of 00:53, 20 November 2024

Problem

Positive integers $x$ and $y$ satisfy the equation $\sqrt{x} + \sqrt{y} = \sqrt{1183}$ . What is the minimum possible value of $x+y$ ?

$\textbf{(A) } 585 \qquad\textbf{(B) } 595 \qquad\textbf{(C) } 623 \qquad\textbf{(D) } 700 \qquad\textbf{(E) } 791$

Solution 1

Note that $\sqrt{1183}=13\sqrt7$ . Since $x$ and $y$ are positive integers, and $\sqrt{x}+\sqrt{y}=\sqrt{1183}$ , we can represent each value of $\sqrt{x}$ and $\sqrt{y}$ as the product of a positive integer and $\sqrt7$ . Let's say that $\sqrt{x}=m\sqrt7$ and $\sqrt{y}=n\sqrt7$ , where $m$ and $n$ are positive integers. This implies that $x+y=(\sqrt{x})^2+(\sqrt{y})^2=7m^2+7n^2=7(m^2+n^2)$ and that $m+n=13$ . WLOG, assume that ${m}\geq{n}$ . It is not hard to see that $x+y$ reaches its minimum when $m^2+n^2$ reaches its minimum. We now apply algebraic manipulation to get that $m^2+n^2=(m+n)^2-2mn$ . Since $m+n$ is determined, we now want $mn$ to reach its maximum. Since $m$ and $n$ are positive integers, we can use the AM-GM inequality to get that: $\frac{m+n}{2}\geq{\sqrt{mn}}$ . When $mn$ reaches its maximum, $\frac{m+n}{2}={\sqrt{mn}}$ . This implies that $m=n=\frac{13}{2}$ . However, this is not possible since $m$ and $n$ and integers. Under this constraint, we can see that $mn$ reaches its maximum when $m=7$ and $n=6$ . Therefore, the minimum possible value of $x+y$ is $7(m^2+n^2)=7(7^2+6^2)=\boxed{\textbf{(B)}595}$

~Bloggish

Solution 2 (Guessing & Answer Choices)

Set $x=y$ , giving the minimum possible values. The given equation becomes $\sqrt{x}=\sqrt{y}=\dfrac{\sqrt{1183}}{2}\implies x=y=\dfrac{1183}{4}.$ This means that $x+y=\dfrac{1183}{2}=591.5.$ Since this is closest to answer choice $\text{(B)}$ , the answer is $\boxed{\textbf{(B) }595}$ ~Neoronean ~Tacos_are_yummy_1 (latex)

Video Solution 1 by Pi Academy (Fast and Easy ⚡🚀)

https://youtu.be/YqKmvSR1Ckk?feature=shared

~ Pi Academy

Video Solution 2 by SpreadTheMathLove

https://www.youtube.com/watch?v=24EZaeAThuE

@@ Line 1: / Line 1: @@
+==Problem==
+Positive integers <math>x</math> and <math>y</math> satisfy the equation <math>\sqrt{x} + \sqrt{y} = \sqrt{1183}</math>. What is the minimum possible value of <math>x+y</math>?
+<math>\textbf{(A) } 585 \qquad\textbf{(B) } 595 \qquad\textbf{(C) } 623 \qquad\textbf{(D) } 700 \qquad\textbf{(E) } 791</math>
+==Solution 1==
+Note that <math>\sqrt{1183}=13\sqrt7</math>. Since <math>x</math> and <math>y</math> are positive integers, and <math>\sqrt{x}+\sqrt{y}=\sqrt{1183}</math>, we can represent each value of <math>\sqrt{x}</math> and <math>\sqrt{y}</math> as the product of a positive integer and <math>\sqrt7</math>. Let's say that <math>\sqrt{x}=m\sqrt7</math> and <math>\sqrt{y}=n\sqrt7</math>, where <math>m</math> and <math>n</math> are positive integers. This implies that <math>x+y=(\sqrt{x})^2+(\sqrt{y})^2=7m^2+7n^2=7(m^2+n^2)</math> and that <math>m+n=13</math>. WLOG, assume that <math>{m}\geq{n}</math>. It is not hard to see that <math>x+y</math> reaches its minimum when <math>m^2+n^2</math> reaches its minimum. We now apply algebraic manipulation to get that <math>m^2+n^2=(m+n)^2-2mn</math>. Since <math>m+n</math> is determined, we now want <math>mn</math> to reach its maximum. Since <math>m</math> and <math>n</math> are positive integers, we can use the AM-GM inequality to get that: <math>\frac{m+n}{2}\geq{\sqrt{mn}}</math>. When <math>mn</math> reaches its maximum, <math>\frac{m+n}{2}={\sqrt{mn}}</math>. This implies that <math>m=n=\frac{13}{2}</math>. However, this is not possible since <math>m</math> and <math>n</math> and integers. Under this constraint, we can see that <math>mn</math> reaches its maximum when <math>m=7</math> and <math>n=6</math>. Therefore, the minimum possible value of <math>x+y</math> is <math>7(m^2+n^2)=7(7^2+6^2)=\boxed{\textbf{(B)}595}</math>
+~[[User:Bloggish|Bloggish]]
+==Solution 2 (Guessing & Answer Choices)==
+Set <math>x=y</math>, giving the minimum possible values. The given equation becomes <cmath>\sqrt{x}=\sqrt{y}=\dfrac{\sqrt{1183}}{2}\implies x=y=\dfrac{1183}{4}.</cmath>This means that <cmath>x+y=\dfrac{1183}{2}=591.5.</cmath>Since this is closest to answer choice <math>\text{(B)}</math>, the answer is <math>\boxed{\textbf{(B) }595}</math> ~Neoronean ~Tacos_are_yummy_1 (latex)
+==Video Solution 1 by Pi Academy (Fast and Easy ⚡🚀)==
+https://youtu.be/YqKmvSR1Ckk?feature=shared
+~ Pi Academy
+==Video Solution 2 by SpreadTheMathLove==
+https://www.youtube.com/watch?v=24EZaeAThuE
+==See also==
+{{AMC10 box|year=2024|ab=B|num-b=12|num-a=14}}
+{{MAA Notice}}

2024 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
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