2024 AMC 10B Problems/Problem 13

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

A similar method is to take $y=1183-26\sqrt{7x}-x^2$ , then noting $x=7a^2$ and bashing to find the value of a where x is closest to y.

~meihk_neiht

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)

Note: If using a solution similar to this one it is recommended to still find valid x and y that add to 595.

~meikh neiht

Solution 3 (quick, no guessing)

Square both sides of the equation to get $2\cdot \sqrt{xy} + (x + y) = 1183.$ We can plug in possible values for $x + y$ based on the answer choices. If we do $585,$ we get $\sqrt{xy} = 299 \implies xy = 299^2.$ So $y = \frac{299^2}{x}.$ Then we have $x^2 + 299^2 = 585x,$ which has no real solutions for $x$ . Following the same process for $595$ we obtain the equation $x^2 + 294^2 = 595x,$ which can be factored as $(x - 252)(x - 343) = 0,$ so this equation has two integer solutions for $x$ (these represent the least possible values of $x$ and $y$ ): $252$ and $343.$ So the answer must be $\boxed{\textbf{(B) }595}$ .

~grogg007

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

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

~ Pi Academy

Video Solution 2 by SpreadTheMathLove

https://youtu.be/7ZKvxU6c75g?si=KvM1x612u6fig_Xc

2024 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

Page

Toolbox

Search