Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2017 AMC 10A Problems/Problem 17"
Line 1: Line 1:
 
==Problem==
 
==Problem==
−
Distinct points <math>P</math>, <math>Q</math>, <math>R</math>, <math>S</math> lie on the circle <math>x^2+y^2=25</math> and have integer coordinates. The distances <math>PQ</math> and <math>RS</math>  are irrational numbers. What is the greatest possible value of the ratio <math>\frac{PQ}{RS}</math>?
+
Distinct points <math>P</math>, <math>Q</math>, <math>R</math>, <math>S</math> lie on the circle <math>x^{2}+y^{2}=25</math> and have integer coordinates. The distances <math>PQ</math> and <math>RS</math>  are irrational numbers. What is the greatest possible value of the ratio <math>\frac{PQ}{RS}</math>?
  
 
<math>\mathrm{(A)}\ 3\qquad\mathrm{(B)}\ 5\qquad\mathrm{(C)}\ 3\sqrt{5}\qquad\mathrm{(D)}\ 7\qquad\mathrm{(E)}\ 5\sqrt{2}</math>
 
<math>\mathrm{(A)}\ 3\qquad\mathrm{(B)}\ 5\qquad\mathrm{(C)}\ 3\sqrt{5}\qquad\mathrm{(D)}\ 7\qquad\mathrm{(E)}\ 5\sqrt{2}</math>
Line 6: Line 6:
 
==Solution==
 
==Solution==
  
−
Because <math>P</math>, <math>Q</math>, <math>R</math>, and <math>S</math> are integers there are only a few coordinates that actually satisfy the equation. The coordinates are <math>(\pm 3,\pm 4), (\pm 4, \pm 3), (0,\pm 5),</math> and <math>(\pm 5,0).</math>
+
Because <math>P</math>, <math>Q</math>, <math>R</math>, and <math>S</math> are integers there are only a few coordinates that actually satisfy the equation. The coordinates are <math>(\pm 3,\pm 4), (\pm 4, \pm 3), (0,\pm 5),</math> and <math>(\pm 5,0).</math> We want to maximize <math>PQ</math> and minimize <math>RS.</math> They also have to be the square root of something, because they are both irrational. The greatest value of <math>PQ</math> happens when it <math>P</math> and <math>Q</math> are almost directly across from each other and are in different quadrants. For example, the endpoints of the segment could be <math>(5,0)</math> and <math>(-4,3)</math> because the two points are almost across from each other. The least value of <math>RS</math> is when the two endpoints are in the same quadrant and are very close to each other. This can occur when, for example, <math>R</math> is <math>(3,4)</math> and <math>S</math> is <math>(4,3).</math> They are in the same quadrant and no other point on the circle with integer coordinates is closer to the point <math>(3,4)</math> than <math>(4,3).</math> Using the distance formula, we get that <math>PQ</math> is <math>\sqrt{90}</math> and that <math>RS</math> is <math>\sqrt{2}.</math> <math>\frac{\sqrt{90}}{\sqrt{2}}=\sqrt{45}=\boxed{\mathrm{(C)}\ 3\sqrt{5}}</math>
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2017|ab=A|num-b=16|num-a=18}}
 
{{AMC10 box|year=2017|ab=A|num-b=16|num-a=18}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 17:21, 8 February 2017

Problem

Distinct points $P$, $Q$, $R$, $S$ lie on the circle $x^{2}+y^{2}=25$ and have integer coordinates. The distances $PQ$ and $RS$ are irrational numbers. What is the greatest possible value of the ratio $\frac{PQ}{RS}$?

$\mathrm{(A)}\ 3\qquad\mathrm{(B)}\ 5\qquad\mathrm{(C)}\ 3\sqrt{5}\qquad\mathrm{(D)}\ 7\qquad\mathrm{(E)}\ 5\sqrt{2}$

Solution

Because $P$, $Q$, $R$, and $S$ are integers there are only a few coordinates that actually satisfy the equation. The coordinates are $(\pm 3,\pm 4), (\pm 4, \pm 3), (0,\pm 5),$ and $(\pm 5,0).$ We want to maximize $PQ$ and minimize $RS.$ They also have to be the square root of something, because they are both irrational. The greatest value of $PQ$ happens when it $P$ and $Q$ are almost directly across from each other and are in different quadrants. For example, the endpoints of the segment could be $(5,0)$ and $(-4,3)$ because the two points are almost across from each other. The least value of $RS$ is when the two endpoints are in the same quadrant and are very close to each other. This can occur when, for example, $R$ is $(3,4)$ and $S$ is $(4,3).$ They are in the same quadrant and no other point on the circle with integer coordinates is closer to the point $(3,4)$ than $(4,3).$ Using the distance formula, we get that $PQ$ is $\sqrt{90}$ and that $RS$ is $\sqrt{2}.$ $\frac{\sqrt{90}}{\sqrt{2}}=\sqrt{45}=\boxed{\mathrm{(C)}\ 3\sqrt{5}}$

See Also

2017 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_10A_Problems/Problem_17&oldid=82999"