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 "2007 iTest Problems/Problem 16"

(Created page with "== Problem == How many lattice points lie within or on the border of the circle in the <math>xy</math>-plane defined by the equation <cmath>x^2+y^2=100</cmath> <math>\text{(A) }...")
 
(Graph for Problem 16)
 
(2 intermediate revisions by the same user not shown)
Line 21: Line 21:
  
 
== Solution ==
 
== Solution ==
 +
Use [[casework]] to divide the problem into two cases -- points on the coordinate axes and points not on the coordinate axes.
 +
 +
* For points on the axes, there are <math>10</math> points on each ray plus the origin, making a total of <math>41</math> points.
 +
* For points not on the axes, [[symmetry]] can be used by focusing on one quadrant then multiplying by four because the equation is a [[circle]] where the center is the origin.  Because the points are within the circle, <math>x^2 + y^2 \le 100</math>.  If <math>x = 9</math>, then <math>y \le 4</math>.  If <math>x = 8</math>, then <math>y \le 6</math>.  If <math>x = 7</math>, then <math>y \le 7</math>.  If <math>x = 6</math> or <math>x = 5</math>, then <math>y \le 8</math>.  Finally, if <math>x \le 4</math>, then <math>y \le 9</math>.  Altogether, there are a total of <math>9(4) + 8(2) + 7 + 6 + 4 = 69</math> points in the first quadrant, so there are a total of <math>69 \cdot 4 = 276</math> points not on the coordinate axes.
 +
 +
<asy>
 +
 +
import graph; size(9.22 cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black;
 +
real xmin=-1.2,xmax=10.2,ymin=-1.2,ymax=10.2;
 +
pen cqcqcq=rgb(0.75,0.75,0.75), evevff=rgb(0.9,0.9,1), zzttqq=rgb(0.6,0.2,0);
 +
 +
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy=1;
 +
for(real i=ceil(xmin/gx)*gx;i<=floor(xmax/gx)*gx;i+=gx) draw((i,ymin)--(i,ymax),gs); for(real i=ceil(ymin/gy)*gy;i<=floor(ymax/gy)*gy;i+=gy) draw((xmin,i)--(xmax,i),gs);
 +
Label laxis; laxis.p=fontsize(10);
 +
xaxis(xmin,xmax,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); yaxis(ymin,ymax,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true);
 +
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
 +
 +
draw(Arc((0,0),10,-5,95));
 +
for (int i=1; i < 10; ++i)
 +
{
 +
  for (int j=1; j < 10; ++j)
 +
  {
 +
    if (i*i + j*j <= 100)
 +
    {
 +
      dot((i,j));
 +
    }
 +
  }
 +
}
 +
 +
</asy>
 +
 +
In total, there are <math>276 + 41 = \boxed{\textbf{(M) } 317}</math> points within the circle.
 +
 +
==See Also==
 +
{{iTest box|year=2007|num-b=15|num-a=17}}
 +
 +
[[Category:Introductory Geometry Problems]]
 +
[[Category:Introductory Combinatorics Problems]]

Latest revision as of 20:28, 17 June 2018

Problem

How many lattice points lie within or on the border of the circle in the $xy$-plane defined by the equation \[x^2+y^2=100\]

$\text{(A) }1\qquad \text{(B) }2\qquad \text{(C) }4\qquad \text{(D) }5\qquad \text{(E) }41\qquad \text{(F) }42\qquad \text{(G) }69\qquad \text{(H) }76\qquad \text{(I) }130\qquad \\ \text{(J) }133\qquad \text{(K) }233\qquad \text{(L) }311\qquad \text{(M) }317\qquad \text{(N) }420\qquad \text{(O) }520\qquad \text{(P) }2007$

Solution

Use casework to divide the problem into two cases -- points on the coordinate axes and points not on the coordinate axes.

  • For points on the axes, there are $10$ points on each ray plus the origin, making a total of $41$ points.
  • For points not on the axes, symmetry can be used by focusing on one quadrant then multiplying by four because the equation is a circle where the center is the origin. Because the points are within the circle, $x^2 + y^2 \le 100$. If $x = 9$, then $y \le 4$. If $x = 8$, then $y \le 6$. If $x = 7$, then $y \le 7$. If $x = 6$ or $x = 5$, then $y \le 8$. Finally, if $x \le 4$, then $y \le 9$. Altogether, there are a total of $9(4) + 8(2) + 7 + 6 + 4 = 69$ points in the first quadrant, so there are a total of $69 \cdot 4 = 276$ points not on the coordinate axes.

[asy] import graph; size(9.22 cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-1.2,xmax=10.2,ymin=-1.2,ymax=10.2; pen cqcqcq=rgb(0.75,0.75,0.75), evevff=rgb(0.9,0.9,1), zzttqq=rgb(0.6,0.2,0); /*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy=1; for(real i=ceil(xmin/gx)*gx;i<=floor(xmax/gx)*gx;i+=gx) draw((i,ymin)--(i,ymax),gs); for(real i=ceil(ymin/gy)*gy;i<=floor(ymax/gy)*gy;i+=gy) draw((xmin,i)--(xmax,i),gs); Label laxis; laxis.p=fontsize(10); xaxis(xmin,xmax,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); yaxis(ymin,ymax,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); draw(Arc((0,0),10,-5,95)); for (int i=1; i < 10; ++i) { for (int j=1; j < 10; ++j) { if (i*i + j*j <= 100) { dot((i,j)); } } } [/asy]

In total, there are $276 + 41 = \boxed{\textbf{(M) } 317}$ points within the circle.

See Also

2007 iTest (Problems, Answer Key)
Preceded by:
Problem 15 		Followed by:
Problem 17
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 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • TB1 • TB2 • TB3 • TB4
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2007_iTest_Problems/Problem_16&oldid=95253"