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 "2016 AMC 12B Problems/Problem 11"

m (→‎Solution)
m (typo fix)
 
(8 intermediate revisions by 7 users not shown)
Line 11: Line 11:
 
==Solution==
 
==Solution==
 
Solution by e_power_pi_times_i
 
Solution by e_power_pi_times_i
 +
Revised by Kinglogic and RJ5303707
  
 +
<asy>
 +
Label l;
 +
l.p=fontsize(8);
 +
xaxis(-1,8,Ticks(l, 1.0));
 +
yaxis(-1,16,Ticks(l, 1.0));
 +
real f(real x)
 +
{
 +
return x * pi;
 +
}
 +
D(graph(f,-1/pi,5.1));
 +
D((5.1,-1)--(5.1,16));
 +
D((-1,-0.1)--(8,-0.1));
 +
for(int x = 0; x < 5.1; ++x) {
 +
for(int y = 0; y < 16; ++y) {
 +
    if(x * pi > y) {
 +
        D((x,y));
 +
        }
 +
    }
 +
}
 +
</asy>
 +
(red shows lattice points within the triangle)
  
−
(Note: diagram is needed)
+
If we draw a picture showing the triangle, we see that it would be easier to count the squares vertically and not horizontally. The upper bound is <math>16</math> squares <math>(y=5.1*\pi)</math>, and the limit for the <math>x</math>-value is <math>5</math> squares. First we count the <math>1*1</math> squares. In the rightmost column, there are <math>12</math> squares with length <math>1</math> because <math>y=4*\pi</math> generates squares from <math>(4,0)</math> to <math>(4,4\pi)</math>, and continuing on we have <math>9</math>, <math>6</math>, and <math>3</math> for <math>x</math>-values for <math>1</math>, <math>2</math>, and <math>3</math> in the equation <math>y=\pi x</math>. So there are <math>12+9+6+3 = 30</math> squares with length <math>1</math> in the figure.
  
−
If we draw a picture showing the triangle, we see that it would be easier to count the squares vertically and not horizontally. The upper bound is <math>16</math> squares <math>(y=5.1*\pi)</math>, and the limit for the <math>x</math>-value is <math>5</math> squares. First we count the <math>1*1</math> squares. In the back row, there are <math>12</math> squares with length <math>1</math> because <math>y=4*\pi</math> generates squares from <math>(4,0)</math> to <math>(4,4\pi)</math>, and continuing on we have <math>9</math>, <math>6</math>, and <math>3</math> for <math>x</math>-values for <math>1</math>, <math>2</math>, and <math>3</math> in the equation <math>y=\pi x</math>. So there are <math>12+9+6+3 = 30</math> squares with length <math>1</math> in the figure. For <math>2*2</math> squares, each square takes up <math>2</math> un left and <math>2</math> un up. Squares can also overlap. For <math>2*2</math> squares, the back row stretches from <math>(3,0)</math> to <math>(3,3\pi)</math>, so there are <math>8</math> squares with length <math>2</math> in a <math>2</math> by <math>9</math> box. Repeating the process, the next row stretches from <math>(2,0)</math> to <math>(2,2\pi)</math>, so there are <math>5</math> squares. Continuing and adding up in the end, there are <math>8+5+2=15</math> squares with length <math>2</math> in the figure. Squares with length <math>3</math> in the back row start at <math>(2,0)</math> and end at <math>(2,2\pi)</math>, so there are <math>4</math> such squares in the back row. As the front row starts at <math>(1,0)</math> and ends at <math>(1,\pi)</math> there are <math>4+1=5</math> squares with length <math>3</math>. As squares with length <math>4</math> would not fit in the triangle, the answer is <math>30+15+5</math> which is <math>\boxed{\textbf{B)}\ 50}</math>.
+
For <math>2*2</math> squares, each square takes up <math>2</math> units left and <math>2</math> units up. Squares can also overlap. For <math>2*2</math> squares, the rightmost column stretches from <math>(3,0)</math> to <math>(3,3\pi)</math>, so there are <math>8</math> squares with length <math>2</math> in a <math>2</math> by <math>9</math> box. Repeating the process, the next column stretches from <math>(2,0)</math> to <math>(2,2\pi)</math>, so there are <math>5</math> squares. Continuing and adding up in the end, there are <math>8+5+2=15</math> squares with length <math>2</math> in the figure.  
 +
 
 +
Squares with length <math>3</math> in the rightmost column start at <math>(2,0)</math> and end at <math>(2,2\pi)</math>, so there are <math>4</math> such squares in the right column. As the left row starts at <math>(1,0)</math> and ends at <math>(1,\pi)</math> there are <math>4+1=5</math> squares with length <math>3</math>. As squares with length <math>4</math> would not fit in the triangle, the answer is <math>30+15+5</math> which is <math>\boxed{\textbf{(D)}\ 50}</math>.
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2016|ab=B|num-b=10|num-a=12}}
 
{{AMC12 box|year=2016|ab=B|num-b=10|num-a=12}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 17:05, 20 October 2024

Problem

How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x$, the line $y=-0.1$ and the line $x=5.1?$

$\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 41 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\ 57$

Solution

Solution by e_power_pi_times_i Revised by Kinglogic and RJ5303707

[asy] Label l; l.p=fontsize(8); xaxis(-1,8,Ticks(l, 1.0)); yaxis(-1,16,Ticks(l, 1.0)); real f(real x) { return x * pi; } D(graph(f,-1/pi,5.1)); D((5.1,-1)--(5.1,16)); D((-1,-0.1)--(8,-0.1)); for(int x = 0; x < 5.1; ++x) { for(int y = 0; y < 16; ++y) { if(x * pi > y) { D((x,y)); } } } [/asy] (red shows lattice points within the triangle)

If we draw a picture showing the triangle, we see that it would be easier to count the squares vertically and not horizontally. The upper bound is $16$ squares $(y=5.1*\pi)$, and the limit for the $x$-value is $5$ squares. First we count the $1*1$ squares. In the rightmost column, there are $12$ squares with length $1$ because $y=4*\pi$ generates squares from $(4,0)$ to $(4,4\pi)$, and continuing on we have $9$, $6$, and $3$ for $x$-values for $1$, $2$, and $3$ in the equation $y=\pi x$. So there are $12+9+6+3 = 30$ squares with length $1$ in the figure.

For $2*2$ squares, each square takes up $2$ units left and $2$ units up. Squares can also overlap. For $2*2$ squares, the rightmost column stretches from $(3,0)$ to $(3,3\pi)$, so there are $8$ squares with length $2$ in a $2$ by $9$ box. Repeating the process, the next column stretches from $(2,0)$ to $(2,2\pi)$, so there are $5$ squares. Continuing and adding up in the end, there are $8+5+2=15$ squares with length $2$ in the figure.

Squares with length $3$ in the rightmost column start at $(2,0)$ and end at $(2,2\pi)$, so there are $4$ such squares in the right column. As the left row starts at $(1,0)$ and ends at $(1,\pi)$ there are $4+1=5$ squares with length $3$. As squares with length $4$ would not fit in the triangle, the answer is $30+15+5$ which is $\boxed{\textbf{(D)}\ 50}$.

See Also

2016 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
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 12 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=2016_AMC_12B_Problems/Problem_11&oldid=230194"