Art of Problem Solving
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
During AMC 10A/12A testing, the AoPS Wiki is in read-only mode and no edits can be made.

2001 AMC 10 Problems/Problem 5: Difference between revisions

← Older edit
Mathloveryeah (talk | contribs)
editor
266 edits
 
(10 intermediate revisions by 9 users not shown)
Line 1: Line 1:
== Problem ==
== Problem ==


How many of the twelve pentominoes pictured below at least one line of symmetry?
How many of the twelve pentominoes pictured below have at least one line of symmetry?


<math> \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7 </math>
<asy>
unitsize(5mm);
defaultpen(linewidth(1pt));
draw(shift(2,0)*unitsquare);
draw(shift(2,1)*unitsquare);
draw(shift(2,2)*unitsquare);
draw(shift(1,2)*unitsquare);
draw(shift(0,2)*unitsquare);
draw(shift(2,4)*unitsquare);
draw(shift(2,5)*unitsquare);
draw(shift(2,6)*unitsquare);
draw(shift(1,5)*unitsquare);
draw(shift(0,5)*unitsquare);
draw(shift(4,8)*unitsquare);
draw(shift(3,8)*unitsquare);
draw(shift(2,8)*unitsquare);
draw(shift(1,8)*unitsquare);
draw(shift(0,8)*unitsquare);
draw(shift(6,8)*unitsquare);
draw(shift(7,8)*unitsquare);
draw(shift(8,8)*unitsquare);
draw(shift(9,8)*unitsquare);
draw(shift(9,9)*unitsquare);
draw(shift(6,5)*unitsquare);
draw(shift(7,5)*unitsquare);
draw(shift(8,5)*unitsquare);
draw(shift(7,6)*unitsquare);
draw(shift(7,4)*unitsquare);
draw(shift(6,1)*unitsquare);
draw(shift(7,1)*unitsquare);
draw(shift(8,1)*unitsquare);
draw(shift(6,0)*unitsquare);
draw(shift(7,2)*unitsquare);
draw(shift(11,8)*unitsquare);
draw(shift(12,8)*unitsquare);
draw(shift(13,8)*unitsquare);
draw(shift(14,8)*unitsquare);
draw(shift(13,9)*unitsquare);
draw(shift(11,5)*unitsquare);
draw(shift(12,5)*unitsquare);
draw(shift(13,5)*unitsquare);
draw(shift(11,6)*unitsquare);
draw(shift(13,4)*unitsquare);
draw(shift(11,1)*unitsquare);
draw(shift(12,1)*unitsquare);
draw(shift(13,1)*unitsquare);
draw(shift(13,2)*unitsquare);
draw(shift(14,2)*unitsquare);
draw(shift(16,8)*unitsquare);
draw(shift(17,8)*unitsquare);
draw(shift(18,8)*unitsquare);
draw(shift(17,9)*unitsquare);
draw(shift(18,9)*unitsquare);
draw(shift(16,5)*unitsquare);
draw(shift(17,6)*unitsquare);
draw(shift(18,5)*unitsquare);
draw(shift(16,6)*unitsquare);
draw(shift(18,6)*unitsquare);
draw(shift(16,0)*unitsquare);
draw(shift(17,0)*unitsquare);
draw(shift(17,1)*unitsquare);
draw(shift(18,1)*unitsquare);
draw(shift(18,2)*unitsquare);</asy>
 
<math>\textbf{(A) } 3 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } 6 \qquad\textbf{(E) } 7</math>


== Solution ==
== Solution ==


Here is the picture: http://www.artofproblemsolving.com/Forum/download/file.php?id=6659&&mode=view
[[File:Pentonimoes.gif]]


The ones with lines over the shapes have at least one line of symmetry. Counting the number of shapes that have line(s) on them,
The ones with lines over the shapes have at least one line of symmetry. Counting the number of shapes that have line(s) on them,
we find <math> \boxed{\textbf{(D)} 6} </math> pentominoes.
we find <math> \boxed{\textbf{(D)}\ 6} </math> pentominoes.
 
==Video Solution by Daily Dose of Math==
 
https://youtu.be/svFpNvUUY7E?si=CloMWtqbbhBNgWy_
 
~Thesmartgreekmathdude


== Solution ==
== See Also ==
 
{{AMC10 box|year=2001|num-b=4|num-a=6}}
{{MAA Notice}}
[[Category: Introductory Geometry Problems]]

Latest revision as of 16:13, 18 October 2025

Problem

How many of the twelve pentominoes pictured below have at least one line of symmetry?

[asy] unitsize(5mm); defaultpen(linewidth(1pt)); draw(shift(2,0)*unitsquare); draw(shift(2,1)*unitsquare); draw(shift(2,2)*unitsquare); draw(shift(1,2)*unitsquare); draw(shift(0,2)*unitsquare); draw(shift(2,4)*unitsquare); draw(shift(2,5)*unitsquare); draw(shift(2,6)*unitsquare); draw(shift(1,5)*unitsquare); draw(shift(0,5)*unitsquare); draw(shift(4,8)*unitsquare); draw(shift(3,8)*unitsquare); draw(shift(2,8)*unitsquare); draw(shift(1,8)*unitsquare); draw(shift(0,8)*unitsquare); draw(shift(6,8)*unitsquare); draw(shift(7,8)*unitsquare); draw(shift(8,8)*unitsquare); draw(shift(9,8)*unitsquare); draw(shift(9,9)*unitsquare); draw(shift(6,5)*unitsquare); draw(shift(7,5)*unitsquare); draw(shift(8,5)*unitsquare); draw(shift(7,6)*unitsquare); draw(shift(7,4)*unitsquare); draw(shift(6,1)*unitsquare); draw(shift(7,1)*unitsquare); draw(shift(8,1)*unitsquare); draw(shift(6,0)*unitsquare); draw(shift(7,2)*unitsquare); draw(shift(11,8)*unitsquare); draw(shift(12,8)*unitsquare); draw(shift(13,8)*unitsquare); draw(shift(14,8)*unitsquare); draw(shift(13,9)*unitsquare); draw(shift(11,5)*unitsquare); draw(shift(12,5)*unitsquare); draw(shift(13,5)*unitsquare); draw(shift(11,6)*unitsquare); draw(shift(13,4)*unitsquare); draw(shift(11,1)*unitsquare); draw(shift(12,1)*unitsquare); draw(shift(13,1)*unitsquare); draw(shift(13,2)*unitsquare); draw(shift(14,2)*unitsquare); draw(shift(16,8)*unitsquare); draw(shift(17,8)*unitsquare); draw(shift(18,8)*unitsquare); draw(shift(17,9)*unitsquare); draw(shift(18,9)*unitsquare); draw(shift(16,5)*unitsquare); draw(shift(17,6)*unitsquare); draw(shift(18,5)*unitsquare); draw(shift(16,6)*unitsquare); draw(shift(18,6)*unitsquare); draw(shift(16,0)*unitsquare); draw(shift(17,0)*unitsquare); draw(shift(17,1)*unitsquare); draw(shift(18,1)*unitsquare); draw(shift(18,2)*unitsquare);[/asy]

$\textbf{(A) } 3 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 5 \qquad\textbf{(D) } 6 \qquad\textbf{(E) } 7$

Solution

Error creating thumbnail: Unable to save thumbnail to destination

The ones with lines over the shapes have at least one line of symmetry. Counting the number of shapes that have line(s) on them, we find $\boxed{\textbf{(D)}\ 6}$ pentominoes.

Video Solution by Daily Dose of Math

https://youtu.be/svFpNvUUY7E?si=CloMWtqbbhBNgWy_

~Thesmartgreekmathdude

See Also

2001 AMC 10 (Problems • Answer Key • Resources)
Preceded by
Problem 4 		Followed by
Problem 6
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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: Unable to save thumbnail to destination

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2001_AMC_10_Problems/Problem_5&oldid=263421"