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 "2021 AMC 12B Problems/Problem 15"
(→‎Problem 15)
Line 1: Line 1:
−
==Problem 15==
+
==Problem 20==
−
The number <math>2021</math> is expressed in the form <cmath>2021=\frac{a_1!a_2!...a_m!}{b_1!b_2!...b_n!},</cmath> where <math>a_1 \geq a_2 \geq \cdots \geq a_m</math> and <math>b_1 \geq b_2 \geq \cdots \geq b_n</math> are positive integers and <math>a_1+b_1</math> is as small as possible. What is <math>|a_1 - b_1|</math>?
+
The figure is constructed from <math>11</math> line segments, each of which has length <math>2</math>. The area of pentagon <math>ABCDE</math> can be written is <math>\sqrt{m} + \sqrt{n}</math>, where <math>m</math> and <math>n</math> are positive integers. What is <math>m + n ?</math>
 +
<asy>
 +
/* Made by samrocksnature */
 +
pair A=(-2.4638,4.10658);
 +
pair B=(-4,2.6567453480756127);
 +
pair C=(-3.47132,0.6335248637894945);
 +
pair D=(-1.464483379039766,0.6335248637894945);
 +
pair E=(-0.956630463955801,2.6567453480756127);
 +
pair F=(-2,2);
 +
pair G=(-3,2);
 +
draw(A--B--C--D--E--A);
 +
draw(A--F--A--G);
 +
draw(B--F--C);
 +
draw(E--G--D);
 +
label("A",A,N);
 +
label("B",B,W);
 +
label("C",C,W);
 +
label("D",D,E);
 +
label("E",E,dir(0));
 +
dot(A^^B^^C^^D^^E^^F^^G);
 +
</asy>
  
−
<math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5</math>
+
<math>\textbf{(A)} ~20 \qquad\textbf{(B)} ~21 \qquad\textbf{(C)} ~22 \qquad\textbf{(D)} ~23 \qquad\textbf{(E)} ~24</math>

Revision as of 19:48, 11 February 2021

Problem 20

The figure is constructed from $11$ line segments, each of which has length $2$. The area of pentagon $ABCDE$ can be written is $\sqrt{m} + \sqrt{n}$, where $m$ and $n$ are positive integers. What is $m + n ?$ [asy] /* Made by samrocksnature */ pair A=(-2.4638,4.10658); pair B=(-4,2.6567453480756127); pair C=(-3.47132,0.6335248637894945); pair D=(-1.464483379039766,0.6335248637894945); pair E=(-0.956630463955801,2.6567453480756127); pair F=(-2,2); pair G=(-3,2); draw(A--B--C--D--E--A); draw(A--F--A--G); draw(B--F--C); draw(E--G--D); label("A",A,N); label("B",B,W); label("C",C,W); label("D",D,E); label("E",E,dir(0)); dot(A^^B^^C^^D^^E^^F^^G); [/asy]

$\textbf{(A)} ~20 \qquad\textbf{(B)} ~21 \qquad\textbf{(C)} ~22 \qquad\textbf{(D)} ~23 \qquad\textbf{(E)} ~24$

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2021_AMC_12B_Problems/Problem_15&oldid=145427"