Problem 2

Happy Thanksgiving! Please note that there are no classes November 25th-December 1st.

Contests & Programs AMC and other contests, summer programs, etc.
AMC and other contests, summer programs, etc.
3 M G
BBookmark  VNew Topic kLocked
Contests & Programs AMC and other contests, summer programs, etc.
AMC and other contests, summer programs, etc.
3 M G
BBookmark  VNew Topic kLocked
G
Topic
First Poster
Last Poster
k a AMC 10/12 A&B Coming up Soon!
jlacosta   0
Nov 1, 2024
There is still time to train for the November 6th and November 12th AMC 10A/12A and AMC 10B/12B, respectively! Enroll in our weekend seminars to be held on November 2nd and 3rd (listed below) and you will learn problem strategies, test taking techniques, and be able to take a full practice test! Note that the “B” seminars will have different material from the “A” seminars which were held in October.

[list][*]Special AMC 10 Problem Seminar B
[*]Special AMC 12 Problem Seminar B[/list]
For those who want to take a free practice test before the AMC 10/12 competitions, you can simulate a real competition experience by following this link. As you assess your performance on these exams, be sure to gather data!

[list][*]Which problems did you get right?
[list][*]Was the topic a strength (e.g. number theory, geometry, counting/probability, algebra)?
[*]How did you prepare?
[*]What was your confidence level?[/list]
[*]Which problems did you get wrong?
[list][list][*]Did you make an arithmetic error?
[*]Did you misread the problem?
[*]Did you have the foundational knowledge for the problem?
[*]Which topics require more fluency through practice (e.g. number theory, geometry, counting/probability, algebra)?
[*]Did you run out of time?[/list][/list]
Once you have analyzed the results with the above questions, you will have a plan of attack for future contests! BEST OF LUCK to all competitors at this year’s AMC 10 and AMC 12!

Did you know that the day after both the AMC 10A/12A and AMC 10B/12B you can join a free math jam where our AoPS team will go over the most interesting problems? Find the schedule below under “Mark your calendars”.

Mark your calendars for these upcoming free math jams!
[list][*]November 20th: Amherst College Info Session, 7:30 pm ET: Matt McGann, Dean of Admission and Financial Aid at Amherst College, and Nathan Pflueger, math professor at Amherst College, will host an info session exploring both Amherst College specifically and liberal arts colleges generally. Topics include opportunities in math, the admission process, and financial aid for both US and international students.
[*]November 7th: 2024 AMC 10/12 A Discussion, Thursday, 7:30 pm ET:
[*]AoPS instructors will discuss problems from the AMC 10/12 A, administered November 6. We will discuss some of the most interesting problems from each test!
[*]November 13th: 2024 AMC 10/12 B Discussion, Wednesday, 7:30 pm ET:
[*]AoPS instructors will discuss problems from the AMC 10/12 B, administered November 12. We will discuss some of the most interesting problems from each test![/list]
AoPS Spring classes are open for enrollment. Get a jump on the New Year and enroll in our math, contest prep, coding, and science classes today! Need help finding the right plan for your goals? Check out our recommendations page!

Don’t forget: Highlight your AoPS Education on LinkedIn!
Many of you are beginning to build your education and achievements history on LinkedIn. Now, you can showcase your courses from Art of Problem Solving (AoPS) directly on your LinkedIn profile!

Whether you've taken our classes at AoPS Online or AoPS Academies or reached the top echelons of our competition training with our Worldwide Online Olympiad Training (WOOT) program, you can now add your AoPS experience to the education section on your LinkedIn profile.

Don't miss this opportunity to stand out and connect with fellow problem-solvers in the professional world and be sure to follow us at: https://www.linkedin.com/school/art-of-problem-solving/mycompany/ Check out our job postings, too, if you are interested in either full-time, part-time, or internship opportunities!

Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

Introductory: Grades 5-10

Prealgebra 1
Sunday, Nov 3 - Mar 9
Tuesday, Nov 5 - Mar 11
Friday, Dec 6 - Apr 4
Sunday, Jan 5 - Apr 20
Wednesday, Jan 15 - Apr 30
Monday, Feb 3 - May 19
Sunday, Mar 2 - Jun 22
Friday, Mar 28 - Jul 18
Sunday, Apr 13 - Aug 10

Prealgebra 1 Self-Paced

Prealgebra 2
Thursday, Nov 7 - Mar 13
Monday, Dec 2 - Mar 31
Wednesday, Jan 8 - Apr 23
Sunday, Jan 19 - May 4 (1:00 - 2:15 pm ET/10:00 - 11:15 am PT)
Monday, Jan 27 - May 12
Tuesday, Jan 28 - May 13 (4:30 - 5:45 pm ET/1:30 - 2:45 pm PT)
Sunday, Feb 16 - Jun 8
Tuesday, Mar 25 - Jul 8
Sunday, Apr 13 - Aug 10

Prealgebra 2 Self-Paced

Introduction to Algebra A
Friday, Nov 8 - Mar 14
Wednesday, Dec 11 - Apr 9
Tuesday, Jan 7 - Apr 22
Wednesday, Jan 29 - May 14
Sunday, Feb 16 - Jun 8 (3:30 - 5:00 pm ET/12:30 - 2:00 pm PT)
Sunday, Mar 23 - Jul 20
Monday, Apr 7 - Jul 28

Introduction to Algebra A Self-Paced

Introduction to Counting & Probability
Thursday, Dec 5 - Mar 6
Wednesday, Jan 8 - Mar 26
Thursday, Jan 30 - Apr 17
Sunday, Feb 9 - Apr 27 (3:30 - 5:00 pm ET/12:30 - 2:00 pm PT)
Sunday, Mar 16 - Jun 8
Wednesday, Apr 16 - Jul 2

Introduction to Counting & Probability Self-Paced

Introduction to Number Theory
Monday, Dec 2 - Mar 3
Tuesday, Jan 28 - Apr 15
Sunday, Feb 16 - May 4
Monday, Mar 17 - Jun 9
Thursday, Apr 17 - Jul 3

Introduction to Algebra B
Wednesday, Dec 11 - Apr 9
Tuesday, Jan 28 - May 13
Thursday, Feb 13 - May 29
Sunday, Mar 2 - Jun 22
Monday, Mar 17 - Jul 7
Wednesday, Apr 16 - Jul 30

Introduction to Geometry
Monday, Nov 11 - May 12
Wednesday, Nov 13 - May 14 (9:30 - 11:00 pm ET/6:30 - 8:00 pm PT)
Tuesday, Dec 10 - Jun 3
Wednesday, Jan 8 - Jun 18
Thursday, Jan 30 - Jul 10
Friday, Feb 14 - Aug 1
Tuesday, Mar 4 - Aug 12
Sunday, Mar 23 - Sep 21
Wednesday, Apr 23 - Oct 1

Paradoxes and Infinity
Sat & Sun, Nov 16 - Nov 17 (4:00 - 7:00 pm ET/1:00 - 4:00 pm PT)

Intermediate: Grades 8-12

Intermediate Algebra
Sunday, Nov 10 - May 11
Tuesday, Dec 3 - May 27
Friday, Jan 17 - Jun 27
Wednesday, Feb 12 - Jul 23
Sunday, Mar 16 - Sep 14
Tuesday, Mar 25 - Sep 2
Monday, Apr 21 - Oct 13

Intermediate Counting & Probability
Thursday, Nov 7 - Mar 27
Monday, Feb 10 - Jun 16
Sunday, Mar 23 - Aug 3

Intermediate Number Theory
Thursday, Feb 20 - May 8
Friday, Apr 11 - Jun 27

Precalculus
Sunday, Nov 10 - Apr 27
Tuesday, Dec 10 - May 20
Wednesday, Jan 8 - Jun 4
Tuesday, Feb 25 - Jul 22
Sunday, Mar 16 - Aug 24
Wednesday, Apr 9 - Sep 3

Advanced: Grades 9-12

Olympiad Geometry
Wednesday, Mar 5 - May 21

Calculus
Tuesday, Dec 10 - Jun 10
Friday, Feb 28 - Aug 22
Sunday, Mar 30 - Oct 5

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Mon, Wed & Fri, Dec 2 - Jan 10 (meets three times each week!)
Tuesday, Feb 4 - Apr 22
Sunday, Mar 23 - Jun 15
Wednesday, Apr 16 - Jul 2

MATHCOUNTS/AMC 8 Advanced
Tuesday, Nov 5 - Feb 11
Mon, Wed & Fri, Dec 2 - Jan 10 (meets three times each week!)
Tue, Thurs & Sun, Dec 10 - Jan 19 (meets three times each week!)
Sunday, Feb 16 - May 4
Friday, Apr 11 - Jun 27

Special AMC 8 Problem Seminar A
Sat & Sun, Jan 11 - Jan 12 (4:00 - 7:00 pm ET/1:00 - 4:00 pm PT)

Special AMC 8 Problem Seminar B
Sat & Sun, Jan 18 - Jan 19 (4:00 - 7:00 pm ET/1:00 - 4:00 pm PT)

AMC 10 Problem Series
Sunday, Feb 9 - Apr 27
Tuesday, Mar 4 - May 20
Monday, Mar 31 - Jun 23

AMC 10 Final Fives
Sunday, Feb 9 - Mar 2 (3:30 - 5:00 pm ET/12:30 - 2:00 pm PT)

Special AMC 10 Problem Seminar B
Sat & Sun, Nov 2 - Nov 3 (4:00 - 7:00 pm ET/1:00 - 4:00 pm PT)

AMC 12 Problem Series
Sunday, Feb 23 - May 11

AMC 12 Final Fives
Sunday, Feb 9 - Mar 2 (3:30 - 5:00 pm ET/12:30 - 2:00 pm PT)

Special AMC 12 Problem Seminar B
Sat & Sun, Nov 2 - Nov 3 (4:00 - 7:00 pm ET/1:00 - 4:00 pm PT)

AIME Problem Series A
Tue, Thurs & Sun, Jan 7 - Feb (meets three times each week!)

AIME Problem Series B
Mon, Wed & Fri, Jan 6 - Jan 31 (meets three times each week!)

Special AIME Problem Seminar A
Sat & Sun, Jan 25 - Jan 26 (4:00 - 7:00 pm ET/1:00 - 4:00 pm PT)

Special AIME Problem Seminar B
Sat & Sun, Feb 1 - Feb 2 (4:00 - 7:00 pm ET/1:00 - 4:00 pm PT)

F=ma Problem Series
Wednesday, Feb 19 - May 7

Programming

Introduction to Programming with Python
Monday, Dec 2 - Mar 3
Friday, Jan 17 - Apr 4
Sunday, Feb 16 - May 4
Monday, Mar 24 - Jun 16

Intermediate Programming with Python
Tuesday, Feb 25 - May 13

Science

Introduction to Physics
Tuesday, Dec 10 - Mar 11
Friday, Feb 7 - Apr 25
Sunday, Mar 30 - Jun 22

Physics 1: Mechanics
Sunday, Feb 9 - Aug 3
Tuesday, Mar 25 - Sep 2

Relativity
Sat & Sun, Dec 14 - Dec 15 (4:00 - 7:00 pm ET/1:00 - 4:00pm PT)
0 replies
jlacosta
Nov 1, 2024
0 replies
9 How well do you think you will do on the AMC 8
TQ_Math   32
N 4 minutes ago by vincentwant
I know it's November, but it's never not too late to prep, right?!

40 VOTES!!!
60 votess!
100 VOTES
120 votesdfgjkafjd
32 replies
TQ_Math
Nov 17, 2024
vincentwant
4 minutes ago
jmo cutoff
cakeiswaybetterthancookie   37
N 6 minutes ago by studymoremath
Source: me
You guys NEED to stop freaking out about JMO cutoffs Its likely going to be in the early-mid 210's assuming an average level aime( 2021 aime?) So if you have a 110+ on the amc 10, you are in contention for USAJMO- its not LIKELY but its possible. My guess is maybe 15ish percent? really, too many people are freaking out about cutoffs- at the end of the day, most of you have next year and the year after that and can always keep improving. You got this!
37 replies
+1 w
cakeiswaybetterthancookie
Today at 1:10 AM
studymoremath
6 minutes ago
9 When To Start Grinding Math Again
Existing_Human1   142
N 7 minutes ago by MathNerdRabbit103
For context, I got a 103.5 on AMC 10B, so I am not sure if I made AIME. I have AMC 8 and Mathcoutns coming up as competitions
142 replies
Existing_Human1
Nov 23, 2024
MathNerdRabbit103
7 minutes ago
how to prep for AIME
CharviA   4
N 15 minutes ago by orangebear
I'm in 8th grade and I got a 129 on the 10b. For the record I got rly lucky because I guessed e for the last two questions since there were practically no e's on the sheet and there was a lot of combinatorics, which ,fortunately, im relatively good at. This is my first time qualifying for aime and I am currently mocking 3-5. How should I prepare for aime to get at least an 8? thanks
4 replies
CharviA
an hour ago
orangebear
15 minutes ago
No more topics!
Problem 2
evt917   50
N Nov 17, 2024 by sadas123
Source: 2024 AMC 12B #2 / AMC 10B #2
What is $10! - 7! \cdot 6!$?

$
\textbf{(A) }-120 \qquad
\textbf{(B) }0 \qquad
\textbf{(C) }120 \qquad
\textbf{(D) }600 \qquad
\textbf{(E) }720 \qquad
$
50 replies
evt917
Nov 13, 2024
sadas123
Nov 17, 2024
Problem 2
G H J
Source: 2024 AMC 12B #2 / AMC 10B #2
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
evt917
1584 posts
#1
Y by
What is $10! - 7! \cdot 6!$?

$
\textbf{(A) }-120 \qquad
\textbf{(B) }0 \qquad
\textbf{(C) }120 \qquad
\textbf{(D) }600 \qquad
\textbf{(E) }720 \qquad
$
This post has been edited 1 time. Last edited by LauraZed, Nov 13, 2024, 5:47 PM
Reason: adding AMC 10 to source
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Jack_w
61 posts
#2 • 40 Y
Y by mathlearner2357, weihang, eg4334, vsamc, EpicBird08, allenshen1226, pog, Sedro, alexanderhamilton124, AbhayAttarde01, mahaler, Amkan2022, Challengees24, Firebreather14, Awesomeness_in_a_bun, cweu001, Metavaria, aidan0626, Ilikeminecraft, bot1132, forestcaller2010, alan9535, Andyluo, ap246, happypi31415, suvamkonar, studymoremath, UnearthedCyclone, NoSignOfTheta, kafuu_chino, centslordm, ihatemath123, giratina3, Jndd, goldenuni678, OronSH, evt917, KLBBC, AlexWin0806, Turtwig113
Divisible by $6!$ so $\textbf{(E) }720$
This post has been edited 1 time. Last edited by Jack_w, Nov 13, 2024, 5:13 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
HonestCat
950 posts
#3 • 3 Y
Y by studymoremath, alexanderhamilton124, MATHMAN9999
Jack_w wrote:
Divisible by $6!$ so $\textbf{(E) }720$

I got 0
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
HumanCalculator9
6230 posts
#4 • 1 Y
Y by golue3120
Simply notice that only 0 is a multiple of 7, and as obviously $10!$ and $7!$ are both multiples of 7, the answer must be $0$

What was MAA thinking with the answer choices bro :skull:
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
yambe2002
1659 posts
#5
Y by
$6!=8\cdot9\cdot10$ so $0$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
BS2012
558 posts
#6
Y by
$$6!*7(10*9*8-6!)=0$$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
megarnie
5324 posts
#7 • 1 Y
Y by Mathandski
Divisible by $7!$, so $\boxed{\textbf{(B)}\ 0}$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
titanlord
197 posts
#8 • 2 Y
Y by Soccerstar9, X423anp1
Jack_w wrote:
Divisible by $6!$ so $\textbf{(E) }720$

bruh oh my lord this is tragic
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Squidget
380 posts
#9
Y by
I actually multiplied it out because I was too paranoid to go along with my original answer.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
paganiniana
204 posts
#10
Y by
note that $6!=720=8\cdot9\cdot10$, and $10!=7!\cdot8\cdot9\cdot10$ so 0
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
alexanderhamilton124
250 posts
#11
Y by
Jack_w wrote:
Divisible by $6!$ so $\textbf{(E) }720$

Bro got cooked so hard

10! - 7! 6! = 6!(10*9*8*7 - 7!), and 10*9*8*7 = 7!, so is zero
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
NoSignOfTheta
1590 posts
#12
Y by
Rip Jack_w
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
pqr.
147 posts
#13
Y by
Jack_w wrote:
Divisible by $6!$ so $\textbf{(E) }720$

This guy bruh
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
ChromeRaptor777
1890 posts
#14
Y by
evt917 wrote:
What is $10! - 7! \cdot 6!$?

$
\textbf{(A) }-120 \qquad
\textbf{(B) }0 \qquad
\textbf{(C) }120 \qquad
\textbf{(D) }600 \qquad
\textbf{(E) }720 \qquad
$

sol1
sol2
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
MathRook7817
285 posts
#15
Y by
just factor 6! or 7! out and it become trivial
Z K Y
G
H
=
a