Stay ahead of learning milestones! Enroll in a class over the summer!

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 May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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 Self-Paced

Prealgebra 1
Tuesday, May 13 - Aug 26
Thursday, May 29 - Sep 11
Sunday, Jun 15 - Oct 12
Monday, Jun 30 - Oct 20
Wednesday, Jul 16 - Oct 29

Prealgebra 2 Self-Paced

Prealgebra 2
Wednesday, May 7 - Aug 20
Monday, Jun 2 - Sep 22
Sunday, Jun 29 - Oct 26
Friday, Jul 25 - Nov 21

Introduction to Algebra A Self-Paced

Introduction to Algebra A
Sunday, May 11 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, May 14 - Aug 27
Friday, May 30 - Sep 26
Monday, Jun 2 - Sep 22
Sunday, Jun 15 - Oct 12
Thursday, Jun 26 - Oct 9
Tuesday, Jul 15 - Oct 28

Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Thursday, May 15 - Jul 31
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Wednesday, Jul 9 - Sep 24
Sunday, Jul 27 - Oct 19

Introduction to Number Theory
Friday, May 9 - Aug 1
Wednesday, May 21 - Aug 6
Monday, Jun 9 - Aug 25
Sunday, Jun 15 - Sep 14
Tuesday, Jul 15 - Sep 30

Introduction to Algebra B Self-Paced

Introduction to Algebra B
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
Sunday, May 11 - Nov 9
Tuesday, May 20 - Oct 28
Monday, Jun 16 - Dec 8
Friday, Jun 20 - Jan 9
Sunday, Jun 29 - Jan 11
Monday, Jul 14 - Jan 19

Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)

Intermediate: Grades 8-12

Intermediate Algebra
Sunday, Jun 1 - Nov 23
Tuesday, Jun 10 - Nov 18
Wednesday, Jun 25 - Dec 10
Sunday, Jul 13 - Jan 18
Thursday, Jul 24 - Jan 22

Intermediate Counting & Probability
Wednesday, May 21 - Sep 17
Sunday, Jun 22 - Nov 2

Intermediate Number Theory
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
Friday, May 16 - Oct 24
Sunday, Jun 1 - Nov 9
Monday, Jun 30 - Dec 8

Advanced: Grades 9-12

Olympiad Geometry
Tuesday, Jun 10 - Aug 26

Calculus
Tuesday, May 27 - Nov 11
Wednesday, Jun 25 - Dec 17

Group Theory
Thursday, Jun 12 - Sep 11

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Friday, May 23 - Aug 15
Monday, Jun 2 - Aug 18
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

MATHCOUNTS/AMC 8 Advanced
Sunday, May 11 - Aug 10
Tuesday, May 27 - Aug 12
Wednesday, Jun 11 - Aug 27
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Problem Series
Friday, May 9 - Aug 1
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Tuesday, Jun 17 - Sep 2
Sunday, Jun 22 - Sep 21 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Jun 23 - Sep 15
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Final Fives
Sunday, May 11 - Jun 8
Tuesday, May 27 - Jun 17
Monday, Jun 30 - Jul 21

AMC 12 Problem Series
Tuesday, May 27 - Aug 12
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Wednesday, Aug 6 - Oct 22

AMC 12 Final Fives
Sunday, May 18 - Jun 15

AIME Problem Series A
Thursday, May 22 - Jul 31

AIME Problem Series B
Sunday, Jun 22 - Sep 21

F=ma Problem Series
Wednesday, Jun 11 - Aug 27

WOOT Programs
Visit the pages linked for full schedule details for each of these programs!


MathWOOT Level 1
MathWOOT Level 2
ChemWOOT
CodeWOOT
PhysicsWOOT

Programming

Introduction to Programming with Python
Thursday, May 22 - Aug 7
Sunday, Jun 15 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Tuesday, Jun 17 - Sep 2
Monday, Jun 30 - Sep 22

Intermediate Programming with Python
Sunday, Jun 1 - Aug 24
Monday, Jun 30 - Sep 22

USACO Bronze Problem Series
Tuesday, May 13 - Jul 29
Sunday, Jun 22 - Sep 1

Physics

Introduction to Physics
Wednesday, May 21 - Aug 6
Sunday, Jun 15 - Sep 14
Monday, Jun 23 - Sep 15

Physics 1: Mechanics
Thursday, May 22 - Oct 30
Monday, Jun 23 - Dec 15

Relativity
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
0 replies
jlacosta
May 1, 2025
0 replies
A box contains 5 chips numbered 1, 2, 3, 4, 5
CobbleHead   23
N 28 minutes ago by superhuman233
Source: 2018 AMC 10B #6
A box contains $5$ chips, numbered $1$, $2$, $3$, $4$, and $5$. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds $4$. What is the probability that $3$ draws are required?

$\textbf{(A)} \frac{1}{15} \qquad \textbf{(B)} \frac{1}{10} \qquad \textbf{(C)} \frac{1}{6} \qquad \textbf{(D)} \frac{1}{5} \qquad \textbf{(E)} \frac{1}{4}$
23 replies
CobbleHead
Feb 16, 2018
superhuman233
28 minutes ago
[$10K+ IN PRIZES] Poolesville Math Tournament (PVMT) 2025
qwerty123456asdfgzxcvb   11
N 43 minutes ago by Ruegerbyrd
Hi everyone!

After the resounding success of the first three years of PVMT, the Poolesville High School Math Team is excited to announce the fourth annual Poolesville High School Math Tournament (PVMT)! The PVMT team includes a MOPper and multiple USA(J)MO and AIME qualifiers!

PVMT is open to all 6th-9th graders in the country (including rising 10th graders). Students will compete in teams of up to 4 people, and each participant will take three subject tests as well as the team round. The contest is completely free, and will be held virtually on June 7, 2025, from 10:00 AM to 4:00 PM (EST).

Additionally, thanks to our sponsors, we will be awarding approximately $10K+ worth of prizes (including gift cards, Citadel merch, AoPS coupons, Wolfram licenses) to top teams and individuals. More details regarding the actual prizes will be released as we get closer to the competition date.

Further, newly for this year we might run some interesting mini-events, which we will announce closer to the competition date, such as potentially a puzzle hunt and integration bee!

If you would like to register for the competition, the registration form can be found at https://pvmt.org/register.html or https://tinyurl.com/PVMT25.

Additionally, more information about PVMT can be found at https://pvmt.org

If you have any questions not answered in the below FAQ, feel free to ask in this thread or email us at falconsdomath@gmail.com!

We look forward to your participation!

FAQ
11 replies
qwerty123456asdfgzxcvb
Apr 5, 2025
Ruegerbyrd
43 minutes ago
ranttttt
alcumusftwgrind   39
N an hour ago by Ruegerbyrd
rant
39 replies
alcumusftwgrind
Apr 30, 2025
Ruegerbyrd
an hour ago
Centroids form Equilateral Triangle
Generic_Username   22
N 2 hours ago by Tetra_scheme
Source: 2019 AMC 12B #25
Let $ABCD$ be a convex quadrilateral with $BC=2$ and $CD=6.$ Suppose that the centroids of $\triangle ABC,\triangle BCD,$ and $\triangle ACD$ form the vertices of an equilateral triangle. What is the maximum possible value of the area of $ABCD$?

$\textbf{(A) } 27 \qquad\textbf{(B) } 16\sqrt3 \qquad\textbf{(C) } 12+10\sqrt3 \qquad\textbf{(D) } 9+12\sqrt3 \qquad\textbf{(E) } 30$
22 replies
Generic_Username
Feb 14, 2019
Tetra_scheme
2 hours ago
geometry
JetFire008   0
5 hours ago
Given four concyclic points. For each subset of three points take the incenter. Show that the four incentres form a rectangle.
0 replies
JetFire008
5 hours ago
0 replies
36x⁴ + 12x² - 36x + 13 > 0
fxandi   0
5 hours ago
Prove that for any real $x \geq 0$ holds inequality $36x^4 + 12x^2 - 36x + 13 > 0.$
0 replies
fxandi
5 hours ago
0 replies
Inequalities
sqing   11
N Today at 3:02 PM by sqing
Let $a,b,c> 0$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1.$ Prove that
$$  (1-abc) (1-a)(1-b)(1-c)  \ge 208 $$$$ (1+abc) (1-a)(1-b)(1-c)  \le -224 $$$$(1+a^2b^2c^2) (1-a)(1-b)(1-c)  \le -5840 $$
11 replies
sqing
Jul 12, 2024
sqing
Today at 3:02 PM
A rather difficult question
BeautifulMath0926   3
N Today at 2:23 PM by evankuang
I got a difficult equation for users to solve:
Find all functions f: R to R, so that to all real numbers x and y,
1+f(x)f(y)=f(x+y)+f(xy)+xy(x+y-2) holds.
3 replies
BeautifulMath0926
Apr 13, 2025
evankuang
Today at 2:23 PM
The return of an inequality
giangtruong13   4
N Today at 1:26 PM by sqing
Let $a,b,c$ be real positive number satisfy that: $a+b+c=1$. Prove that: $$\sum_{cyc} \frac{a}{b^2+c^2} \geq \frac{3}{2}$$
4 replies
giangtruong13
Mar 18, 2025
sqing
Today at 1:26 PM
Looking for users and developers
derekli   10
N Today at 1:22 PM by Jackson0423
Guys I've been working on a web app that lets you grind high school lvl math. There's AMCs, AIME, BMT, HMMT, SMT etc. Also, it's infinite practice so you can keep grinding without worrying about finding new problems. Please consider helping me out by testing and also consider joining our developer team! :P :blush:

Link: https://stellarlearning.app/competitive
10 replies
derekli
Yesterday at 12:57 AM
Jackson0423
Today at 1:22 PM
Polynomial
kellyelliee   1
N Today at 1:19 PM by Jackson0423
Let the polynomial $f(x)=x^2+ax+b$, where $a,b$ integers and $k$ is a positive integer. Suppose that the integers
$m,n,p$ satisfy: $f(m), f(n), f(p)$ are divisible by k. Prove that:
$(m-n)(n-p)(p-m)$ is divisible by k
1 reply
kellyelliee
Today at 3:57 AM
Jackson0423
Today at 1:19 PM
Sum of digits is 18
Ecrin_eren   14
N Today at 1:11 PM by jestrada
How many 5 digit numbers are there such that sum of its digits is 18
14 replies
Ecrin_eren
May 3, 2025
jestrada
Today at 1:11 PM
IOQM 2022-23 P-7
lifeismathematics   2
N Today at 12:09 PM by Adywastaken
Find the number of ordered pairs $(a,b)$ such that $a,b \in \{10,11,\cdots,29,30\}$ and
$\hspace{1cm}$ $GCD(a,b)+LCM(a,b)=a+b$.
2 replies
lifeismathematics
Oct 30, 2022
Adywastaken
Today at 12:09 PM
Inequalities
sqing   7
N Today at 10:31 AM by sqing
Let $ a,b,c>0 $ and $ a+b\leq 16abc. $ Prove that
$$ a+b+kc^3\geq\sqrt[4]{\frac{4k} {27}}$$$$ a+b+kc^4\geq\frac{5} {8}\sqrt[5]{\frac{k} {2}}$$Where $ k>0. $
$$ a+b+3c^3\geq\sqrt{\frac{2} {3}}$$$$ a+b+2c^4\geq \frac{5} {8}$$
7 replies
sqing
Yesterday at 12:46 PM
sqing
Today at 10:31 AM
The Empty Set Exists
Archimedes15   37
N Apr 3, 2025 by lpieleanu
Source: 2021 AIME II P6
For any finite set $S$, let $|S|$ denote the number of elements in $S$. FInd the number of ordered pairs $(A,B)$ such that $A$ and $B$ are (not necessarily distinct) subsets of $\{1,2,3,4,5\}$ that satisfy
$$|A| \cdot |B| = |A \cap B| \cdot |A \cup B|$$
37 replies
Archimedes15
Mar 19, 2021
lpieleanu
Apr 3, 2025
The Empty Set Exists
G H J
G H BBookmark kLocked kLocked NReply
Source: 2021 AIME II P6
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Archimedes15
1491 posts
#1 • 2 Y
Y by megarnie, cubres
For any finite set $S$, let $|S|$ denote the number of elements in $S$. FInd the number of ordered pairs $(A,B)$ such that $A$ and $B$ are (not necessarily distinct) subsets of $\{1,2,3,4,5\}$ that satisfy
$$|A| \cdot |B| = |A \cap B| \cdot |A \cup B|$$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Jzhang21
308 posts
#2 • 1 Y
Y by cubres
I got 454.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
HamstPan38825
8859 posts
#3 • 1 Y
Y by cubres
forgot to count the intersection, luckily found my error in the last ~10 minutes

Basically $A \subseteq B$ or $B \subseteq A$ then subtract $A=B$

$3^5 \cdot 2 - 2^5=454$
This post has been edited 1 time. Last edited by HamstPan38825, Mar 19, 2021, 5:15 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
i_equal_tan_90
34 posts
#4 • 1 Y
Y by cubres
Click to reveal hidden text
This post has been edited 2 times. Last edited by i_equal_tan_90, Mar 19, 2021, 7:57 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
UnicornPanda
105 posts
#5 • 1 Y
Y by cubres
I got $455$. Can someone please explain how to get $454$?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
eduD_looC
6610 posts
#6 • 4 Y
Y by ike.chen, mathking999, Mango247, cubres
$|A|+|B|-|A \cap B| = |A \cup B|$, so substituting gives

\begin{align*}
|A| \cdot |B| &= |A \cap B|(|A| + |B| - |A \cap B|)\\
|A||B| - |A \cap B||A| - |A \cap B||B| + |A \cap B| &= 0\\
(|A| - |A \cap B|)(|B| - |A \cap B|) &= 0.\\
\end{align*}
Therefore, we must have $|A| = |A \cap B|$ or $|B| = |A \cap B|$, so this implies $A \subseteq B$ or $B \subseteq A$.

Then do some more PIE.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
JimY
627 posts
#7 • 4 Y
Y by Mango247, Mango247, Mango247, cubres
Casework on the value of $|A|, $ and realize that either $ B \subseteq A$ or $A \subseteq B$ :

$\binom{5}{0}2^5 + \binom{5}{1}2^4 + \binom{5}{2}2^3 + \binom{5}{3}2^2 + \binom{5}{4}2 + \binom{5}{5} - 2^5 = 211$ possibilities of $B$ where $|B| > |A|. $ Times that by $2$ then add in the possibilities where $|A| = |B|$ to get $422 + 2^5 = \boxed{454}. $
This post has been edited 1 time. Last edited by JimY, Mar 19, 2021, 5:27 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
djmathman
7938 posts
#8 • 12 Y
Y by HamstPan38825, 631558, AMC_Kid, bluelinfish, samrocksnature, peace09, kavya.rajesh, megarnie, rayfish, ike.chen, programjames1, cubres
By PIE, $|A| + |B| = |A\cap B| + |A\cup B|$, so $\{|A|, |B|\} = \{|A\cap B|, |A\cup B|\}$ (use Vieta). This implies either $A\subseteq B$ or $B\subseteq A$.

In the former case, each element of $\{1,2,3,4,5\}$ has three possibilities: in both $A$ and $B$, in $A$ but not $B$, or in neither $A$ nor $B$. This gives $3^5$ possibilities in that case. Analogously, $3^5$ possibilities in the second case.

Subtracting off the $2^5 = 32$ cases where $A = B$ gives a final answer of $2\cdot 3^5 - 2^5 = 454$.
This post has been edited 1 time. Last edited by djmathman, Mar 19, 2021, 5:21 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Archimedes15
1491 posts
#9 • 1 Y
Y by cubres
A lot of people I talked to seem to have missed the empty set.

Interestingly enough this appears to be one of the first AIME problems ever where non-empty was not specified.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
ilovepizza2020
12156 posts
#10 • 4 Y
Y by megarnie, Mango247, Mango247, cubres
Another reading exercise on AIME.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
khina
993 posts
#11 • 3 Y
Y by CyclicISLscelesTrapezoid, Mango247, cubres
Darn so in the contest I thought this would be killed by this.
This post has been edited 1 time. Last edited by khina, Mar 19, 2021, 5:24 PM
Reason: fix url
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
dogsareawesome123
131 posts
#12 • 1 Y
Y by cubres
I was kinda worried abt empty vs nonempty lol, I just assumed empty was ok thankfully
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
kred9
1022 posts
#13 • 4 Y
Y by Inconsistent, greenturtle3141, BakedPotato66, cubres
Draw a Venn Diagram, and call $a$ the number of elements in $A$ but not $B$, $b$ the number of elements in $B$ but not $A$, and $c$ the number of elements $A$ and $B$. Then we get $(a+c)(b+c) = (a+c+b)c$ so $ab=0$.

If $a=0$, then each of the remaining numbers must be in $b$ or $c$ or neither, meaning $3^5$ ways.
If $b=0$, it is also $3^5$ ways.
If $a=b=0$, each number can either be in $c$ or not in $c$, so $2^5$.

So the answer is $2*3^5-2^5 = 486 - 32 = 454$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
etotheipiplusone
35 posts
#14 • 1 Y
Y by cubres
One way you can think about this is that $|A| + |B| = |A \cup B| + |A \cap B|$ all the time, and if $x_1x_2 = y_1y_2$ and $x_1 + x_2 = y_1 + y_2$ then $\{x_1,x_2\} = \{y_1,y_2\}$ so either $|A|$ or $|B|$ is equal to $|A \cap B|$ and then proceed as others said.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sae123
692 posts
#15 • 1 Y
Y by cubres
Use PIE, factor with SFFT, and find that $A \subseteq B$ or the opposite. Then count.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
v_Enhance
6877 posts
#16 • 12 Y
Y by djmathman, insertionsort, Imayormaynotknowcalculus, HamstPan38825, khina, scrabbler94, samrocksnature, Mathematicsislovely, megarnie, rayfish, Ritwin, cubres
This was my problem. I'm surprised that people "missed" the empty set since (AFAIK) you need to do extra work to exclude it, assuming you have the nice solution.

In my head, since $|A|+|B| = |A\cap B| + |A \cup B|$, if the products are equal as well, then by Vieta formula it follows that $|A|$ and $|B|$ are roots of same polynomial as $|A \cap B|$ and $|A \cup B|$, hence the main claim.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
math31415926535
5617 posts
#17 • 1 Y
Y by cubres
UnicornPanda wrote:
I got $455$. Can someone please explain how to get $454$?

probably you forgot the empty set
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sidchukkayapally
669 posts
#18 • 1 Y
Y by cubres
Neat
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
eduD_looC
6610 posts
#19 • 1 Y
Y by cubres
math31415926535 wrote:
UnicornPanda wrote:
I got $455$. Can someone please explain how to get $454$?

probably you forgot the empty set

No. If @UnicornPanda forgot the empty set, the answer gotten would've been $453$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
ilovepizza2020
12156 posts
#20 • 1 Y
Y by cubres
eduD_looC wrote:
math31415926535 wrote:
UnicornPanda wrote:
I got $455$. Can someone please explain how to get $454$?

probably you forgot the empty set

No. If @UnicornPanda forgot the empty set, the answer gotten would've been $453$.

They forgot to include the empty set
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
ike.chen
1162 posts
#21 • 1 Y
Y by cubres
I'm pretty late (just mocked this now) but the factorization instantly reminded me of JMC 10 #18 (https://artofproblemsolving.com/community/c1465090h2420752p19921359) for some reason.

Chden saved my life again lol.
This post has been edited 1 time. Last edited by ike.chen, Mar 21, 2021, 5:50 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
PlaneGod
452 posts
#22 • 1 Y
Y by cubres
doesnt everyone know the empty set exists?
what does the title mean
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
ilovepizza2020
12156 posts
#23 • 1 Y
Y by cubres
PlaneGod wrote:
doesnt everyone know the empty set exists?
what does the title mean

Duuude most problems say to disregard the empty set but this one needs you to not disregard it.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
aaja3427
1918 posts
#24 • 1 Y
Y by cubres
PlaneGod wrote:
doesnt everyone know the empty set exists?
what does the title mean

I don't know if you got the joke or not but basically the title says that since a lot of people missed the empty set.
It's kinda like saying "Complex solutions exist" if someone says that the sum of the real solutions of $x^2+4x+333=0$ is -4.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sugar_rush
1341 posts
#25 • 2 Y
Y by Mango247, cubres
I'm getting $391$
solution
Can someone please explain why $391$ is incorrect?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
franchester
1487 posts
#26 • 1 Y
Y by cubres
sugar_rush wrote:
I'm getting $391$
solution
Can someone please explain why $391$ is incorrect?

$A$ and $B$ dont have to be nonempty
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
megarnie
5604 posts
#27 • 1 Y
Y by cubres
The worst solution.

Case 1: $A$ is empty.
Then this is always true, giving a total of $32$.

Case 2: $A$ has one element.
WLOG $A=\{1\}$.

Then if $B$ is not empty, then $1\in B$. Any such $B$ works. So there are $16$ ways.

If $b$ is empty, then there is $1$ way.

Total for this case is $17\cdot 5=85$.

Case 3: $A$ has $2$ elements.
WLOG $A=\{1,2\}$.

If $B$ contains both $1$ and $2$, then all such $B$ work, giving $8$.

If $B$ doesn't contain $1$ or $2$, then $B$ is empty, giving $1$.

if $B$ contains $1$ but not $2$, then $2|B|=|B|+1\implies |B|=1$. Thus, there is $1$ way.

If $B$ contains $2$ but not $1$, then there is $1$ way.

Total is $10\cdot 11=110$.

Case 4: $A$ has $3$ elements.
Then WLOG $A=\{1,2,3\}$.

If $|A\cup B|=0$, then $B$ is empty. $1$ way.

If $|A\cup B|=1$, then $3|B|=|B|+2\implies |B|=1$. $3$ ways.

If $|A\cup B|=2$, then $3|B|=2(|B|+1)\implies |B|=2$. $3$ ways.

If $|A\cup B|=3$, then there are $4$ total ways, all work.

Thus, total for this case is $11\cdot 10=110$.

Case 5: $|A|=4$.
WLOG $A=\{1,2,3,4\}$.

If $|A\cup B|=0$, then $B$ is empty. $1$ way.

If $|A\cup B|=1$, then $|B|=1$. $4$ ways.

If $|A\cup B|=2$, then $|B|=2$. $6$ ways.

If $|A\cup B|=3$, then $|B|=3$. $4$ ways.

If $|A\cup B|=4$, then there are $2$ ways.

So total for this case is $5\cdot 17=85$.

Case 6: $|A|=5$.
Then $A=\{1,2,3,4,5\}$.

Any subset for $B$ works, giving a total of $32$.


Answer is $2(32+85+110)=\boxed{454}$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
megarnie
5604 posts
#28 • 1 Y
Y by cubres
Looks like I mixed up $\cup$ with $\cap$ in my previous solution.

We can check that either $A\subseteq B$ or $B\subseteq A$ and everything of this form works.

Case 1: $A\subseteq B$.
Then there are $2^5+5\cdot 2^4+10\cdot 2^3+10\cdot 2^2+5\cdot 2^1+1=(2+1)^5=243$ ways.

Case 2: $B\subseteq A$.
Then there are also $243$ ways.

Case 3: $A=B$.
Then there are $32$ ways.

So the answer is $243+243-32=\boxed{454}$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
brainfertilzer
1831 posts
#29 • 1 Y
Y by cubres
We use PIE: $|A\cup B| = |A| + |B| - |A\cap B|$. Hence
\[ |A|\cdot |B| = |A\cap B|\cdot (|A| + |B| - |A\cap B|).\]
Let $|A| = a$, $|B| = b$, and $|A\cap B| = i$. Then
\[ ab = ai +bi - i^2\]\[ a(b-i) = bi - i^2 = i(b-i).\]
Hence $i = b$ or $i = a$. The only way for this to happen is either if $A\subseteq B$ or $B\subseteq A$. WLOG, let $A\subseteq B$ (we will deal with some complications later). If $B$ has $j$ elements, there are $\binom{5}{j}$ ways to choose $B$. Then there are $2^j$ ways to choose $A$. The total number of ways is
\[ \sum_{j = 0}^{5}2^j\binom{5}{j} = 1 + 10 + 40 + 80 + 80 + 32 = 243.\]
Now, there should $2\cdot 243 = 486$ ways, but we overcounted the case $A = B$ once, which has $2^5 = 32$ ways of happening. The answer is $486 - 32 = \boxed{454}$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
samrocksnature
8791 posts
#30 • 1 Y
Y by cubres
Archimedes15 wrote:
For any finite set $S$, let $|S|$ denote the number of elements in $S$. FInd the number of ordered pairs $(A,B)$ such that $A$ and $B$ are (not necessarily distinct) subsets of $\{1,2,3,4,5\}$ that satisfy
$$|A| \cdot |B| = |A \cap B| \cdot |A \cup B|$$

typo in the first word of the second sentence help I'm catching typoes everywhere I go more and more I think there's something wrong with me
This post has been edited 1 time. Last edited by samrocksnature, Jan 23, 2022, 6:02 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
RunyangWang
192 posts
#31 • 1 Y
Y by cubres
I bashed this problem. Got the correct answer 5 minutes before the end of the test (I did other problems first and came back to this one)
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
daijobu
526 posts
#32 • 1 Y
Y by cubres
Video Solution + review of set theory
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
ChromeRaptor777
1889 posts
#33 • 2 Y
Y by ryanbear, cubres
Realize that $A \subseteq B.$ Do a bit of casework to get $211.$ Multiply by $2$ because $(A, B) \neq (B, A).$ We've counted $A=B$ twice here because it doesn't replicate; subtract by the number of times once ($2^5$ for correction), and we're done.
Took me a while to catch everything but the first two sentences, but solve.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Shreyasharma
682 posts
#34 • 1 Y
Y by cubres
The main insight is that either $A$ or $B$ is a subset of the other. How I found this was letting $|A \cup B| = |A| + |B| - |A \cap B|$. Then denoting $|A| = a$, $|B| = b$ and $|A \cap B| = c$ we have,
\begin{align*}
ab &= c(a+b-c)\\
ab &= ac + bc - c^2\\
a(b-c) + c(c - b)&= 0\\
(a-c)(b-c) &= 0
\end{align*}and hence either $a = c$, $b = c$ or $a = b = c$. Now its just a simple counting problem.

We have two cases.

First if $a = b$ then we must have $A = B$. This can be done in $\sum \binom{5}{i} = 32$ ways.

Then assume that they are not equal and WLOG $B$ is a subset of $A$. Then do casework on the size of $a$.

If $a = 0$ then $b = 0$, we've already counted this.

If $a = 1$ we have $1$ choice for $b$ adding $5$ to our count.

If $a = 2$ we have $2^2 - 1 = 3$ choices for $b$ giving $\binom{5}{2} \cdot 3 = 30$ to our count.

If $a = 3$ we have $2^3 - 1 = 7$ choices for $b$ giving $\binom{5}{3} \cdot 7 = 70$ to our count.

If $a = 4$ we have $2^4 - 1 = 15$ choices for $b$ giving $\binom{5}{4} \cdot 15= 75$ to our count.

Finally if $a = 5$ we have $2^5 - 1 = 31$ choices for $b$ giving $31$ to our count.

Summing these cases up and multiplying by $2$ we have $422$ in this case.

Then adding our final count is $422 + 32 = \boxed{454}$.
This post has been edited 1 time. Last edited by Shreyasharma, Jan 6, 2024, 12:17 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Ritwin
156 posts
#35 • 2 Y
Y by LLL2019, cubres
Interesting that no one else has posted this, but I found the main claim in a slightly different way: after recalling $|A| + |B| = |{A \cup B}| + |{A \cap B}|$, because \[ |{A \cup B}| \geq |A|, |B| \geq |{A \cap B}| \]for inequality reasons we are forced to have $\{|A|, |B|\} = \{|{A \cup B}|, |{A \cap B}|\}$ meaning either $A \subseteq B$ or $B \subseteq A$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
blueprimes
351 posts
#36 • 1 Y
Y by cubres
We will solve for the general case for subsets of $\{1, 2, \dots, n \}$ where $n$ is a positive integer.

Let $x = |A \cap B|, a = |A| - x, b = |B| - x$. The equation provided in the problem is equivalent to
$$(a + x)(b + x) = x(a + b + x) \implies ab = 0.$$Suppose $a = 0$. This implies that $(A \cap B) \subseteq A$. Observe that for every element in $\{1, 2, \dots, n \}$, it is either in $A$, the set $A$ when $(A \cap B)$ is removed, or in neither. Hence, we obtain $3^n$ possibilities. Similarly, $b = 0$ gives $3^n$. Now we subtract off the duplicates. If $a = b = 0$, $A = B$, so there are $2^n$ ways to choose the common subset. We obtain our final answer of $\boxed{2 \cdot 3^n - 2^n}$. (It can easily be checked that $n = 5$ yields $454$, the answer to the original problem.)
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
ProMaskedVictor
43 posts
#37 • 1 Y
Y by cubres
$|A \cup B|=|A|+|B|-|A\cap B|$.
Using this and by manipulating the expression, we get that $|A|=|A \cap B|$ or$|B|=|A\cap B|$
For $|A|=|A \cap B|$, we get that $A \subseteq B$. So, number of pairs $=3^5$.
Similarly for the other part, there are $3^5$ pairs.
There are $2^5$ such pairs where $A=B$.
Hence, total number of ordered pairs $=2 \cdot 3^5 -2^5 = \boxed{454}$.
This post has been edited 1 time. Last edited by ProMaskedVictor, Aug 19, 2024, 1:26 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
lpieleanu
2994 posts
#38 • 1 Y
Y by cubres
Solution
Z K Y
N Quick Reply
G
H
=
a