Happy Memorial Day! Please note that AoPS Online is closed May 24-26th.

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
k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
Serbian selection contest for the IMO 2025 - P6
OgnjenTesic   4
N 22 minutes ago by GreenTea2593
Source: Serbian selection contest for the IMO 2025
For an $n \times n$ table filled with natural numbers, we say it is a divisor table if:
- the numbers in the $i$-th row are exactly all the divisors of some natural number $r_i$,
- the numbers in the $j$-th column are exactly all the divisors of some natural number $c_j$,
- $r_i \ne r_j$ for every $i \ne j$.

A prime number $p$ is given. Determine the smallest natural number $n$, divisible by $p$, such that there exists an $n \times n$ divisor table, or prove that such $n$ does not exist.

Proposed by Pavle Martinović
4 replies
OgnjenTesic
May 22, 2025
GreenTea2593
22 minutes ago
Integral
Martin.s   3
N 33 minutes ago by Figaro
$$\int_0^{\pi/6}\arcsin\Bigl(\sqrt{\cos(3\psi)\cos\psi}\Bigr)\,d\psi.$$
3 replies
Martin.s
May 14, 2025
Figaro
33 minutes ago
pairs (m, n) such that a fractional expression is an integer
cielblue   2
N 39 minutes ago by cielblue
Find all pairs $(m,\ n)$ of positive integers such that $\frac{m^3-mn+1}{m^2+mn+2}$ is an integer.
2 replies
cielblue
Yesterday at 8:38 PM
cielblue
39 minutes ago
Sociable set of people
jgnr   23
N an hour ago by quantam13
Source: RMM 2012 day 1 problem 1
Given a finite number of boys and girls, a sociable set of boys is a set of boys such that every girl knows at least one boy in that set; and a sociable set of girls is a set of girls such that every boy knows at least one girl in that set. Prove that the number of sociable sets of boys and the number of sociable sets of girls have the same parity. (Acquaintance is assumed to be mutual.)

(Poland) Marek Cygan
23 replies
jgnr
Mar 3, 2012
quantam13
an hour ago
diophantine equation
m4thbl3nd3r   0
an hour ago
Find all positive integers $n,k$ such that $$5^{2n+1}-5^n+1=k^2$$
0 replies
m4thbl3nd3r
an hour ago
0 replies
A geometry problem
Lttgeometry   1
N an hour ago by Funcshun840
Triangle $ABC$ has two isogonal conjugate points $P$ and $Q$. The circle $(BPC)$ intersects circle $(AP)$ at $R \neq P$, and the circle $(BQC)$ intersects circle $(AQ)$ at $S\neq Q$. Prove that $R$ and $S$ are isogonal conjugates in triangle $ABC$.
Note: Circle $(AP)$ is the circle with diameter $AP$, Circle $(AQ)$ is the circle with diameter $AQ$.
1 reply
Lttgeometry
Today at 4:03 AM
Funcshun840
an hour ago
Reducing the exponents for good
RobertRogo   1
N 2 hours ago by RobertRogo
Source: The national Algebra contest (Romania), 2025, Problem 3/Abstract Algebra (a bit generalized)
Let $A$ be a ring with unity such that for every $x \in A$ there exist $t_x, n_x \in \mathbb{N}^*$ such that $x^{t_x+n_x}=x^{n_x}$. Prove that
a) If $t_x \cdot 1 \in U(A), \forall x \in A$ then $x^{t_x+1}=x, \forall x \in A$
b) If there is an $x \in A$ such that $t_x \cdot 1 \notin U(A)$ then the result from a) may no longer hold.

Authors: Laurențiu Panaitopol, Dorel Miheț, Mihai Opincariu, me, Filip Munteanu
1 reply
RobertRogo
May 20, 2025
RobertRogo
2 hours ago
Functional equation
shobber   19
N 3 hours ago by Unique_solver
Source: Canada 2002
Let $\mathbb N = \{0,1,2,\ldots\}$. Determine all functions $f: \mathbb N \to \mathbb N$ such that
\[ xf(y) + yf(x) = (x+y) f(x^2+y^2)  \]
for all $x$ and $y$ in $\mathbb N$.
19 replies
shobber
Mar 5, 2006
Unique_solver
3 hours ago
Prove the inequality
Butterfly   0
3 hours ago
Let $a,b,c$ be real numbers such that $a+b+c=3$. Prove $$a^3b+b^3c+c^3a\le \frac{9}{32}(63+5\sqrt{105}).$$
0 replies
Butterfly
3 hours ago
0 replies
Functional equation
shactal   1
N 3 hours ago by ariopro1387
Let $f:\mathbb R\to \mathbb R$ a function satifying $$f(x+2xy) = f(x) + 2f(xy)$$for all $x,y\in \mathbb R$.
If $f(1991)=a$, then what is $f(1992)$, the answer is in terms of $a$.
1 reply
shactal
5 hours ago
ariopro1387
3 hours ago
interesting diophantiic fe in natural numbers
skellyrah   5
N 3 hours ago by skellyrah
Find all functions \( f : \mathbb{N} \to \mathbb{N} \) such that for all \( m, n \in \mathbb{N} \),
\[
mn + f(n!) = f(f(n))! + n \cdot \gcd(f(m), m!).
\]
5 replies
skellyrah
Yesterday at 8:01 AM
skellyrah
3 hours ago
Non-linear Recursive Sequence
amogususususus   3
N 3 hours ago by SunnyEvan
Given $a_1=1$ and the recursive relation
$$a_{i+1}=a_i+\frac{1}{a_i}$$for all natural number $i$. Find the general form of $a_n$.

Is there any way to solve this problem and similar ones?
3 replies
amogususususus
Jan 24, 2025
SunnyEvan
3 hours ago
Sequence divisible by infinite primes - Brazil Undergrad MO
rodamaral   5
N 3 hours ago by cursed_tangent1434
Source: Brazil Undergrad MO 2017 - Problem 2
Let $a$ and $b$ be fixed positive integers. Show that the set of primes that divide at least one of the terms of the sequence $a_n = a \cdot 2017^n + b \cdot 2016^n$ is infinite.
5 replies
rodamaral
Nov 1, 2017
cursed_tangent1434
3 hours ago
Reduction coefficient
zolfmark   2
N 4 hours ago by wh0nix

find Reduction coefficient of x^10

in(1+x-x^2)^9
2 replies
zolfmark
Jul 17, 2016
wh0nix
4 hours ago
A Ball-Drawing problem
Vivacious_Owl   9
N Apr 28, 2025 by Vivacious_Owl
Source: Inspired by a certain daily routine of mine
There are N identical black balls in a bag. I randomly take one ball out of the bag. If it is a black ball, I throw it away and put a white ball back into the bag instead. If it is a white ball, I simply throw it away and do not put anything back into the bag. The probability of getting any ball is the same.
Questions:
1. How many times will I need to reach into the bag to empty it?
2. What is the ratio of the expected maximum number of white balls in the bag to N in the limit as N goes to infinity?
9 replies
Vivacious_Owl
Apr 24, 2025
Vivacious_Owl
Apr 28, 2025
A Ball-Drawing problem
G H J
G H BBookmark kLocked kLocked NReply
Source: Inspired by a certain daily routine of mine
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Vivacious_Owl
4 posts
#1
Y by
There are N identical black balls in a bag. I randomly take one ball out of the bag. If it is a black ball, I throw it away and put a white ball back into the bag instead. If it is a white ball, I simply throw it away and do not put anything back into the bag. The probability of getting any ball is the same.
Questions:
1. How many times will I need to reach into the bag to empty it?
2. What is the ratio of the expected maximum number of white balls in the bag to N in the limit as N goes to infinity?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
alexheinis
10624 posts
#2 • 1 Y
Y by Vivacious_Owl
1. Suppose we have $k$ black and $l$ white balls in the bag. The quantity $f:=2k+l$ decreases by exactly 1 with each step. Hence we need exactly $2N$ steps to empty the bag. I will think about 2 at a later time.
I think we can use the reflection principle to count paths with a given maximum.
This post has been edited 2 times. Last edited by alexheinis, Apr 24, 2025, 5:00 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Vivacious_Owl
4 posts
#3
Y by
Correct!
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
solyaris
649 posts
#4 • 2 Y
Y by GreenKeeper, Saucitom
Nice problem! What is the real source of the problem? ('Inspired by a certain daily routine' seems somewhat implausible: The problem is certainly very carefully designed.) Do you have references for the problem (and maybe a solution)?

I suspect a rigorous answer to part 2 will be tricky and technical: I don't think reflection principle will be useful due to the lack of symmetry.

Let me give some heuristics that are hopefully correct and should even give a stronger result. I will write $n = N$ for convenience. First of all we will represent the urn model by a random walk: Let $S_k$ denote the number of white balls in the bag at time $k$ (i.e. after $k$ steps). Note that $S_k = j$ implies that at time $k$ there are $j$ white balls and $n- \frac{k+j} 2$ black balls in the urn (which is consistent with alexheinis answer to part 1). Thus we have
$$
P(S_{k+1} = j+1| S_k = j) = \frac{n - \frac 1 2 (j+k)}{n + \frac 1 2 (j-k)} =: p_{k,j}
$$and $P(S_{k+1} = j-1| S_k = j) = 1 - p_{k,j}$. Thus $S_k$, $0 \le k \le 2n$ is a random walk (on nonnegative integers) with increments $\pm 1$. The increments however are not independent, and also depend on the time, so this is messy. So far this is rigorous. Now we want to consider $n \to \infty$ and scale time and space by $n$.

If we had a simple random walk, scaling time by $n$ and space by $\sqrt n$ gives Brownian motion. In our random walk we locally should also expect random fluctuations to be of size $\sqrt n$, so in the space scaling of $n$, we should expect all randomness to dissapear, i.e. the paths of our random walk should converge to a deterministic curve. The slope of this curve should correspond to the local drift of the random walk, which is given by $2 p_{k,j} - 1$. Introducing $t := \frac k n$ and $x := \frac j n$, the above one-step-transition thus gives the differential equation
$$
x' = 2 \frac{1 - \frac 1 2 (x+t)}{1 + \frac 1 2(x-t) }-1 = \frac{2-t-3x}{2-t+x}.
$$We also have the initial value $x(0) = 0$. The maximum of $x$ on the interval $[0,2]$ should give the ratio of the maximum number of white balls in the bag to $n$. Now this is a purely analytic problem. I don't know if one can solve the ODE by hand, but with help of wolfram we seem to get an implicit description of $x$, namely
$$
\ln(1-\frac 1 2 (x+t)) +  \frac{x}{1 - \frac 1 2 (x+t)} = 0. 
$$At the maximum of $x$ we have $x' = 0$, i.e. $2-t-3x = 0$ and using this to eliminate $t$ in the above implicit description we get a maximum avlue of $x = \frac 1 e$. I didn't double check my calculations and I also didn't perform simulations in order to see whether this is plausible, but it seems reasonable.

Note that (if there are no computational mistakes above), we get something stronger: The scaled random walk paths should converge to the above deterministic curve, so the scaled
maximum should not just converge to $\frac 1 e$ in expectation, but also in distribution. (If true, this should also be visible in simulations.)

The above obviously is not a proof. It would require some effort to show the claimed convergence of the random walk paths, but in priciple it should be possible.There also might be simpler arguments for the convergence in expectation, e.g. with arguments of a combinatorial flavor.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
GreenKeeper
1699 posts
#5
Y by
solyaris wrote:
I don't know if one can solve the ODE by hand
https://en.wikipedia.org/wiki/Homogeneous_differential_equation#Special_case
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
solyaris
649 posts
#6
Y by
Thanks for pointing that out! So indeed the ODE can be solved by hand (and I would hope that this reproduces the solution given by Wolfram alpha).
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
GreenKeeper
1699 posts
#7 • 1 Y
Y by Saucitom
Yeah the solution of the ODE is correct. Also I wrote a simple simulation in Python:
import numpy as np
 
N = 10**4
SIMS = 10**3
 
maxs = []
 
for _ in range(SIMS):
    blacks = N
    whites = 0
    whites_max = 0
 
    for _ in range(2 * N):
        random_ball = np.random.randint(blacks + whites)
 
        if random_ball < blacks:
            blacks -= 1
            whites += 1
        else:
            whites -= 1
 
        whites_max = max(whites_max, whites)
 
    maxs.append(whites_max)
 
print(np.mean(maxs) / N)
print(np.exp(-1))

The results look promising. For $N=10^4$ and $10^3$ simulations I got approximately $0.3702$, a decent match for $1/e\doteq0.3679$. I might try the distribution later if nobody beats me to it.

UPDATE: For $N=10^5$ and $10^4$ simulations I got approximately $0.3684$, so it seems that $1/e$ is indeed correct.
This post has been edited 3 times. Last edited by GreenKeeper, Apr 26, 2025, 7:36 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Vivacious_Owl
4 posts
#8 • 1 Y
Y by Saucitom
Glad you like the problem! I came up with it while taking vitamin pills. Dose of one pill was too high so I would take half a pill and put the other back in the bottle. I noticed that at first when the bottle was new I mostly got a whole pill each time, but then as the halves accumulated they started to appear more often and after a little it’s mostly them you get with occasional wholes until the end. This was counterintuitive to me at the time. Initially I assumed that they would equalize in number and remain in this equilibrium until the very end. But as often the case with probabilities our intuition fails. If you plot the graphs of white and black balls you’ll see that after crossing the white ball curve stays above the black one until the very end. Moreover the difference between them after crossing grows for a while, reaches max at a point and starts decreasing towards the end.

Considering the continuous case in your notation the differential equations of the process are dy/dt=-y/(x+y) and dx/dt=(y-x)/(x+y). where y is the ratio of the expected number of black balls in the bag to N in the limit as N goes to infinity. From them follows the invariant t+x+2y=2 and the solution t=2+y(ln(y)-2) (considering the initial conditions t=0, x=0, y=1). Now dx/dt=0 when y=x which gives x=y=1/e at t=2-3/e

BTW the invariant could be deduced directly from the problem. Consider the number of times you need to touch the balls to empty the bag. Think of it not as if you replace one ball for another but simply change its color by touching it. In the middle of the process it’s T times you have already touched the balls + X times for each white ball you have left and 2Y times for each black one. It’s 2N in total. So T+X+2Y=2N

The max difference between the white and black balls I was referring to at beginning can be inferred from the differential equations (x-y)'=(2y-x)/(x+y)=0 so x=2y and y=1/e^2, x=2/e^2, t=2-4/e^2
This post has been edited 1 time. Last edited by Vivacious_Owl, Apr 26, 2025, 1:48 AM
Reason: typo
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Saucitom
87 posts
#9
Y by
Very nice discussion! This problem reminds me of a variation of Polya Urn, which appears last year at Miklos-Schweitzer (P11). In both cases, one can approximate the solution with an ODE.
This post has been edited 1 time. Last edited by Saucitom, Apr 26, 2025, 3:34 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Vivacious_Owl
4 posts
#10
Y by
The next step would be to generalize it to an arbitrary number of ball colors (or better in this case mark them with numbers maybe). Say initially there are N balls marked 0 in the bag. Then you randomly pick a ball and replace k-th ball number with a ball marked k+1. It could be a separate post with a beautiful and well known result!
Z K Y
N Quick Reply
G
H
=
a