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Contests & Programs AMC and other contests, summer programs, etc.
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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]
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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
Close to JMO, but not close enough
isache   12
N 4 minutes ago by LearnMath_105
Im currently a freshman in hs, and i rlly wanna make jmo in sophmore yr. Ive been cooking at in-person competitions recently (ucsd hmc, scmc, smt, mathcounts) but I keep fumbling jmo. this yr i had a 133.5 on 10b and a 9 on aime. How do i get that up by 20 points to a 240?
12 replies
2 viewing
isache
Yesterday at 11:37 PM
LearnMath_105
4 minutes ago
Bring Back Downvotes
heheman   99
N 29 minutes ago by heheman
i would like to start a petition to bring back downvote, it you agree then write "bbd $    $" in threads
99 replies
1 viewing
heheman
Yesterday at 7:21 PM
heheman
29 minutes ago
n^6 + 5n^3 + 4n + 116 is the product of two or more consecutive numbers
Amir Hossein   2
N an hour ago by KTYC
Source: Bulgaria JBMO TST 2018, Day 1, Problem 3
Find all positive integers $n$ such that the number
$$n^6 + 5n^3 + 4n + 116$$is the product of two or more consecutive numbers.
2 replies
Amir Hossein
Jun 25, 2018
KTYC
an hour ago
[TEST RELEASED] OMMC Year 5
DottedCaculator   180
N an hour ago by fuzimiao2013
Test portal: https://ommc-test-portal-2025.vercel.app/

Hello to all creative problem solvers,

Do you want to work on a fun, untimed team math competition with amazing questions by MOPpers and IMO & EGMO medalists? $\phantom{You lost the game.}$
Do you want to have a chance to win thousands in cash and raffle prizes (no matter your skill level)?

Check out the fifth annual iteration of the

Online Monmouth Math Competition!

Online Monmouth Math Competition, or OMMC, is a 501c3 accredited nonprofit organization managed by adults, college students, and high schoolers which aims to give talented high school and middle school students an exciting way to develop their skills in mathematics.

Our website: https://www.ommcofficial.org/

This is not a local competition; any student 18 or younger anywhere in the world can attend. We have changed some elements of our contest format, so read carefully and thoroughly. Join our Discord or monitor this thread for updates and test releases.

How hard is it?

We plan to raffle out a TON of prizes over all competitors regardless of performance. So just submit: a few minutes of your time will give you a great chance to win amazing prizes!

How are the problems?

You can check out our past problems and sample problems here:
https://www.ommcofficial.org/sample
https://www.ommcofficial.org/2022-documents
https://www.ommcofficial.org/2023-documents
https://www.ommcofficial.org/ommc-amc

How will the test be held?/How do I sign up?

Solo teams?

Test Policy

Timeline:
Main Round: May 17th - May 24th
Test Portal Released. The Main Round of the contest is held. The Main Round consists of 25 questions that each have a numerical answer. Teams will have the entire time interval to work on the questions. They can submit any time during the interval. Teams are free to edit their submissions before the period ends, even after they submit.

Final Round: May 26th - May 28th
The top placing teams will qualify for this invitational round (5-10 questions). The final round consists of 5-10 proof questions. Teams again will have the entire time interval to work on these questions and can submit their proofs any time during this interval. Teams are free to edit their submissions before the period ends, even after they submit.

Conclusion of Competition: Early June
Solutions will be released, winners announced, and prizes sent out to winners.

Scoring:

Prizes:

I have more questions. Whom do I ask?

We hope for your participation, and good luck!

OMMC staff

OMMC’S 2025 EVENTS ARE SPONSORED BY:

[list]
[*]Nontrivial Fellowship
[*]Citadel
[*]SPARC
[*]Jane Street
[*]And counting!
[/list]
180 replies
DottedCaculator
Apr 26, 2025
fuzimiao2013
an hour ago
IMO Shortlist 2009 - Problem G3
April   49
N 2 hours ago by Ilikeminecraft
Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram.
Prove that $GR=GS$.

Proposed by Hossein Karke Abadi, Iran
49 replies
April
Jul 5, 2010
Ilikeminecraft
2 hours ago
[$10K+ IN PRIZES] Poolesville Math Tournament (PVMT) 2025
qwerty123456asdfgzxcvb   19
N 2 hours 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
19 replies
qwerty123456asdfgzxcvb
Apr 5, 2025
Ruegerbyrd
2 hours ago
Four tangent lines concur on the circumcircle
v_Enhance   36
N 2 hours ago by Ilikeminecraft
Source: USA TSTST 2018 Problem 3
Let $ABC$ be an acute triangle with incenter $I$, circumcenter $O$, and circumcircle $\Gamma$. Let $M$ be the midpoint of $\overline{AB}$. Ray $AI$ meets $\overline{BC}$ at $D$. Denote by $\omega$ and $\gamma$ the circumcircles of $\triangle BIC$ and $\triangle BAD$, respectively. Line $MO$ meets $\omega$ at $X$ and $Y$, while line $CO$ meets $\omega$ at $C$ and $Q$. Assume that $Q$ lies inside $\triangle ABC$ and $\angle AQM = \angle ACB$.

Consider the tangents to $\omega$ at $X$ and $Y$ and the tangents to $\gamma$ at $A$ and $D$. Given that $\angle BAC \neq 60^{\circ}$, prove that these four lines are concurrent on $\Gamma$.

Evan Chen and Yannick Yao
36 replies
v_Enhance
Jun 26, 2018
Ilikeminecraft
2 hours ago
Inspired by m4thbl3nd3r
sqing   1
N 2 hours ago by sqing
Source: Own
Let $  a, b,c>0,b+c>a$. Prove that$$\sqrt{\frac{a}{b+c-a}}-\frac{2a^2-b^2-c^2}{(a+b)(a+c)}\geq 1$$$$\frac{a}{b+c-a}-\frac{2a^2-b^2-c^2}{(a+b)(a+c)} \geq  \frac{4\sqrt 2}{3}-1$$
1 reply
sqing
3 hours ago
sqing
2 hours ago
2-var inequality
sqing   6
N 3 hours ago by sqing
Source: Own
Let $ a,b> 0 $ and $2a+2b+ab=5. $ Prove that
$$ \frac{a^2}{b^2}+\frac{1}{a^2}-a^2\geq  1$$$$ \frac{a^3}{b^3}+\frac{1}{a^3}-a^3\geq  1$$
6 replies
sqing
Yesterday at 1:19 PM
sqing
3 hours ago
Inequality
Amin12   8
N 3 hours ago by A.H.H
Source:  Iran 3rd round-2017-Algebra final exam-P3
Let $a,b$ and $c$ be positive real numbers. Prove that
$$\sum_{cyc} \frac {a^3b}{(3a+2b)^3} \ge \sum_{cyc} \frac {a^2bc}{(2a+2b+c)^3} $$
8 replies
1 viewing
Amin12
Sep 2, 2017
A.H.H
3 hours ago
Every subset of size k has sum at most N/2
orl   50
N 4 hours ago by de-Kirschbaum
Source: USAMO 2006, Problem 2, proposed by Dick Gibbs
For a given positive integer $k$ find, in terms of $k$, the minimum value of $N$ for which there is a set of $2k + 1$ distinct positive integers that has sum greater than $N$ but every subset of size $k$ has sum at most $\tfrac{N}{2}.$
50 replies
orl
Apr 20, 2006
de-Kirschbaum
4 hours ago
Inspired by a9opsow_
sqing   2
N 4 hours ago by sqing
Source: Own
Let $ a,b > 0  .$ Prove that
$$ \frac{(ka^2 - kab-b)^2 + (kb^2 - kab-a)^2 + (ab-ka-kb )^2}{ (ka+b)^2 + (kb+a)^2+(a - b)^2 }\geq  \frac {1}{(k+1)^2}$$Where $ k\geq 0.37088 .$
$$\frac{(a^2 - ab-b)^2 + (b^2 - ab-a)^2 + ( ab-a-b)^2}{a^2 +b^2+(a - b)^2 } \geq 1$$$$ \frac{(2a^2 - 2ab-b)^2 + (2b^2 - 2ab-a)^2 + (ab-2a-2b )^2}{ (2a+b)^2 + (2b+a)^2+(a - b)^2 }\geq  \frac 19$$
2 replies
sqing
4 hours ago
sqing
4 hours ago
Cute NT Problem
M11100111001Y1R   5
N 5 hours ago by compoly2010
Source: Iran TST 2025 Test 4 Problem 1
A number \( n \) is called lucky if it has at least two distinct prime divisors and can be written in the form:
\[
n = p_1^{\alpha_1} + \cdots + p_k^{\alpha_k}
\]where \( p_1, \dots, p_k \) are distinct prime numbers that divide \( n \). (Note: it is possible that \( n \) has other prime divisors not among \( p_1, \dots, p_k \).) Prove that for every prime number \( p \), there exists a lucky number \( n \) such that \( p \mid n \).
5 replies
M11100111001Y1R
Tuesday at 7:20 AM
compoly2010
5 hours ago
3 var inequality
SunnyEvan   13
N 5 hours ago by Nguyenhuyen_AG
Let $ a,b,c \in R $ ,such that $ a^2+b^2+c^2=4(ab+bc+ca)$Prove that :$$ \frac{7-2\sqrt{14}}{48} \leq \frac{a^3b+b^3c+c^3a}{(a^2+b^2+c^2)^2} \leq \frac{7+2\sqrt{14}}{48} $$
13 replies
SunnyEvan
May 17, 2025
Nguyenhuyen_AG
5 hours ago
Jumping on Lily Pads to Avoid a Snake
brandbest1   53
N Apr 29, 2025 by ESAOPS
Source: 2014 AMC 10B #25 & 2014 AMC 12B #22
In a small pond there are eleven lily pads in a row labeled $0$ through $10$. A frog is sitting on pad $1$. When the frog is on pad $N$, $0<N<10$, it will jump to pad $N-1$ with probability $\frac{N}{10}$ and to pad $N+1$ with probability $1-\frac{N}{10}$. Each jump is independent of the previous jumps. If the frog reaches pad $0$ it will be eaten by a patiently waiting snake. If the frog reaches pad $10$ it will exit the pond, never to return. What is the probability that the frog will escape being eaten by the snake?

$ \textbf {(A) } \frac{32}{79} \qquad \textbf {(B) } \frac{161}{384} \qquad \textbf {(C) } \frac{63}{146} \qquad \textbf {(D) } \frac{7}{16} \qquad \textbf {(E) } \frac{1}{2} $
53 replies
brandbest1
Feb 20, 2014
ESAOPS
Apr 29, 2025
Jumping on Lily Pads to Avoid a Snake
G H J
Source: 2014 AMC 10B #25 & 2014 AMC 12B #22
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mathwizard888
1635 posts
#40 • 2 Y
Y by Adventure10, Mango247
We know that is the answer because the problem asked for the probability the frog survives starting on Pad 1.
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hshiems
769 posts
#41 • 2 Y
Y by Adventure10, Mango247
hshiems wrote:
I have a question:

I'm confused about this statement.

sidenote

Edit: Here is the link to the AMC10/12B Math Jam: http://www.artofproblemsolving.com/School/mathjams.php?mj_id=353
FlakeLCR wrote:
@hshiems that formula was given in the problem :P
What do you mean by "that formula was given in the problem"?

How do you derive the formula anyway? Do you use recursion? Does the formula have anything to do with averages?

Edit: I've noticed that this problem is based on intuition. We use our intuition about averages to build the equation $ p_i=(1-\frac{i}{10})p_{i+1}+\frac{i}{10}p_{i-1} $ and we use our intuition about symmetry to find that $p_5=\frac{1}{2}$.

Edit: Why is it that the formula is $ p_i=(1-\frac{i}{10})p_{i+1}+\frac{i}{10}p_{i-1} $ instead of $ 2p_i=(1-\frac{i}{10})p_{i+1}+\frac{i}{10}p_{i-1} $?
This post has been edited 1 time. Last edited by hshiems, Apr 13, 2014, 6:11 PM
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pinetree1
1207 posts
#42 • 2 Y
Y by Adventure10, Mango247
The statement says that the probability of surviving is the probability of going to the previous pad and surviving plus the probability of going to the next pad and surviving. This is just a formula derived from the problem statement.
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mathtastic
3258 posts
#43 • 1 Y
Y by Adventure10
Well just think about it. The probability that you survive is the same as the probability you go to the right then survive plus the probability that you go to the left then survive.
Z K Y
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brandbest1
259 posts
#44 • 2 Y
Y by Adventure10, Mango247
Anyone want to post a solution to this using steady-state Markov chains, even though it's completely unecessary? I'm trying to get a hold on Markov chains with this problem, and I can't seem to get an answer.
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zero.destroyer
813 posts
#45 • 1 Y
Y by Adventure10
^ you literally would need to find 1+M+M^2+M^3+... infinite series for the markov matrix, which would involve finding (I -M)^(-1). Then multiply that by the (0,1,0,0,...) vector, and look at the 10th entry after multiplication.
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hamup1
380 posts
#46 • 1 Y
Y by Adventure10
Can someone explain to me how they would quickly arrive at the answer from the system of equations? Thanks!
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v_Enhance
6878 posts
#47 • 12 Y
Y by hamup1, MSTang, Mudkipswims42, AstrapiGnosis, Tawan, daniellionyang, candiru, pad, Ultroid999OCPN, HamstPan38825, Adventure10, Mango247
hamup1 wrote:
Can someone explain to me how they would quickly arrive at the answer from the system of equations? Thanks!

A year late, but apparently no one wrote it out, so...

Letting $p_i$ be the probability from lily pad $i$ (so $p_0=0$, $p_{10}=1$),
the system is rewritten as
\begin{align*}
	p_2 - p_1 &= \frac19(p_1-p_0) = \frac{1}{\binom91} p_1 \\
	p_3 - p_2 &= \frac28(p_2-p_1) = \frac{1}{\binom92} p_1\\
	p_4 - p_3 &= \frac37(p_3-p_2) = \frac{1}{\binom93} p_1 \\
	&\vdots \\
	p_{10} - p_9 &= \frac91(p_3-p_2) = \frac{1}{\binom99} p_1 \\
\end{align*}Adding them all, we get
\[ 1 = p_{10} = \left(
	\frac{1}{\binom90} + \frac{1}{\binom91}
	+ \frac{1}{\binom92} + \dots + \frac{1}{\binom99}
	\right) p_1. \]Equivalently, one can also add up to just $p_5$ to derive the equivalent
\[ \frac12 = p_{5} = \left(
	\frac{1}{\binom90} + \frac{1}{\binom91}
	+ \frac{1}{\binom92} + \frac{1}{\binom93} + \frac{1}{\binom94}
	\right) p_1. \]In any case, we have
\[
	p_1 = 
	\frac{1}{2\sum_{k=0}^4 \binom9k ^{-1} } = \frac{63}{146}.
\]
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Porky623
124 posts
#48 • 2 Y
Y by Adventure10, Mango247
Sadly enough, Einstein1 still has not posted another problem this year. Well, it was worth a try to look and see! :P
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Porky623
124 posts
#49 • 2 Y
Y by Adventure10, Mango247
Also, if a positive integer choose another positive integer less than or equal to the former is an integer, how would you get anything other than 1 as the numerator?
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vmaddur
864 posts
#50 • 1 Y
Y by Adventure10
@v_Enhance Could you break down the process of jumping from $p_1 = 9p_2/10+p_0/10$ and similar equations to what you have above?
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daniellionyang
1840 posts
#51 • 3 Y
Y by vmaddur, Adventure10, Mango247
$p_1 = \frac{9p_2}{10}+\frac{p_0}{10}$
$p_2-p_1=p_2-(\frac{9p_2}{10}+\frac{p_0}{10})=\frac{1}{10}p_2-\frac{1}{10}p_0$
$p_1-p_0=\frac{9p_2}{10}+\frac{p_0}{10}-p_0=\frac{9p_2}{10}-\frac{9p_0}{10}$
Therefore, $p_2-p_1=\frac{1}{9}(p_1-p_0) $.
This post has been edited 7 times. Last edited by daniellionyang, Sep 14, 2016, 2:37 AM
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vmaddur
864 posts
#52 • 2 Y
Y by Adventure10, Mango247
From there, is it just intuition to see that it is equal to $\frac{1}{\binom91} p_1$?
This post has been edited 2 times. Last edited by vmaddur, Sep 14, 2016, 2:37 AM
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daniellionyang
1840 posts
#53 • 2 Y
Y by vmaddur, Adventure10
I believe so. But from this state, its not at all hard to efficient bash to get to $p_5$.
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ESAOPS
263 posts
#54
Y by
this problem is bashy
sol
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N Quick Reply
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a