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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)!

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[*]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
Help my diagram has too many points
MarkBcc168   29
N 35 minutes ago by VideoCake
Source: IMO Shortlist 2023 G6
Let $ABC$ be an acute-angled triangle with circumcircle $\omega$. A circle $\Gamma$ is internally tangent to $\omega$ at $A$ and also tangent to $BC$ at $D$. Let $AB$ and $AC$ intersect $\Gamma$ at $P$ and $Q$ respectively. Let $M$ and $N$ be points on line $BC$ such that $B$ is the midpoint of $DM$ and $C$ is the midpoint of $DN$. Lines $MP$ and $NQ$ meet at $K$ and intersect $\Gamma$ again at $I$ and $J$ respectively. The ray $KA$ meets the circumcircle of triangle $IJK$ again at $X\neq K$.

Prove that $\angle BXP = \angle CXQ$.

Kian Moshiri, United Kingdom
29 replies
MarkBcc168
Jul 17, 2024
VideoCake
35 minutes ago
A problem with series
Pena317   1
N 37 minutes ago by venhancefan777
Source: P5, Mexico Center Regional Olympiad 2019
A serie of positive integers $a_{1}$,$a_{2}$,. . . ,$a_{n}$ is $auto-delimited$ if for every index $i$ that holds $1\leq i\leq n$, there exist at least $a_{i}$ terms of the serie such that they are all less or equal to $i$.
Find the maximum value of the sum $a_{1}+a_{2}+\cdot \cdot \cdot+a_{n}$, where $a_{1}$,$a_{2}$,. . . ,$a_{n}$ is an $auto-delimited$ serie.
1 reply
Pena317
Nov 28, 2019
venhancefan777
37 minutes ago
IMO 2009, problem 4
ZetaX   60
N 44 minutes ago by FarrukhBurzu
Let $ ABC$ be a triangle with $ AB = AC$ . The angle bisectors of $ \angle C AB$ and $ \angle AB C$ meet the sides $ B C$ and $ C A$ at $ D$ and $ E$ , respectively. Let $ K$ be the incentre of triangle $ ADC$. Suppose that $ \angle B E K = 45^\circ$ . Find all possible values of $ \angle C AB$ .

Jan Vonk, Belgium, Peter Vandendriessche, Belgium and Hojoo Lee, Korea
60 replies
ZetaX
Jul 16, 2009
FarrukhBurzu
44 minutes ago
Tennis tournament with rotating courts
v_Enhance   6
N an hour ago by Blast_S1
Source: ELMO Shortlist 2013: Problem C10, by Ray Li
Let $N\ge2$ be a fixed positive integer. There are $2N$ people, numbered $1,2,...,2N$, participating in a tennis tournament. For any two positive integers $i,j$ with $1\le i<j\le 2N$, player $i$ has a higher skill level than player $j$. Prior to the first round, the players are paired arbitrarily and each pair is assigned a unique court among $N$ courts, numbered $1,2,...,N$.

During a round, each player plays against the other person assigned to his court (so that exactly one match takes place per court), and the player with higher skill wins the match (in other words, there are no upsets). Afterwards, for $i=2,3,...,N$, the winner of court $i$ moves to court $i-1$ and the loser of court $i$ stays on court $i$; however, the winner of court 1 stays on court 1 and the loser of court 1 moves to court $N$.

Find all positive integers $M$ such that, regardless of the initial pairing, the players $2, 3, \ldots, N+1$ all change courts immediately after the $M$th round.

Proposed by Ray Li
6 replies
v_Enhance
Jul 23, 2013
Blast_S1
an hour ago
∑(a-b)(a-c)/(2a^2 + (b+c)^2) >= 0
Zhero   24
N an hour ago by RevolveWithMe101
Source: ELMO Shortlist 2010, A2
Let $a,b,c$ be positive reals. Prove that
\[ \frac{(a-b)(a-c)}{2a^2 + (b+c)^2} + \frac{(b-c)(b-a)}{2b^2 + (c+a)^2} + \frac{(c-a)(c-b)}{2c^2 + (a+b)^2} \geq 0. \]

Calvin Deng.
24 replies
Zhero
Jul 5, 2012
RevolveWithMe101
an hour ago
i am not abel to prove or disprove
frost23   8
N an hour ago by frost23
Source: made on my own
sorrrrrry
8 replies
frost23
3 hours ago
frost23
an hour ago
points on sides of a triangle, intersections, extensions, ratio of areas wanted
parmenides51   1
N an hour ago by FrancoGiosefAG
Source: Mexican Mathematical Olympiad 1997 OMM P5
Let $P,Q,R$ be points on the sides $BC,CA,AB$ respectively of a triangle $ABC$. Suppose that $BQ$ and $CR$ meet at $A', AP$ and $CR$ meet at $B'$, and $AP$ and $BQ$ meet at $C'$, such that $AB' = B'C', BC' =C'A'$, and $CA'= A'B'$. Compute the ratio of the area of $\triangle PQR$ to the area of $\triangle ABC$.
1 reply
parmenides51
Jul 28, 2018
FrancoGiosefAG
an hour ago
starting with intersecting circles, line passes through midpoint wanted
parmenides51   2
N 2 hours ago by EmersonSoriano
Source: Peru Ibero TST 2014
Circles $C_1$ and $C_2$ intersect at different points $A$ and $B$. The straight lines tangents to $C_1$ that pass through $A$ and $B$ intersect at $T$. Let $M$ be a point on $C_1$ that is out of $C_2$. The $MT$ line intersects $C_1$ at $C$ again, the $MA$ line intersects again to $C_2$ in $K$ and the line $AC$ intersects again to the circumference $C_2$ in $L$. Prove that the $MC$ line passes through the midpoint of the $KL$ segment.
2 replies
parmenides51
Jul 23, 2019
EmersonSoriano
2 hours ago
An inequality
Rushil   14
N 2 hours ago by frost23
Source: Indian RMO 1994 Problem 8
If $a,b,c$ are positive real numbers such that $a+b+c = 1$, prove that \[ (1+a)(1+b)(1+c) \geq 8 (1-a)(1-b)(1-c) . \]
14 replies
Rushil
Oct 25, 2005
frost23
2 hours ago
3 var inequality
SunnyEvan   6
N 2 hours ago by JARP091
Let $ a,b,c \in R $ ,such that $ a^2+b^2+c^2=4(ab+bc+ca)$Prove that :$$ \frac{7-2\sqrt{14}}{48} \leq \frac{a^3b+b^3c+c^3a}{(a^2+b^2+c^2)^2} \leq \frac{7+2\sqrt{14}}{48} $$
6 replies
SunnyEvan
May 17, 2025
JARP091
2 hours ago
collinearity as a result of perpendicularity and equality
parmenides51   2
N 2 hours ago by FrancoGiosefAG
Source: Mexican Mathematical Olympiad 1996 OMM P6
In a triangle $ABC$ with $AB < BC < AC$, points $A' ,B' ,C'$ are such that $AA' \perp BC$ and $AA' = BC, BB' \perp  CA$ and $BB'=CA$, and $CC' \perp AB$ and $CC'= AB$, as shown on the picture. Suppose that $\angle AC'B$ is a right angle. Prove that the points $A',B' ,C' $ are collinear.
2 replies
parmenides51
Jul 28, 2018
FrancoGiosefAG
2 hours ago
3 var inequality
JARP091   6
N 2 hours ago by JARP091
Source: Own
Let \( x, y, z \in \mathbb{R}^+ \). Prove that
\[
\sum_{\text{cyc}} \frac{x^3}{y^2 + z^2} \geq \frac{x + y + z}{2}
\]without using the Rearrangement Inequality or Chebyshev's Inequality.
6 replies
JARP091
Today at 8:54 AM
JARP091
2 hours ago
Helplooo
Bet667   1
N 2 hours ago by Lil_flip38
Let $ABC$ be an acute angled triangle.And altitudes $AD$ and $BE$ intersects at point $H$.Let $F$ be a point on ray $AD$ such that $DH=DF$.Circumcircle of $AEF$ intersects line $BC$ at $K$ and $L$ so prove that $BK=BL$
1 reply
Bet667
3 hours ago
Lil_flip38
2 hours ago
Cyclic sum of 1/(a+1/b+1)
v_Enhance   22
N 3 hours ago by Rayvhs
Source: ELMO Shortlist 2013: Problem A2, by David Stoner
Prove that for all positive reals $a,b,c$,
\[\frac{1}{a+\frac{1}{b}+1}+\frac{1}{b+\frac{1}{c}+1}+\frac{1}{c+\frac{1}{a}+1}\ge \frac{3}{\sqrt[3]{abc}+\frac{1}{\sqrt[3]{abc}}+1}. \]Proposed by David Stoner
22 replies
v_Enhance
Jul 23, 2013
Rayvhs
3 hours ago
Sequence and prime factors
USJL   7
N Apr 18, 2025 by MathLuis
Source: 2025 Taiwan TST Round 2 Independent Study 1-N
Let $a_0,a_1,\ldots$ be a sequence of positive integers with $a_0=1$, $a_1=2$ and
\[a_n = a_{n-1}^{a_{n-1}a_{n-2}}-1\]for all $n\geq 2$. Show that if $p$ is a prime less than $2^k$ for some positive integer $k$, then there exists $n\leq k+1$ such that $p\mid a_n$.
7 replies
USJL
Mar 26, 2025
MathLuis
Apr 18, 2025
Sequence and prime factors
G H J
G H BBookmark kLocked kLocked NReply
Source: 2025 Taiwan TST Round 2 Independent Study 1-N
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USJL
540 posts
#1 • 1 Y
Y by MS_asdfgzxcvb
Let $a_0,a_1,\ldots$ be a sequence of positive integers with $a_0=1$, $a_1=2$ and
\[a_n = a_{n-1}^{a_{n-1}a_{n-2}}-1\]for all $n\geq 2$. Show that if $p$ is a prime less than $2^k$ for some positive integer $k$, then there exists $n\leq k+1$ such that $p\mid a_n$.
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Know1Math
4 posts
#2
Y by
how to solve this????? :(
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internationalnick123456
135 posts
#3 • 1 Y
Y by MS_asdfgzxcvb
Claim. $\nu_2(a_{n})\geq  n,\forall n\in\mathbb N$, $n$ is odd.
Now, we prove the statement by induction on \( k \).
When \( k\leq 2 \), the statement is obvious. Assume the statement holds for \( k-1 \). We prove it for \( k \).
Assume there exists an odd prime \( p \), \( p<2^k \), such that \( p \) does not divide \( a_1a_2\ldots a_{k+1} \).
We prove \( p-1\mid a_{k}a_{k-1} \). Indeed, let $$p-1=2^s q_1^{\ell_1} \ldots q_t^{\ell_t}$$where \( q_1,\ldots,q_t \) are odd primes. Then, by claim, \( 2^s\mid a_ka_{k-1} \). We only need to prove \( q_i^{\ell_i}\mid a_ka_{k-1} \).
Consider \( m \) as the smallest integer such that \( q_i\leq 2^m \). Then \( m\leq k-1 \).
By the inductive hypothesis, \( q_i\mid a_j \) for some \( 2\leq j\leq m+1 \).
Applying the LTE lemma, we have
\[
\nu_{q_i}(a_{j+2}) =\nu_{q_i}\left(a_{j+1}^{a_{j+1}a_j} - 1\right) = \nu_{q_i}(a_j) + \nu_{q_i}(a_{j+1} + 1) = \nu_{q_i}(a_j)\left(1 + a_{j}a_{j-1}\right)\geq 7\nu_{q_i}(a_j)
\]$\Rightarrow \nu_{q_i}(a_{j+2n})\geq 7^n\nu_{q_i}(a_j)\geq 7^n\Rightarrow \nu_{q_i}(a_ka_{k-1})\geq 7^{\left\lfloor (k-j)/2\right\rfloor}$
Moreover, $
q_i^{\ell_i}\leq \frac{p-1}{2}\leq 2^{k-1}$ and $q_i>2^{m-1}$, hence, $\ell_i\leq \frac{k-1}{m-1}\leq 7^{\left\lfloor (k-j)/2\right\rfloor}
$
$\Rightarrow q_i^{\ell_i}\mid a_ka_{k-1}\Rightarrow p-1\mid a_ka_{k-1} \Rightarrow p\mid a_{k+1}$.
Contradiction. The statement is proved.
This post has been edited 1 time. Last edited by internationalnick123456, Mar 31, 2025, 9:39 AM
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zRevenant
14 posts
#4
Y by
:oops_sign:
This post has been edited 2 times. Last edited by zRevenant, Mar 31, 2025, 10:03 AM
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internationalnick123456
135 posts
#5
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@above I also thought about this approach before, however, I recognized that $\phi(\phi(p))\mid a_{n-2}a_{n-3}$ is not enough, we must also have $(\phi(p),a_{n-2}) = 1$!
This post has been edited 1 time. Last edited by internationalnick123456, Apr 1, 2025, 3:19 PM
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zRevenant
14 posts
#6 • 1 Y
Y by internationalnick123456
internationalnick123456 wrote:
@above I also thought about this approach before, however, I recognized that if $\phi(\phi(p))\mid a_{n-2}a_{n-3}$ is not enough, we must also have $(\phi(p),a_{n-2}) = 1$!

True, thx brother. Really be forgetting about this detail, cost me p2 IMO :(
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shanelin-sigma
164 posts
#7
Y by
USJL wrote:
Let $a_0,a_1,\ldots$ be a sequence of positive integers with $a_0=1$, $a_1=2$ and
\[a_n = a_{n-1}^{a_{n-1}a_{n-2}}-1\]for all $n\geq 2$. Show that if $p$ is a prime less than $2^k$ for some positive integer $k$, then there exists $n\leq k+1$ such that $p\mid a_n$.

Before getting start, let's prove a lemma:
Lemma: $a_{n-2}^{a_{n-2}a_{n-3}}|a_n$ for all $n$
Proof: first notice that $2|a_n \iff 2\nmid n$ (which can be proved by easy induction)
$\because a_n=a_{n-1}^{a_{n-1}a_{n-2}}-1=(a_{n-2}^{a_{n-2}a_{n-3}}-1)^{a_{n-1}a_{n-2}}-1 \overset{2|a_{n-1}a_{n-2}}{\implies} (a_{n-2}^{a_{n-2}a_{n-3}}-1)^2-1|a_n \overset{(x-1)^2-1=x(x-2)}{\implies} a_{n-2}^{a_{n-2}a_{n-3}}|a_n$
Now, back to the original problem, let's induct on $k$. For $k=0,1,2,3,4$ we can prove by easy calculation. Now suppose our claim is true for $k\ge m$ ($m\geq 4$) and let’s consider $k=m+1$
From Fermat’s little theorem it’s sufficient to show that $\forall$ prime $p$, $2^m<p<2^{m+1}$, $\exists n\leq m+2$ such that $p-1|a_{n-2}a_{n-1}$.
Now, for every prime divisor $q$ of $p-1$, let’s prove that $\nu_q(p-1)\leq \nu_q(a_ma_{m+1})$
$\because p>2$ so $2|p-1$, therefore $q\le \frac p2 \le 2^m$. I'll split it into two cases:
  • for $q<2^{m-1}$,from induction hypothesis we know $q|a_i$ for some $i\le m-2$ and the Lemma we know at least one of $a_{m-2}$,$a_{m-1}$ is divisible by $q$, $\implies \nu_q(a_ma_{m+1})\ge \nu_q(a_{m-2}^{a_{m-2}a_{m-3}}a_{m-1}^{a_{m-1}a_{m-2}})\geq a_{m-2}\nu_q(a_{m-2}a_{m-1}) \geq a_{m-2}$
    and since $m\geq 4$ so $a_{m-2} \geq m+1\geq log_2(2^{m+1}) > log_2(p-1)\geq \nu_q(p-1)$, combine the two inequality we can reach our conclusion.
  • for $2^{m-1}<q<2^m$, we must have $p-1=2q$ (otherwise, $p>2^{m+1}$) and from induction hypothesis and theLemma we know $2q=p-1| a_ma_{m+1}$
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MathLuis
1546 posts
#8 • 1 Y
Y by teomihai
Notice that $a_n=a_{n-1}^{a_{n-1}a_{n-2}}-1=(a_{n-2}^{a_{n-2}a_{n-3}}-1)^{a_{n-1}a_{n-2}}-1$ and since consecutive terms of $a_i$ alter in parity we must have that $a_n \equiv (-1)^{a_{n-1}a_{n-2}}-1 \equiv 0 \pmod{a_{n-2}^{a_{n-2}a_{n-3}}}$ this is more than sufficient to stablish that $\nu_2(a_n) \ge n$ for all odd $n$ (in fact it will grow faster than exponentially).
Now we will prove the statement by strong induction on $k$ where it is trivial for $k=1$, now if it were true for $k$ we prove it for $k+1$, so suppose that there exists a prime $2^k<p<2^{k+1}$ (since other cases are covered) such that $p \not \; \mid a_1a_2 \cdots a_{k+1}$ then we prove that $p-1 \mid a_ka_{k+1}$ which would be enough to conclude $p \mid a_{k+2}$ by FLT.
Let $p-1=2^{a} \cdot q_1^{\alpha_1} \cdots q_i^{\alpha_i}$ for odd primes $q_j$, then $\nu_2(a_ka_{k+1}) \ge 2k+1>a$ so it remains to check that for any index $1 \le j \le i$ it happens that $\nu_{q_j}(a_ka_{k+1}) \ge \nu_{q_j}(p-1)$.
The way to do this is to notice that every single $q_j<2^{\frac{k}{\alpha_j}}$ by definition of $p$ which shows the existence of some $a_j'$ for which $q_j \mid a_j'$ where $j' \le \frac{k}{\alpha_j}+1$ by inductive hypothesis and so say that next up we have $a_{j'}^{a_{j'}a_{j'-1}} \mid a_{j'+2}$ which shows that $\nu_{q_j}(a_{j'+2}) \ge (a_{j'}a_{j'-1})\nu_{q_j}(a_{j'})$ this is suficient to prove it whenever $\alpha_j=1$, now for $\alpha_j \ge 2$ notice that $\nu_{q_i}(a_{j'+2m}) \ge a_{j'+2m-2} \cdots a_{j'-1}$ and obviously for $\alpha_j=2$ this solves it by checking initial terms even, for $\alpha_j \ge 3$ it now happens (for $k \ge 5$)(check base cases $k=2,3$ just to be safe lol) that:
\[ \nu_{q_i}(a_{j'+2m}) \ge 6 \cdot (728)^{\frac{2(k-2)}{3}-2} \ge k+1 \ge \log_2(p-1)>\nu_{q_j}(p-1) \]Where here we get $j'+2m$ to be $k$ or $k+1$ but when $k=4$ we can ignore the rest of terms and only consider having two consecutives whose product is at least $6>5>4$ so it also works in this case and for $k=3$ too, so we have it all covered and from the $\nu_{q_j}$'s we have that the inductive step is complete, thus we are done :cool:.
This post has been edited 2 times. Last edited by MathLuis, Apr 18, 2025, 8:53 PM
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