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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.

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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
Mathcounts Stories
Eyed   174
N 2 minutes ago by valisaxieamc
Because Mathcounts season is over, and I think we are free to discuss the problems, I think we can post our stories without fear of sharing information that is not allowed.
Hopefully this does not become too spammy, and I hope to see good quality stories. For example, I am not looking to see stories like this, or this.
Instead I would like to have higher quality stories.
Mods if this is not ok, then feel free to lock, I just hope we can all share our reflections on this contest, since mathcounts is now over.
Anyway, here's mine (At the request of Stormersyle)

old locked mathcounts stories thread
174 replies
Eyed
Jun 3, 2019
valisaxieamc
2 minutes ago
MATHCOUNTS
ILOVECATS127   2
N 18 minutes ago by ILOVECATS127
Hi,

I am looking to get on my school MATHCOUNTS team next year in 7th grade, and I had a question: Where do the school round questions come from? (Sprint, Chapter, Team, Countdown)
2 replies
ILOVECATS127
an hour ago
ILOVECATS127
18 minutes ago
How to get a 300+ on the NWEA MAP MATH test (URGENT)
nmlikesmath   18
N 44 minutes ago by snoopylover
I have 4 days till this test, I'm wondering how do I get a 300+ and what do I need to know, thank you.
18 replies
nmlikesmath
May 3, 2025
snoopylover
44 minutes ago
Acute Angle Altitudes... say that ten times fast
Math-lover1   0
an hour ago
In acute triangle $ABC$, points $D$ and $E$ are the feet of the angle bisector and altitude from $A$, respectively. Suppose that $AC-AB=36$ and $DC-DB=24$. Compute $EC-EB$.
0 replies
Math-lover1
an hour ago
0 replies
q(x) to be the product of all primes less than p(x)
orl   18
N 2 hours ago by happypi31415
Source: IMO Shortlist 1995, S3
For an integer $x \geq 1$, let $p(x)$ be the least prime that does not divide $x$, and define $q(x)$ to be the product of all primes less than $p(x)$. In particular, $p(1) = 2.$ For $x$ having $p(x) = 2$, define $q(x) = 1$. Consider the sequence $x_0, x_1, x_2, \ldots$ defined by $x_0 = 1$ and \[ x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} \] for $n \geq 0$. Find all $n$ such that $x_n = 1995$.
18 replies
orl
Aug 10, 2008
happypi31415
2 hours ago
IMO 2018 Problem 5
orthocentre   80
N 2 hours ago by OronSH
Source: IMO 2018
Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of positive integers. Suppose that there is an integer $N > 1$ such that, for each $n \geq N$, the number
$$\frac{a_1}{a_2} + \frac{a_2}{a_3} + \cdots + \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}$$is an integer. Prove that there is a positive integer $M$ such that $a_m = a_{m+1}$ for all $m \geq M$.

Proposed by Bayarmagnai Gombodorj, Mongolia
80 replies
orthocentre
Jul 10, 2018
OronSH
2 hours ago
Line passes through fixed point, as point varies
Jalil_Huseynov   60
N 3 hours ago by Rayvhs
Source: APMO 2022 P2
Let $ABC$ be a right triangle with $\angle B=90^{\circ}$. Point $D$ lies on the line $CB$ such that $B$ is between $D$ and $C$. Let $E$ be the midpoint of $AD$ and let $F$ be the seconf intersection point of the circumcircle of $\triangle ACD$ and the circumcircle of $\triangle BDE$. Prove that as $D$ varies, the line $EF$ passes through a fixed point.
60 replies
Jalil_Huseynov
May 17, 2022
Rayvhs
3 hours ago
Tangent to two circles
Mamadi   2
N 3 hours ago by A22-
Source: Own
Two circles \( w_1 \) and \( w_2 \) intersect each other at \( M \) and \( N \). The common tangent to two circles nearer to \( M \) touch \( w_1 \) and \( w_2 \) at \( A \) and \( B \) respectively. Let \( C \) and \( D \) be the reflection of \( A \) and \( B \) respectively with respect to \( M \). The circumcircle of the triangle \( DCM \) intersect circles \( w_1 \) and \( w_2 \) respectively at points \( E \) and \( F \) (both distinct from \( M \)). Show that the line \( EF \) is the second tangent to \( w_1 \) and \( w_2 \).
2 replies
Mamadi
May 2, 2025
A22-
3 hours ago
Deduction card battle
anantmudgal09   55
N 4 hours ago by deduck
Source: INMO 2021 Problem 4
A Magician and a Detective play a game. The Magician lays down cards numbered from $1$ to $52$ face-down on a table. On each move, the Detective can point to two cards and inquire if the numbers on them are consecutive. The Magician replies truthfully. After a finite number of moves, the Detective points to two cards. She wins if the numbers on these two cards are consecutive, and loses otherwise.

Prove that the Detective can guarantee a win if and only if she is allowed to ask at least $50$ questions.

Proposed by Anant Mudgal
55 replies
anantmudgal09
Mar 7, 2021
deduck
4 hours ago
Geometry
Lukariman   7
N 4 hours ago by vanstraelen
Given circle (O) and point P outside (O). From P draw tangents PA and PB to (O) with contact points A, B. On the opposite ray of ray BP, take point M. The circle circumscribing triangle APM intersects (O) at the second point D. Let H be the projection of B on AM. Prove that $\angle HDM$ = 2∠AMP.
7 replies
Lukariman
Tuesday at 12:43 PM
vanstraelen
4 hours ago
perpendicularity involving ex and incenter
Erken   20
N 5 hours ago by Baimukh
Source: Kazakhstan NO 2008 problem 2
Suppose that $ B_1$ is the midpoint of the arc $ AC$, containing $ B$, in the circumcircle of $ \triangle ABC$, and let $ I_b$ be the $ B$-excircle's center. Assume that the external angle bisector of $ \angle ABC$ intersects $ AC$ at $ B_2$. Prove that $ B_2I$ is perpendicular to $ B_1I_B$, where $ I$ is the incenter of $ \triangle ABC$.
20 replies
Erken
Dec 24, 2008
Baimukh
5 hours ago
Isosceles Triangle Geo
oVlad   4
N 5 hours ago by Double07
Source: Romania Junior TST 2025 Day 1 P2
Consider the isosceles triangle $ABC$ with $\angle A>90^\circ$ and the circle $\omega$ of radius $AC$ centered at $A.$ Let $M$ be the midpoint of $AC.$ The line $BM$ intersects $\omega$ a second time at $D.$ Let $E$ be a point on $\omega$ such that $BE\perp AC.$ Let $N$ be the intersection of $DE$ and $AC.$ Prove that $AN=2\cdot AB.$
4 replies
oVlad
Apr 12, 2025
Double07
5 hours ago
Geometry
Lukariman   1
N 5 hours ago by Primeniyazidayi
Given acute triangle ABC ,AB=b,AC=c . M is a variable point on side AB. The circle circumscribing triangle BCM intersects AC at N.

a)Let I be the center of the circle circumscribing triangle AMN. Prove that I always lies on a fixed line.

b)Let J be the center of the circle circumscribing triangle MBC. Prove that line segment IJ has a constant length.
1 reply
Lukariman
Yesterday at 4:02 PM
Primeniyazidayi
5 hours ago
Kingdom of Anisotropy
v_Enhance   24
N 5 hours ago by deduck
Source: IMO Shortlist 2021 C4
The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a path from $X$ to $Y$ is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called diverse if no road belongs to two or more paths in the collection.

Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$.

Proposed by Warut Suksompong, Thailand
24 replies
v_Enhance
Jul 12, 2022
deduck
5 hours ago
random problem i just thought about one day
ceilingfan404   27
N Apr 29, 2025 by PikaPika999
i don't even know if this is solvable
Prove that there are finite/infinite powers of 2 where all the digits are also powers of 2. (For example, $4$ and $128$ are numbers that work, but $64$ and $1024$ don't work.)
27 replies
ceilingfan404
Apr 20, 2025
PikaPika999
Apr 29, 2025
random problem i just thought about one day
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ceilingfan404
1135 posts
#1 • 4 Y
Y by aidan0626, e_is_2.71828, Exponent11, PikaPika999
i don't even know if this is solvable
Prove that there are finite/infinite powers of 2 where all the digits are also powers of 2. (For example, $4$ and $128$ are numbers that work, but $64$ and $1024$ don't work.)
This post has been edited 1 time. Last edited by ceilingfan404, Apr 20, 2025, 7:55 PM
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huajun78
71 posts
#2 • 1 Y
Y by PikaPika999
well as the number gets bigger, there are more digits, so it's less likely that ALL the digits will be a power of 2 (1, 2, 4, 8).

for the first 20 powers of 2 after $2^{10}$ ($2^{11}$ to $2^{30}$), none of them satisfy the condition (I tested all of them), so it's very unlikely that numbers with even more digits will.

I don't know how to prove this but that fact suggests that there are only a finite number.
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vincentwant
1380 posts
#3 • 1 Y
Y by PikaPika999
if the number is greater than 512 then the last four digits must be 2112, 4112, 8112, 2224, 4224, 8224, 1424, 1824, 2144, 4144, 8144, 1184, 2128, 4128, 8128, 1248, 2448, 4448, 8448, 2848, 4848, 8848, 2288, 4288, 8288, 1488, 1888

dont think this helps
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Yummo
298 posts
#4 • 1 Y
Y by PikaPika999
@above, what about 1024?
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vincentwant
1380 posts
#5 • 1 Y
Y by PikaPika999
Yummo wrote:
@above, what about 1024?

0 is not a power of 2
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e_is_2.71828
222 posts
#6 • 1 Y
Y by PikaPika999
ceilingfan404 wrote:
i don't even know if this is solvable
Prove that there are finite/infinite powers of 2 where all the digits are also powers of 2. (For example, $4$ and $128$ are numbers that work, but $64$ and $1024$ don't work.)

I won't look into it completely, but we can start somewhere. We'll see if it is possible to "generate" a formula for these numbers. So let $n$ be a $k$-digit number such that $n=a_ka_{k-1}...a_2a_1a_0$. Then $n=10^ka_k+10^{k-1}a_k-1...+10a_1+a_0$, and note for all $i$ $a_i=2^b$, for some $b$. So, $n=10^k \cdot 2^{b_k}+10^{k-1}\cdot 2^{b_{k-1}}+...+10\cdot 2^{b_1}+2^{b_0}$. From there we need also $n=2^c$ for some $c$, and presumably we can take the largest $b_i$, factor it out, and we need the remaining sum to also be a power of $2$. Someone can try working it out from here, I think I started it off well enough.
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wangzrpi
159 posts
#7
Y by
See
https://math.stackexchange.com/questions/2238383/how-many-powers-of-2-have-only-0-or-powers-of-2-as-digits
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e_is_2.71828
222 posts
#8
Y by
Definitely not middle school math
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e_is_2.71828
222 posts
#10 • 1 Y
Y by PikaPika999
K1mchi_ wrote:
e_is_2.71828 wrote:
Definitely not middle school math

its fine

doesn’t need to be msm curriculum
just for msm

if u can’t do it skill issue


"This problem is unlikely to have a simple proof, because the following holds:

Theorem. For any k, there exists a power of 2 whose first k digits and last k digits are all either 1 or 2.
Proof. We begin with looking at the last digits, taking 2nmod10k. For sufficiently large n, 2n≡0(mod2k). Since 2 is a primitive root modulo 5 and modulo 52, it is a primitive root modulo 5k for any k (Wikipedia), so we can have 2n≡b(mod10k) for any b such that b≡0(mod2k).

This is possible to accomplish with only 1 and 2 as digits. We start with b1=2 for k=1, and extend bk−1≡0(mod2k−1) to bk≡0(mod2k) by the rule:

If bk−1≡0(mod2k), take bk=2⋅10k−1+bk−1.
If bk−1≡2k−1(mod2k), take bk=10k−1+bk−1.
(This works because 10k−1≡2k−1(mod2)k.)

There is a unique sequence of digits ending …211111212122112 that we obtain in this way; reversed, it is A023396 in the OEIS.

To make sure that 2n ends in bk, there will be some condition along the lines of
n≡c(modϕ(5k))
or n=c+n′ϕ(5k) for some n′. From there, getting the first k digits to be 1 or 2 is easy along the lines of a recently popular question. We might as well aim for the sequence 111…111k, because we can. To do this, we want
log101.11…1<{(c+n′⋅ϕ(5k))log102}<log101.11…2
where {x} denotes the fractional part of x. This translates into a condition of the form
{n⋅log102ϕ(5k)}∈Ik
for some interval Ik, which we know is possible because α=log102ϕ(5k) is irrational, and therefore the sequence {α},{2α},{3α},… is dense in [0,1].

This concludes the proof.

Instead of the digits {1,2} we could have used the digits {1,4} or {1,8} and given a similar proof; if we multiply the solution to one of these by 2 or 4, we get a power of 2 whose first and last digits come from the set {2,4} or {2,8} or {4,8}. (We can't do this with just the set {0,1} or {0,2} or {0,4} or {0,8}, because eventually we can rule these out by a modular condition.)

It's of course still almost certain that there's no large power of 2 entirely made from the digits {0,1,2,4,8}, but you'd have to say something about the "middle digits" of such a power, which is much harder."

From the stack exchange.
This post has been edited 1 time. Last edited by e_is_2.71828, Apr 24, 2025, 6:01 PM
Reason: Added
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Andrew2019
2309 posts
#12 • 2 Y
Y by e_is_2.71828, Demetri
K1mchi_ wrote:
e_is_2.71828 wrote:
Definitely not middle school math

its fine

doesn’t need to be msm curriculum
just for msm

if u can’t do it skill issue

it would be crazy if someone who has only done the amc 8 and sold on it says others have a skill issue
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maromex
184 posts
#13
Y by
There is a related question: Does the base-$3$ expression of $2^n$ always have a digit equal to $2$ for sufficiently large $n$? If I recall correctly, this problem is unsolved.

The problem discussed in this topic seems similar to this question, and I don't see why it would be solvable with currently known techniques.
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e_is_2.71828
222 posts
#14 • 1 Y
Y by mithu542
I wouldn't listen to someone who can't even spell figure ...
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maxamc
568 posts
#15
Y by
e_is_2.71828 wrote:
I wouldn't listen to someone who can't even spell figure ...

K1mchi_ is always right 100000 aura.
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K1mchi_
85 posts
#16
Y by
Andrew2019 wrote:
K1mchi_ wrote:
e_is_2.71828 wrote:
Definitely not middle school math

its fine

doesn’t need to be msm curriculum
just for msm

if u can’t do it skill issue

it would be crazy if someone who has only done the amc 8 and sold on it says others have a skill issue

slander

i just dont do competitive math

hate me if u like
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MathPerson12321
3746 posts
#17 • 2 Y
Y by e_is_2.71828, mithu542
#11
why dont u?
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K1mchi_
85 posts
#18
Y by
MathPerson12321 wrote:
#11
why dont u?

just quote me


i have better things to do with my time than math rn

i’ll do u the service of enlightenment if i ever find time
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Aaronjudgeisgoat
898 posts
#19
Y by
K1mchi_ wrote:
MathPerson12321 wrote:
#11
why dont u?

just quote me


i have better things to do with my time than math rn

i’ll do u the service of enlightenment if i ever find time

you only have 105 posts, but i feel like ive seen you everywhere
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MathPerson12321
3746 posts
#22
Y by
@bove stop trying to say ur better
do i see mop quals trying to bring me down? no
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RollingPanda4616
255 posts
#24 • 1 Y
Y by PikaPika999
K3mchi_ wrote:
MathPerson12321 wrote:
@bove stop trying to say ur better
do i see mop quals trying to bring me down? no

so? im not trying to bring u down u still bring urself down bc ur very sensitive

dont we celebrate intelligence in our society?

alt alert
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RollingPanda4616
255 posts
#26 • 2 Y
Y by PikaPika999, e_is_2.71828
hey

yes
$~~~~~~$
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valisaxieamc
421 posts
#27 • 3 Y
Y by RollingPanda4616, PikaPika999, e_is_2.71828
Bro imagine making alts cause you fear that aops is going to ban you
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RollingPanda4616
255 posts
#28 • 1 Y
Y by PikaPika999
valisaxieamc wrote:
Bro imagine making alts cause you fear that aops is going to ban you

:rotfl:

anyway let's get this thread back on track

I think you might need to break up the digits and use the prime factorization. (like a 3 digit number $abc$ would be broken down into $a \cdot 2^2 5^2 + b \cdot 2^1 5^1 + c$ and since a, b,c are powers of 2, you could just look at the 5s?) idk how to continue though.
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maromex
184 posts
#29
Y by
I'll say this again:
maromex wrote:
There is a related question: Does the base-$3$ expression of $2^n$ always have a digit equal to $2$ for sufficiently large $n$? If I recall correctly, this problem is unsolved.

The problem discussed in this topic seems similar to this question, and I don't see why it would be solvable with currently known techniques.

Unless a problem about digits has good reason to be solvable with currently known techniques, it's probably not solvable, even if the answer seems obviously true/false at first.

Here's another unsolved problem related to the topic of this thread: For $n > 86$, does $2^n$ always have a $0$ in base $10$?
This post has been edited 1 time. Last edited by maromex, Apr 26, 2025, 7:16 PM
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PikaPika999
1709 posts
#30
Y by
K1mchi_ wrote:
e_is_2.71828 wrote:
Definitely not middle school math

its fine

doesn’t need to be msm curriculum
just for msm

if u can’t do it skill issue

but if the forum is literally called msm, then shouldn't it be msm? plus, if it is harder than msm, there are high school math and college math and high school olympiads, and it could've been placed there?

k1mchi_

not nice
valisaxieamc wrote:
Bro imagine making alts cause you fear that aops is going to ban you

lol imo aops should use ip bans
This post has been edited 1 time. Last edited by PikaPika999, Apr 27, 2025, 11:09 PM
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PikaPika999
1709 posts
#31 • 2 Y
Y by RollingPanda4616, Pengu14
K3mchi_ wrote:
MathPerson12321 wrote:
@bove stop trying to say ur better
do i see mop quals trying to bring me down? no

so? im not trying to bring u down u still bring urself down bc ur very sensitive

dont we celebrate intelligence in our society?

1. True intelligence shines through clarity and simplicity, not overcomplication.
2. Intelligence isn’t just about flaunting knowledge—it’s also about understanding, humility, and connection.
3. True intelligence lies not in power over others, but in empowering those around us.
4. Creativity/intelligence isn’t just about thinking outside the box—it’s about reshaping the box entirely.
5. Leadership isn’t a title—it’s the trust you earn and the influence you wield wisely.
6. Intelligence is not in the answers we give, but in the questions we dare to ask.
7. Intelligence grows when we challenge our own assumptions, not just those of others.
8. The hallmark of intelligence is recognizing that there’s always more to learn.
9. Intelligence flourishes in collaboration, not isolation.
This post has been edited 2 times. Last edited by PikaPika999, Apr 27, 2025, 11:15 PM
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valisaxieamc
421 posts
#33 • 1 Y
Y by PikaPika999
I completely agree with PikaPika but like RollingPanda said, we probably should get back on topic. I mean the kimchi dude is finally leaving us alone and hopefully getting a life so I'll take it as a win
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fake123
93 posts
#34
Y by
bro why are you guys raging over some random kid why can't you just ignore him
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PikaPika999
1709 posts
#35
Y by
fake123 wrote:
bro why are you guys raging over some random kid why can't you just ignore him

we're not raging over "some random kid" who can be ignored (sorry if this sounds harsher than it is)

they start flamewars on multiple different threads. This is how my 1000th post thread got locked :furious

also, they created multiple different alts, which is explicitly said to be against the rules (probably because of getting postbanned from this sheriff

sry if this sounds harsher than i meant to be
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