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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Yesterday at 3:18 PM
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

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April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Yesterday at 3:18 PM
0 replies
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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
J
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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
J
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prime factorization formula questions
Soupboy0   8
N an hour ago by martianrunner
If i have a number, say $N$ with prime factorization $p_1^{e_1}p_2^{e_2}...p_n^{e_n}$, and I want to find $3, 4, 5, ..., k$ numbers that multiply to $N$, does anybody know a formula for this?
8 replies
Soupboy0
an hour ago
martianrunner
an hour ago
J
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Game About Passing Pencils
WilliamSChen   0
an hour ago
A group of $n$ children sit in a circle facing inward with $n > 2$, and each child starts with an arbitrary even number of pencils. Each minute, each child simultaneously passes exactly half of all of their pencils to the child to their right. Then, all children that have an odd number of pencils receive one more pencil.
Prove that after a finite amount of time, the children will all have the same number of pencils.

I do not know the source.
0 replies
WilliamSChen
an hour ago
0 replies
J
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An nxn Checkboard
MithsApprentice   26
N an hour ago by NicoN9
Source: USAMO 1999 Problem 1
Some checkers placed on an $n \times n$ checkerboard satisfy the following conditions:

(a) every square that does not contain a checker shares a side with one that does;

(b) given any pair of squares that contain checkers, there is a sequence of squares containing checkers, starting and ending with the given squares, such that every two consecutive squares of the sequence share a side.

Prove that at least $(n^{2}-2)/3$ checkers have been placed on the board.
26 replies
MithsApprentice
Oct 3, 2005
NicoN9
an hour ago
J
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Is this FE solvable?
Mathdreams   4
N an hour ago by Mathdreams
Find all $f:\mathbb{R} \rightarrow \mathbb{R}$ such that \[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]for all reals $x$ and $y$.
4 replies
Mathdreams
Tuesday at 6:58 PM
Mathdreams
an hour ago
J
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Coaxial circles related to Gergon point
Headhunter   0
an hour ago
Source: I tried but can't find the source...
Hi, everyone.

In $\triangle$$ABC$, $Ge$ is the Gergon point and the incircle $(I)$ touch $BC$, $CA$, $AB$ at $D$, $E$, $F$ respectively.
Let the circumcircles of $\triangle IDGe$, $\triangle IEGe$, $\triangle IFGe$ be $O_{1}$ , $O_{2}$ , $O_{3}$ respectively.

Reflect $O_{1}$ in $ID$ and then we get the circle $O'_{1}$
Reflect $O_{2}$ in $IE$ and then the circle $O'_{2}$
Reflect $O_{3}$ in $IF$ and then the circle $O'_{3}$

Prove that $O'_{1}$ , $O'_{2}$ , $O'_{3}$ are coaxial.
0 replies
Headhunter
an hour ago
0 replies
J
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9 MATHCOUNTS STATE difficulty
Eddie_tiger   56
N 2 hours ago by giratina3
I personally thought the problems were much easier than last year, but I didn't really improve as much as I would of liked to improve.
56 replies
Eddie_tiger
Tuesday at 1:13 PM
giratina3
2 hours ago
J
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Equation with powers
a_507_bc   6
N 2 hours ago by EVKV
Source: Serbia JBMO TST 2024 P1
Find all non-negative integers $x, y$ and primes $p$ such that $$3^x+p^2=7 \cdot 2^y.$$
6 replies
a_507_bc
May 25, 2024
EVKV
2 hours ago
J
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2025 MATHCOUNTS State Hub
SirAppel   217
N 2 hours ago by morestuf
Previous Years' "Hubs": (2022) (2023) (2024)Please Read

Now that it's April and we're allowed to discuss, and no one else has made this yet ...
[list=disc]
[*] CA: 43 (45 44 43 43 43 42 42 41 41 41)
[*] NJ: 43 (45 44 44 43 39 42 40 40 39 38) *
[*] NY: 42 (43 42 42 42 41 40)
[*] TX: 42 (43 43 43 42 42 40 40 38 38 38)
[*] MA: 41 (45 43 42 41)
[*] WA: 41 (41 45 42 41 41 41 41 41 41 40) *
[*] FL: 39 (42 41 40 39 38 37 37)
[*] IN: 39 (41 40 40 39 ?? 35)
[*] NC: 39 (42 42 41 39)
[*] IL: 38 (41 40 39 38 38 38)
[*] OR: 38 (44 41? 38 38)
[*] PA: 38 (41 40 40 38 38 37 36 36 34 34) *
[*] MD: 37 (43 39 39 37 37 37)
[*] CT: 36 (44 39? 38 36 34 34 34 34)
[*] MI: 36 (39 41 41 36 37 37 36 36 36 36) *
[*] MN: 36 (40 36 36 36 35 35 35 34)
[*] CO: 35 (41 37 37 35 35 35 ?? 31 31 30) *
[*] GA: 35 (38 37 36 35 34 34 34 34 34 33)
[*] OH: 35 (41 37 36 35)
[*] AR: 34 (46 45 35 34 33 31 31 31 29 29)
[*] WI: 34 (40 37 37 34 35 30 28 29 29 29) *
[*] NH: 31 (42 35 33 31 30)
[*] DE: 30 (34 33 32 30 30 29 28 27 26? 24)
[*] SC: 30 (33 33 31 30)
[*] IA: 29 (33 30 31 29 29 29 29 29 29 29 29 29) *
[*] NE: 28 (34 30 28 28 27 27 26 26 25 25)
[*] SD: 22 (30 29 24 22 22 22 21 21 20 20)
[/list]
Cutoffs Unknown

* means that CDR is official in that state.

Notes

For those asking about the removal of the tiers, I'd like to quote Jason himself:
[quote=peace09]
learn from my mistakes
[/quote]

Help contribute by sharing your state's cutoffs!
As per last year's guidelines, refrain from problem discussion until their official release on the MATHCOUNTS website.
217 replies
SirAppel
Tuesday at 12:22 PM
morestuf
2 hours ago
J
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no numbers of the form 80...01 are squares
Marius_Avion_De_Vanatoare   2
N 2 hours ago by EVKV
Source: Moldova JTST 2024 P5
Prove that a number of the form $80\dots01$ (there is at least 1 zero) can't be a perfect square.
2 replies
Marius_Avion_De_Vanatoare
Jun 10, 2024
EVKV
2 hours ago
J
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f((x XOR f(y)) + y) = (f(x) XOR y) + y
the_universe6626   3
N 2 hours ago by jasperE3
Source: Janson MO 5 P4
Find all functions $f:\mathbb{Z}_{\ge0}\rightarrow\mathbb{Z}_{\ge0}$ such that
\[f((x\oplus f(y))+y)=(f(x)\oplus y)+y\]Note: $\oplus$ denotes the bitwise XOR operation. For example, $1001_2 \oplus 101_2 = 1100_2$.

(Proposed by ja.)
3 replies
the_universe6626
Feb 21, 2025
jasperE3
2 hours ago
J
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2024 8's
Marius_Avion_De_Vanatoare   3
N 2 hours ago by EVKV
Source: Moldova JTST 2024 P2
Prove that the number $ \underbrace{88\dots8}_\text{2024\; \textrm{times}}$ is divisible by 2024.
3 replies
Marius_Avion_De_Vanatoare
Jun 10, 2024
EVKV
2 hours ago
J
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pretty well known
dotscom26   0
2 hours ago
Let $\triangle ABC$ be a scalene triangle such that $\Omega$ is its incircle. $AB$ is tangent to $\Omega$ at $D$. A point $E$ ($E \notin \Omega$) is located on $BC$.

Let $\omega_1$, $\omega_2$, and $\omega_3$ be the incircles of the triangles $BED$, $ADE$, and $AEC$, respectively.

Show that the common tangent to $\omega_1$ and $\omega_3$ is also tangent to $\omega_2$.

0 replies
+1 w
dotscom26
2 hours ago
0 replies
J
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Thanks u!
Ruji2018252   6
N 2 hours ago by jasperE3
Find all $f:\mathbb{R}\to\mathbb{R}$ and
\[ f(x+y)+f(x^2+f(y))=f(f(x))^2+f(x)+f(y)+y,\forall x,y\in\mathbb{R}\]
6 replies
Ruji2018252
Mar 26, 2025
jasperE3
2 hours ago
J
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I cant find one problem
tanujkundu   9
N 2 hours ago by vsarg
Does anybody know which problem is about when a number is a meteor and when a number is a shooting star?
9 replies
tanujkundu
Sep 17, 2024
vsarg
2 hours ago
J
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J
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exponents
Arvind_sn   40
N Apr 1, 2025 by jkim0656
What is the highest possible $e$ in $2^{e}<200$?
40 replies
Arvind_sn
Jul 12, 2006
jkim0656
Apr 1, 2025
J
exponents
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Reveal topic tags
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Arvind_sn
524 posts
Arvind_sn
#1 h  Jul 12, 2006, 9:12 PM • 2 Y
Y by Adventure10, Mango247
What is the highest possible $e$ in $2^{e}<200$?
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math92
2630 posts
math92
#2 h  Jul 12, 2006, 9:23 PM • 2 Y
Y by Adventure10, Mango247
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MCrawford
6325 posts
MCrawford
#3 h  Jul 12, 2006, 9:52 PM • 4 Y
Y by Adventure10, Mango247, Hanruz, MathDolphin95
You should probably mention in the problem that $e$ is an integer.
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José
1828 posts
José
#4 h  Jul 12, 2006, 11:46 PM • 2 Y
Y by Adventure10, Mango247
Arvind_sn wrote:
What is the highest possible $e$ in $2^{e}<200$?

Click to reveal hidden text
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mimar
103 posts
mimar
#5 h  Jul 13, 2006, 2:21 AM • 2 Y
Y by Adventure10, Mango247
By using the guess and check method, we obtain 7 as our final answer!
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math92
2630 posts
math92
#6 h  Jul 13, 2006, 12:23 PM • 1 Y
Y by Adventure10
if e is not an integer
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jhollenbeck
682 posts
jhollenbeck
#7 h  Jul 13, 2006, 3:02 PM • 1 Y
Y by Adventure10
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mad_skillz_aops
363 posts
mad_skillz_aops
#8 h  Jul 14, 2006, 3:11 AM • 2 Y
Y by Adventure10, Mango247
easy
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1=2
4594 posts
1=2
#9 h  Jul 18, 2006, 2:12 PM • 2 Y
Y by Adventure10, Mango247
Do you mean that e is 2.718281828... or e is an unknown?
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SplashD
2631 posts
SplashD
#10 h  Jul 18, 2006, 2:16 PM • 2 Y
Y by Adventure10, Mango247
nutz_for2.718281828 wrote:
Do you mean that e is 2.718281828... or e is an unknown?

In this case, it's obviously an unknown. He should really use a different variable like $x$ or something.
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daermon
1393 posts
daermon
#11 h  Jul 19, 2006, 2:00 AM • 2 Y
Y by Adventure10, Mango247
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alex__ib
161 posts
alex__ib
#12 h  Jul 20, 2006, 10:03 AM • 2 Y
Y by Adventure10, Mango247
Oh, well!You should really mention if it an integar otherwise 7 isn't the right solution :maybe:
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redcomet46
5476 posts
redcomet46
#13 h  Jul 21, 2006, 4:48 PM • 1 Y
Y by Adventure10
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moogra
1330 posts
moogra
#14 h  Sep 6, 2006, 12:37 AM • 2 Y
Y by Adventure10, Mango247
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Chocolate milk
35 posts
Chocolate milk
#15 h  Sep 6, 2006, 3:53 AM • 1 Y
Y by Adventure10
Of course : e=7
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meenamathgirl
200 posts
meenamathgirl
#16 h  Sep 6, 2006, 4:46 PM • 2 Y
Y by Adventure10, Mango247
Does e have to be an integer?
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José
1828 posts
José
#17 h  Sep 6, 2006, 5:03 PM • 2 Y
Y by Adventure10, Mango247
Because, if not, you can't find the highest possible value
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stevenbao
14 posts
stevenbao
#18 h  Sep 6, 2006, 7:29 PM • 1 Y
Y by Adventure10
I believe it is...

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#H34N1
4235 posts
#H34N1
#19 h  Sep 28, 2006, 9:38 PM • 2 Y
Y by Adventure10, Mango247
Arvind_sn wrote:
What is the highest possible $e$ in $2^{e}<200$?

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ProtestanT
669 posts
ProtestanT
#20 h  Sep 29, 2006, 12:52 AM • 2 Y
Y by Adventure10, Mango247
Click to reveal hidden text

Mod edit: Try not to use such complicated methods in a middle school forum. :)
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moogra
1330 posts
moogra
#21 h  Oct 1, 2006, 3:44 AM • 2 Y
Y by Adventure10, Mango247
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MichaelFaraday
202 posts
MichaelFaraday
#22 h  Oct 17, 2006, 10:38 PM • 2 Y
Y by Adventure10, Mango247
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DhruvJha
804 posts
DhruvJha
#23 h  Mar 24, 2025, 2:30 PM
Y by
I got 7. Thats right because 2^7=128 and 2^8 =256 and it needs to be less than 200. so 7. im right, right?
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oliviahung
176 posts
oliviahung
#24 h  Mar 24, 2025, 2:40 PM
Y by
I think its this

Edit: as long as its an integer lmao
This post has been edited 1 time. Last edited by oliviahung, Mar 24, 2025, 2:45 PM
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iwastedmyusername
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iwastedmyusername
#25 h  Mar 24, 2025, 3:17 PM • 1 Y
Y by Exponent11
$e=\log_{2}{200}$
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JamesYMath
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JamesYMath
#26 h  Mar 26, 2025, 10:44 PM
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7.64385618616
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Yihangzh
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Yihangzh
#27 h  Mar 27, 2025, 12:08 AM • 1 Y
Y by jkim0656
DhruvJha wrote:
I got 7. Thats right because 2^7=128 and 2^8 =256 and it needs to be less than 200. so 7. im right, right?

crazy 19 year bump bro
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jkim0656
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jkim0656
#28 h  Mar 27, 2025, 1:19 AM
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Dang this is the biggest bump ive ever seen on aops
although a couple have come close
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maxamc
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maxamc
#29 h  Mar 27, 2025, 1:25 AM
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ProtestanT wrote:
Click to reveal hidden text

Mod edit: Try not to use such complicated methods in a middle school forum. :)

"complicated methods"
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WiseHawkCuteFriendly
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WiseHawkCuteFriendly
#30 h  Mar 27, 2025, 1:47 AM
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$e$ should be $7$, unless you want non-integer numbers.
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nmlikesmath
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nmlikesmath
#31 h  Mar 27, 2025, 2:01 AM
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7, 2^7 = 128
or js sqrt 200 if u have a calculator with you
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PatTheKing806
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PatTheKing806
#32 h  Mar 27, 2025, 2:02 AM
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isn't $e = 2.718281828459045$
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jkim0656
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jkim0656
#33 h  Mar 27, 2025, 2:02 AM
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PatTheKing806 wrote:
isn't $e = 2.718281828459045$

it is
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PatTheKing806
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PatTheKing806
#34 h  Mar 27, 2025, 3:34 AM
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so is $e$ a variable here or $2.7$
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jkim0656
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jkim0656
#35 h  Mar 27, 2025, 3:36 AM
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2.7....
also @pat consider a math story for post 1000
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PatTheKing806
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PatTheKing806
#36 h  Mar 27, 2025, 4:04 AM
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jkim0656 wrote:
2.7....
also @pat consider a math story for post 1000

i will prob
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Yihangzh
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Yihangzh
#37 h  Mar 27, 2025, 4:21 AM
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why people bumping 19 year old threads.
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AVY2024
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AVY2024
#38 h  Mar 27, 2025, 5:01 AM
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7<e<8. greatest possible e is in the middle of 7 and 8 if answer is necessarily not an integer.
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lightsynth123
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lightsynth123
#39 h  Mar 31, 2025, 6:39 AM
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just test out all e's from 1 to 10 and realize that e=7 works
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ohiorizzler1434
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ohiorizzler1434
#40 h  Mar 31, 2025, 9:47 AM
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no you dont! because 2^x is increasing for x increasing you only need to test out all es between 7 and 8.
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jkim0656
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jkim0656
#41 h  Apr 1, 2025, 4:05 AM
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ohiorizzler1434 wrote:
no you dont! because 2^x is increasing for x increasing you only need to test out all es between 7 and 8.

???
how would u know btwn 7 and 8 by just realizing that 2^x is increasing loll
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