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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 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
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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
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USAMO Medals
YauYauFilter   8
N 13 minutes ago by Inaaya
https://maa.org/wp-content/uploads/2025/04/2025-USAMO-Awardees.pdf
8 replies
YauYauFilter
Apr 24, 2025
Inaaya
13 minutes ago
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everyone will get zero marx on this
Th3Numb3rThr33   48
N an hour ago by Blast_S1
Source: JMO 2018 Problem 6
Karl starts with $n$ cards labeled $1,2,3,\dots,n$ lined up in a random order on his desk. He calls a pair $(a,b)$ of these cards swapped if $a>b$ and the card labeled $a$ is to the left of the card labeled $b$. For instance, in the sequence of cards $3,1,4,2$, there are three swapped pairs of cards, $(3,1)$, $(3,2)$, and $(4,2)$.

He picks up the card labeled 1 and inserts it back into the sequence in the opposite position: if the card labeled 1 had $i$ card to its left, then it now has $i$ cards to its right. He then picks up the card labeled $2$ and reinserts it in the same manner, and so on until he has picked up and put back each of the cards $1,2,\dots,n$ exactly once in that order. (For example, the process starting at $3,1,4,2$ would be $3,1,4,2\to 3,4,1,2\to 2,3,4,1\to 2,4,3,1\to 2,3,4,1$.)

Show that, no matter what lineup of cards Karl started with, his final lineup has the same number of swapped pairs as the starting lineup.
48 replies
Th3Numb3rThr33
Apr 19, 2018
Blast_S1
an hour ago
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Find the radius of circle O
TheMaskedMagician   3
N 3 hours ago by fruitmonster97
Source: 1976 AHSME Problem 18
IMAGE

In the adjoining figure, $AB$ is tangent at $A$ to the circle with center $O$; point $D$ is interior to the circle; and $DB$ intersects the circle at $C$. If $BC=DC=3$, $OD=2$, and $AB=6$, then the radius of the circle is

$\textbf{(A) }3+\sqrt{3}\qquad\textbf{(B) }15/\pi\qquad\textbf{(C) }9/2\qquad\textbf{(D) }2\sqrt{6}\qquad \textbf{(E) }\sqrt{22}$
3 replies
TheMaskedMagician
May 18, 2014
fruitmonster97
3 hours ago
J
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Divisibility NT
reni_wee   0
3 hours ago
Source: Japan 1996, ONTCP
Let $m,n$ be relatively prime positive integers. Calculate $gcd(5^m+7^m, 5^n+7^n).$
0 replies
reni_wee
3 hours ago
0 replies
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Modular arithmetic at mod n
electrovector   3
N 3 hours ago by Primeniyazidayi
Source: 2021 Turkey JBMO TST P6
Integers $a_1, a_2, \dots a_n$ are different at $\text{mod n}$. If $a_1, a_2-a_1, a_3-a_2, \dots a_n-a_{n-1}$ are also different at $\text{mod n}$, we call the ordered $n$-tuple $(a_1, a_2, \dots a_n)$ lucky. For which positive integers $n$, one can find a lucky $n$-tuple?
3 replies
electrovector
May 24, 2021
Primeniyazidayi
3 hours ago
J
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MathPath
PatTheKing806   13
N 3 hours ago by Nora2021
Is anybody else going to MathPath?

I haven't gotten in. its been 3+ weeks since they said my application was done.
13 replies
PatTheKing806
Mar 24, 2025
Nora2021
3 hours ago
J
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Sequences problem
BBNoDollar   3
N 5 hours ago by BBNoDollar
Source: Mathematical Gazette Contest
Determine the general term of the sequence of non-zero natural numbers (a_n)n≥1, with the property that gcd(a_m, a_n, a_p) = gcd(m^2 ,n^2 ,p^2), for any distinct non-zero natural numbers m, n, p.

⁡Note that gcd(a,b,c) denotes the greatest common divisor of the natural numbers a,b,c .
3 replies
BBNoDollar
Yesterday at 5:53 PM
BBNoDollar
5 hours ago
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Arbitrary point on BC and its relation with orthocenter
falantrng   33
N 5 hours ago by Thapakazi
Source: Balkan MO 2025 P2
In an acute-angled triangle \(ABC\), \(H\) be the orthocenter of it and \(D\) be any point on the side \(BC\). The points \(E, F\) are on the segments \(AB, AC\), respectively, such that the points \(A, B, D, F\) and \(A, C, D, E\) are cyclic. The segments \(BF\) and \(CE\) intersect at \(P.\) \(L\) is a point on \(HA\) such that \(LC\) is tangent to the circumcircle of triangle \(PBC\) at \(C.\) \(BH\) and \(CP\) intersect at \(X\). Prove that the points \(D, X, \) and \(L\) lie on the same line.

Proposed by Theoklitos Parayiou, Cyprus
33 replies
falantrng
Apr 27, 2025
Thapakazi
5 hours ago
J
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Inequality
lgx57   4
N 5 hours ago by GeoMorocco
Source: Own
$a,b,c>0,ab+bc+ca=1$. Prove that

$$\sum \sqrt{8ab+1} \ge 5$$
(I don't know whether the equality holds)
4 replies
lgx57
Yesterday at 3:14 PM
GeoMorocco
5 hours ago
J
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Rubber bands
v_Enhance   5
N 5 hours ago by lpieleanu
Source: OTIS Mock AIME 2024 #12
Let $\mathcal G_n$ denote a triangular grid of side length $n$ consisting of $\frac{(n+1)(n+2)}{2}$ pegs. Charles the Otter wishes to place some rubber bands along the pegs of $\mathcal G_n$ such that every edge of the grid is covered by exactly one rubber band (and no rubber band traverses an edge twice). He considers two placements to be different if the sets of edges covered by the rubber bands are different or if any rubber band traverses its edges in a different order. The ordering of which bands are over and under does not matter.
For example, Charles finds there are exactly $10$ different ways to cover $\mathcal G_2$ using exactly two rubber bands; the full list is shown below, with one rubber band in orange and the other in blue.
IMAGE
Let $N$ denote the total number of ways to cover $\mathcal G_4$ with any number of rubber bands. Compute the remainder when $N$ is divided by $1000$.

Ethan Lee
5 replies
v_Enhance
Jan 16, 2024
lpieleanu
5 hours ago
J
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Geometry with orthocenter config
thdnder   6
N 5 hours ago by ohhh
Source: Own
Let $ABC$ be a triangle, and let $AD, BE, CF$ be its altitudes. Let $H$ be its orthocenter, and let $O_B$ and $O_C$ be the circumcenters of triangles $AHC$ and $AHB$. Let $G$ be the second intersection of the circumcircles of triangles $FDO_B$ and $EDO_C$. Prove that the lines $DG$, $EF$, and $A$-median of $\triangle ABC$ are concurrent.
6 replies
thdnder
Apr 29, 2025
ohhh
5 hours ago
J
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Strange Inequality
anantmudgal09   40
N 5 hours ago by starchan
Source: INMO 2020 P4
Let $n \geqslant 2$ be an integer and let $1<a_1 \le a_2 \le \dots \le a_n$ be $n$ real numbers such that $a_1+a_2+\dots+a_n=2n$. Prove that$$a_1a_2\dots a_{n-1}+a_1a_2\dots a_{n-2}+\dots+a_1a_2+a_1+2 \leqslant a_1a_2\dots a_n.$$
Proposed by Kapil Pause
40 replies
anantmudgal09
Jan 19, 2020
starchan
5 hours ago
J
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Finding Solutions
MathStudent2002   22
N 5 hours ago by ihategeo_1969
Source: Shortlist 2016, Number Theory 5
Let $a$ be a positive integer which is not a perfect square, and consider the equation \[k = \frac{x^2-a}{x^2-y^2}.\]Let $A$ be the set of positive integers $k$ for which the equation admits a solution in $\mathbb Z^2$ with $x>\sqrt{a}$, and let $B$ be the set of positive integers for which the equation admits a solution in $\mathbb Z^2$ with $0\leq x<\sqrt{a}$. Show that $A=B$.
22 replies
MathStudent2002
Jul 19, 2017
ihategeo_1969
5 hours ago
J
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USAMO 2000 Problem 3
MithsApprentice   10
N 5 hours ago by HamstPan38825
A game of solitaire is played with $R$ red cards, $W$ white cards, and $B$ blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of $R, W,$ and $B,$ the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.
10 replies
MithsApprentice
Oct 1, 2005
HamstPan38825
5 hours ago
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USA(J)MO Index
15Pandabears   9
N Feb 2, 2015 by MSTang
If you get a score of A on the AMC 10A and a score of B on the AMC 10B, where A>B>120, is score A used for your USA(J)MO index?
9 replies
15Pandabears
Jan 27, 2015
MSTang
Feb 2, 2015
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USA(J)MO Index
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15Pandabears
443 posts
15Pandabears
#1 h  Jan 27, 2015, 11:23 PM • 2 Y
Y by Adventure10, Mango247
If you get a score of A on the AMC 10A and a score of B on the AMC 10B, where A>B>120, is score A used for your USA(J)MO index?
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Not_a_Username
1215 posts
Not_a_Username
#2 h  Jan 27, 2015, 11:25 PM • 2 Y
Y by Adventure10, Mango247
The higher score is used for your USAJMO index.
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BOGTRO
5818 posts
BOGTRO
#3 h  Jan 27, 2015, 11:26 PM • 2 Y
Y by Adventure10, Mango247
Yes. The AMC score used for your USA(J)MO index is $\text{max}(\text{AIME-qualifying A score}, \text{AIME-qualifying B score})$. As being above 120 automatically makes a score AIME-qualifying, the larger one would be used.
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MSTang
6012 posts
MSTang
#4 h  Jan 29, 2015, 1:13 AM • 10 Y
Y by 15Pandabears, droid347, Bob_Smith, spartan168, hwl0304, CML, Adventure10, Mango247, and 2 other users
I made a flow chart about this if you have any other questions :)

@BOGTRO: Mistake fixed!
Attachments:
This post has been edited 1 time. Last edited by MSTang, Feb 2, 2015, 3:39 PM
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BOGTRO
5818 posts
BOGTRO
#5 h  Jan 29, 2015, 8:58 PM • 8 Y
Y by niraekjs, zmyshatlp, droid347, MSTang, Adventure10, Mango247, and 2 other users
Unfortunately the above is somewhat misleading. The box containing "Congrats! Which AMC 10/12 test(s) did you take?" should be replaced by "Congrats! Which AMC 10/12 test(s) did you qualify for AIME on?". In particular, if you get something like a 132 on the AMC 10A and a 72 on the AMC 12B (assuming a 72 on the AMC 12B does not make AIME, a very safe assumption), you cannot make USAMO regardless of your AIME score. Similarly, if you get something like a 117 on the AMC 12 and a 96 on the AMC 10, you cannot make USAJMO regardless of your AIME score.
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bestwillcui1
2735 posts
bestwillcui1
#6 h  Jan 29, 2015, 11:15 PM • 3 Y
Y by droid347, Adventure10, Mango247
Wait last year xantho got some score on the AMC 12 that did not qualify for AIME but qualified through his 10 score and proceeded to score a 13 on AIME, and somehow they told him to take AMO.
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CPU-RN
4 posts
CPU-RN
#7 h  Feb 2, 2015, 5:38 AM • 1 Y
Y by Adventure10
That's fairly hard to believe but maybe his 10(AIME score)+his non-qualifying AMC 12 score was a USAMO qualifying index?
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bestwillcui1
2735 posts
bestwillcui1
#8 h  Feb 2, 2015, 9:43 AM • 2 Y
Y by Adventure10, Mango247
Yes that was the case even though he did not theoretically make the AIME with his AMC 12 score.
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xantho
767 posts
xantho
#9 h  Feb 2, 2015, 1:05 PM • 3 Y
Y by mathmind123, Adventure10, Mango247
bestwillcui1 wrote:
Wait last year xantho got some score on the AMC 12 that did not qualify for AIME but qualified through his 10 score and proceeded to score a 13 on AIME, and somehow they told him to take AMO.
As the user who is being spoken of here, I can indeed confirm my index of 84+13(10)=214 qualified for the USAMO index. I too thought that I couldn't make AMO, since I didn't qualify through the 12, and made it because of my 10a score. Apparently, this is not the case. Although a person who does so has to be extraordinarily inconsistent.. Sigh..
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MSTang
6012 posts
MSTang
#10 h  Feb 2, 2015, 3:20 PM • 1 Y
Y by Adventure10
BOGTRO wrote:
Unfortunately the above is somewhat misleading. The box containing "Congrats! Which AMC 10/12 test(s) did you take?" should be replaced by "Congrats! Which AMC 10/12 test(s) did you qualify for AIME on?". In particular, if you get something like a 132 on the AMC 10A and a 72 on the AMC 12B (assuming a 72 on the AMC 12B does not make AIME, a very safe assumption), you cannot make USAMO regardless of your AIME score. Similarly, if you get something like a 117 on the AMC 12 and a 96 on the AMC 10, you cannot make USAJMO regardless of your AIME score.

Oh, right. Thanks, I'll fix it :blush:
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