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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.

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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
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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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camp/class recommendations for incoming freshman
walterboro   11
N an hour ago by Pengu14
hi guys, i'm about to be an incoming freshman, does anyone have recommendations for classes to take next year and camps this summer? i am sure that i can aime qual but not jmo qual yet. ty
11 replies
walterboro
May 10, 2025
Pengu14
an hour ago
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Metamorphosis of Medial and Contact Triangles
djmathman   102
N an hour ago by Mathandski
Source: 2014 USAJMO Problem 6
Let $ABC$ be a triangle with incenter $I$, incircle $\gamma$ and circumcircle $\Gamma$. Let $M,N,P$ be the midpoints of sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ and let $E,F$ be the tangency points of $\gamma$ with $\overline{CA}$ and $\overline{AB}$, respectively. Let $U,V$ be the intersections of line $EF$ with line $MN$ and line $MP$, respectively, and let $X$ be the midpoint of arc $BAC$ of $\Gamma$.

(a) Prove that $I$ lies on ray $CV$.

(b) Prove that line $XI$ bisects $\overline{UV}$.
102 replies
1 viewing
djmathman
Apr 30, 2014
Mathandski
an hour ago
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Gives typical russian combinatorics vibes
Sadigly   3
N an hour ago by AL1296
Source: Azerbaijan Senior MO 2025 P3
You are given a positive integer $n$. $n^2$ amount of people stand on coordinates $(x;y)$ where $x,y\in\{0;1;2;...;n-1\}$. Every person got a water cup and two people are considered to be neighbour if the distance between them is $1$. At the first minute, the person standing on coordinates $(0;0)$ got $1$ litres of water, and the other $n^2-1$ people's water cup is empty. Every minute, two neighbouring people are chosen that does not have the same amount of water in their water cups, and they equalize the amount of water in their water cups.

Prove that, no matter what, the person standing on the coordinates $(x;y)$ will not have more than $\frac1{x+y+1}$ litres of water.
3 replies
Sadigly
May 8, 2025
AL1296
an hour ago
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ranttttt
alcumusftwgrind   41
N 2 hours ago by meyler
rant
41 replies
alcumusftwgrind
Apr 30, 2025
meyler
2 hours ago
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high tech FE as J1?!
imagien_bad   62
N 2 hours ago by jasperE3
Source: USAJMO 2025/1
Let $\mathbb Z$ be the set of integers, and let $f\colon \mathbb Z \to \mathbb Z$ be a function. Prove that there are infinitely many integers $c$ such that the function $g\colon \mathbb Z \to \mathbb Z$ defined by $g(x) = f(x) + cx$ is not bijective.
Note: A function $g\colon \mathbb Z \to \mathbb Z$ is bijective if for every integer $b$, there exists exactly one integer $a$ such that $g(a) = b$.
62 replies
imagien_bad
Mar 20, 2025
jasperE3
2 hours ago
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Triangular Numbers in action
integrated_JRC   29
N 2 hours ago by Aiden-1089
Source: RMO 2018 P5
Find all natural numbers $n$ such that $1+[\sqrt{2n}]~$ divides $2n$.

( For any real number $x$ , $[x]$ denotes the largest integer not exceeding $x$. )
29 replies
integrated_JRC
Oct 7, 2018
Aiden-1089
2 hours ago
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Cute property of Pascal hexagon config
Miquel-point   1
N 3 hours ago by FarrukhBurzu
Source: KoMaL B. 5444
In cyclic hexagon $ABCDEF$ let $P$ denote the intersection of diagonals $AD$ and $CF$, and let $Q$ denote the intersection of diagonals $AE$ and $BF$. Prove that if $BC=CP$ and $DP=DE$, then $PQ$ bisects angle $BQE$.

Proposed by Géza Kós, Budapest
1 reply
Miquel-point
4 hours ago
FarrukhBurzu
3 hours ago
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Number theory problem
Angelaangie   3
N 3 hours ago by megarnie
Source: JBMO 2007
Prove that 7p+3^p-4 it is not a perfect square where p is prime.
3 replies
Angelaangie
Jun 19, 2018
megarnie
3 hours ago
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another n x n table problem.
pohoatza   3
N 3 hours ago by reni_wee
Source: Romanian JBTST III 2007, problem 3
Consider a $n$x$n$ table such that the unit squares are colored arbitrary in black and white, such that exactly three of the squares placed in the corners of the table are white, and the other one is black. Prove that there exists a $2$x$2$ square which contains an odd number of unit squares white colored.
3 replies
pohoatza
May 13, 2007
reni_wee
3 hours ago
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Concurrency from isogonal Mittenpunkt configuration
MarkBcc168   18
N 3 hours ago by ihategeo_1969
Source: Fake USAMO 2020 P3
Let $\triangle ABC$ be a scalene triangle with circumcenter $O$, incenter $I$, and incircle $\omega$. Let $\omega$ touch the sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ at points $D$, $E$, and $F$ respectively. Let $T$ be the projection of $D$ to $\overline{EF}$. The line $AT$ intersects the circumcircle of $\triangle ABC$ again at point $X\ne A$. The circumcircles of $\triangle AEX$ and $\triangle AFX$ intersect $\omega$ again at points $P\ne E$ and $Q\ne F$ respectively. Prove that the lines $EQ$, $FP$, and $OI$ are concurrent.

Proposed by MarkBcc168.
18 replies
MarkBcc168
Apr 28, 2020
ihategeo_1969
3 hours ago
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Anything real in this system must be integer
Assassino9931   8
N 3 hours ago by Abdulaziz_Radjabov
Source: Al-Khwarizmi International Junior Olympiad 2025 P1
Determine the largest integer $c$ for which the following statement holds: there exists at least one triple $(x,y,z)$ of integers such that
\begin{align*} x^2 + 4(y + z) = y^2 + 4(z + x) = z^2 + 4(x + y) = c \end{align*}and all triples $(x,y,z)$ of real numbers, satisfying the equations, are such that $x,y,z$ are integers.

Marek Maruin, Slovakia
8 replies
Assassino9931
May 9, 2025
Abdulaziz_Radjabov
3 hours ago
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Good Permutations in Modulo n
swynca   10
N 3 hours ago by MR.1
Source: BMO 2025 P1
An integer $n > 1$ is called $\emph{good}$ if there exists a permutation $a_1, a_2, a_3, \dots, a_n$ of the numbers $1, 2, 3, \dots, n$, such that:
$(i)$ $a_i$ and $a_{i+1}$ have different parities for every $1 \leq i \leq n-1$;
$(ii)$ the sum $a_1 + a_2 + \cdots + a_k$ is a quadratic residue modulo $n$ for every $1 \leq k \leq n$.
Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.
10 replies
swynca
Apr 27, 2025
MR.1
3 hours ago
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Quadratic + cubic residue => 6th power residue?
Miquel-point   0
4 hours ago
Source: KoMaL B. 5445
Decide whether the following statement is true: if an infinite arithmetic sequence of positive integers includes both a perfect square and a perfect cube, then it also includes a perfect $6$th power.

Proposed by Sándor Róka, Nyíregyháza
0 replies
Miquel-point
4 hours ago
0 replies
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II_a - r_a = R - r implies A = 60
Miquel-point   0
4 hours ago
Source: KoMaL B. 5421
The incenter and the inradius of the acute triangle $ABC$ are $I$ and $r$, respectively. The excenter and exradius relative to vertex $A$ is $I_a$ and $r_a$, respectively. Let $R$ denote the circumradius. Prove that if $II_a=r_a+R-r$, then $\angle BAC=60^\circ$.

Proposed by Class 2024C of Fazekas M. Gyak. Ált. Isk. és Gimn., Budapest
0 replies
Miquel-point
4 hours ago
0 replies
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What's the easiest proof-based math competition?
Muu9   7
N Apr 30, 2025 by Iwowowl253
In terms of the difficulty of the questions, not the level of competition. There's USAJMO, but surely there must be countries with less developed competitive math scenes whose Olympiads are easier.
7 replies
Muu9
Apr 21, 2025
Iwowowl253
Apr 30, 2025
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What's the easiest proof-based math competition?
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Reveal topic tags
proofs
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Muu9
103 posts
Muu9
#1 h  Apr 21, 2025, 2:16 PM
Y by
In terms of the difficulty of the questions, not the level of competition. There's USAJMO, but surely there must be countries with less developed competitive math scenes whose Olympiads are easier.
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EaZ_Shadow
1273 posts
EaZ_Shadow
#2 h  Apr 21, 2025, 2:17 PM • 11 Y
Y by Exponent11, Lhaj3, MisakaMikasa, Aaronjudgeisgoat, zhoujef000, Sedro, jkim0656, RollingPanda4616, EZ588, giratina3, eg4334
Homework
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Konigsberg
2234 posts
Konigsberg
#3 h  Apr 22, 2025, 10:25 PM • 2 Y
Y by OronSH, RollingPanda4616
Check out the contests listed at C1 Proof and C2 proof here: https://tinyurl.com/ContestGuideIntl

There’s enough for hundreds of hours of practice at this level. If you’d want proof questions that are around the lower end of AIME, check out the proof-style contests at level B2.

Proof contests below the JMO level aren’t just from less developed countries— many local US contests also have this format.
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Mummypig
1 post
Mummypig
#4 h  Apr 27, 2025, 3:07 PM
Y by
A lot of difficult questions are just a combination of multiple easier questions in a clever way. So one way to solve your question, is to find a good teacher to help you breakdown the jmo questions into many easier questions, and start from there! gl
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js104
14 posts
js104
#5 h  Apr 27, 2025, 7:27 PM
Y by
Take a look at https://sites.math.washington.edu/~mathcircle/olympiad/archive/. These are intended for younger students (as I understand it, the way it works is that once you solve a problem you go up to a grader and directly verbally explain your solution to them, and they can ask for clarification/etc).
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N3bula
277 posts
N3bula
#6 h  Apr 28, 2025, 1:15 AM
Y by
Can’t give you an absolute easiest but bmo1 could be a decent starting point for very easy level. You can also just try random questions from random countries olympiads.
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N3bula
277 posts
N3bula
#7 h  Apr 28, 2025, 1:15 AM
Y by
js104 wrote:
Take a look at https://sites.math.washington.edu/~mathcircle/olympiad/archive/. These are intended for younger students (as I understand it, the way it works is that once you solve a problem you go up to a grader and directly verbally explain your solution to them, and they can ask for clarification/etc).
Ifl this seems like a way to develop bad habits for proof writing and such
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Iwowowl253
139 posts
Iwowowl253
#8 h  Apr 30, 2025, 9:34 PM • 1 Y
Y by llddmmtt1
I think some questions in the Canadian Contests, the Euclid and CIMC contests are pretty good for practicing writing some proofs.
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