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k a August Highlights and 2025 AoPS Online Class Information
jwelsh   0
Aug 1, 2025
CONGRATULATIONS to all the competitors at this year’s International Mathematical Olympiad (IMO)! The US Team took second place with 5 gold medals and 1 silver - we are proud to say that each member of the 2025 IMO team has participated in an AoPS WOOT (Worldwide Online Olympiad Training) class!

"As a parent, I'm deeply grateful to AoPS. Tiger has taken very few math courses outside of AoPS, except for a local Math Circle that doesn't focus on Olympiad math. AoPS has been one of the most important resources in his journey. Without AoPS, Tiger wouldn't be where he is today — especially considering he's grown up in a family with no STEM background at all."
— Doreen Dai, parent of IMO US Team Member Tiger Zhang

Interested to learn more about our WOOT programs? Check out the course page here or join a Free Scheduled Info Session. Early bird pricing ends August 19th!:
CodeWOOT Code Jam - Monday, August 11th
ChemWOOT Chemistry Jam - Wednesday, August 13th
PhysicsWOOT Physics Jam - Thursday, August 14th
MathWOOT Math Jam - Friday, August 15th

There is still time to enroll in our last wave of summer camps that start in August at the Virtual Campus, our video-based platform, for math and language arts! From Math Beasts Camp 6 (Prealgebra Prep) to AMC 10/12 Prep, you can find an informative 2-week camp before school starts. Plus, our math camps don’t have homework and cover cool enrichment topics like graph theory. Our language arts courses will build the foundation for next year’s challenges, such as Language Arts Triathlon for levels 5-6 and Academic Essay Writing for high school students.

Lastly, Fall is right around the corner! You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US. We’ve opened new Academy locations in San Mateo, CA, Pasadena, CA, Saratoga, CA, Johns Creek, GA, Northbrook, IL, and Upper West Side (NYC), New York.

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0 replies
jwelsh
Aug 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
J
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Fixed Points from Moving Intersections
RANDOM__USER   1
N 18 minutes ago by v4913
Source: Own
Let \(P\) be an arbitrary fixed point and \(X\) an abitrary fixed point on \((ABC)\). Let \(D\) be an abritrary point on \(BC\). Let \(PD\) intersect \(AC\)at \(E\). Let \(XD\) intersect \((ABC)\) at \(G\). Let \((AEG)\) intersect \(AB\) a second time at \(F\). Prove that the line \(DF\) passes through a constant point \(Q\) as \(D\) moves on \(BC\), and that \((AEF)\) passes through a fixed point \(W\).

IMAGE

1 reply
RANDOM__USER
Today at 12:43 PM
v4913
18 minutes ago
J
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gcd(f(m) + n, f(n) + m) bounded for m != n
62861   11
N 42 minutes ago by pi271828
Source: IMO 2015 Shortlist, N7
Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is called $k$-good if $\gcd(f(m) + n, f(n) + m) \le k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function.

Proposed by James Rickards, Canada
11 replies
62861
Jul 7, 2016
pi271828
42 minutes ago
J
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easy sequence
Seungjun_Lee   18
N 43 minutes ago by Kempu33334
Source: KMO 2023 P1
A sequence of positive reals $\{ a_n \}$ is defined below. $$a_0 = 1, a_1 = 3, a_{n+2} = \frac{a_{n+1}^2+2}{a_n}$$Show that for all nonnegative integer $n$, $a_n$ is a positive integer.
18 replies
Seungjun_Lee
Nov 4, 2023
Kempu33334
43 minutes ago
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|z+2024|<2024
MathMaxGreat   1
N an hour ago by ThE-dArK-lOrD
Source: 2024 China Summer NSMO
Let $z_0$ be a root of $P(z)=\frac{(z+1)(z+2)\cdot\cdot\cdot (z+2024)-2024!}{z}$.
Prove: $z_0\in \{z|z\in\mathbb{C},|z+2024|<2024\}$
1 reply
MathMaxGreat
Today at 4:45 AM
ThE-dArK-lOrD
an hour ago
J
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No. of possible equalities
old_csk_mo   1
N an hour ago by blug
Source: Czech and Slovak Olympiad 2025, National Round, Problem 1
Real numbers $a,b,c,d$ satisfy \[a+b+c+d=0,\qquad \frac1a+\frac1b+\frac1c+\frac1d=0.\]How many equalities \[ab=cd,\qquad ac=bd,\qquad ad=bc\]can simultaneously hold? Determine all possibilities.
1 reply
old_csk_mo
3 hours ago
blug
an hour ago
J
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Equal lengths of segments
old_csk_mo   1
N 2 hours ago by Sir_Cumcircle
Source: Czech and Slovak Olympiad 2025, National Round, Problem 6
Let $ABC$ be an acute triangle, $H$ its orthocenter, $O$ circumcenter and $M$ midpoint of $BC.$ Denote $D\neq A$ the intersection of line $AH$ and the circumcircle $\omega,$ $E\neq D$ intersection of line $DM$ and $\omega.$ Finally, let $F\neq E$ be the intersection of line $AE$ and the circumcircle of $OME.$ Show that $FH=FA.$
1 reply
old_csk_mo
2 hours ago
Sir_Cumcircle
2 hours ago
J
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21 distinct real numbers
old_csk_mo   0
2 hours ago
Source: CPSJ 2025 team competition p4
Is it always possible to choose (different) elements $x,y$ of a set of 21 distinct real numbers such that \[20|x-y|<(x+1)(y+1)?\]
0 replies
old_csk_mo
2 hours ago
0 replies
J
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Coloring of a square grid
old_csk_mo   0
2 hours ago
Source: Czech and Slovak Olympiad 2025, National Round, Problem 5
Determine all positive integers $n$ such that $2n$ cells of $n\times n$ square grid can be colored in a way where no two dyed squares share a point and there are exactly two dyed squares in every column and every row.
0 replies
1 viewing
old_csk_mo
2 hours ago
0 replies
J
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Counting + Number theory
urfinalopp   1
N 2 hours ago by megarnie
Source: Hai Phong VMO TST 2020-2021
Given a prime $p \equiv 1$ (mod 4), determine the number of ordered integer triplets $(a_1; a_2; a_3)$ such that
\begin{align*} a_1a_2 + a_3^2 + 1 \vdots p^2 \end{align*}($a_1, a_2, a_3$ are not necessarily different from each other)
1 reply
urfinalopp
5 hours ago
megarnie
2 hours ago
J
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Primes on a circle
old_csk_mo   0
2 hours ago
Source: Czech and Slovak Olympiad 2025, National Round, Problem 4
At least three primes are written on a circle, all of them distinct. Compute greatest prime divisors of sums of any two neighbors. Suppose that we received the same primes as already written (up to ordering). Determine all possible input sets of primes.
0 replies
old_csk_mo
2 hours ago
0 replies
J
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Integer terms of recurrent sequence
old_csk_mo   2
N 2 hours ago by blug
Source: CPSJ 2025 team competition p5
Let $\left(a_n\right)_{n\ge1}$ be a sequence of positive numbers such that \[a_{n+1}=a_n+\frac{1}{a_n}\]for all positive integers $n.$ Determine the greatest integer $N$ such that exactly $N$ terms of the sequence are integers (for some $a_1$).
2 replies
old_csk_mo
Today at 10:23 AM
blug
2 hours ago
J
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Unique sums of divisors
old_csk_mo   0
2 hours ago
Source: Czech and Slovak Olympiad 2025, National Round, Problem 3
Let $n>1$ be a positive integer and $p$ its greatest prime divisor. For each non-empty subset of divisors of $n,$ write the sum of its elements on the board. Assume that more than $p$ numbers from the set $\{1,2,\ldots,p+2\}$ are written and any of them occurs at most once. Show that all numbers on the board are distinct.
0 replies
old_csk_mo
2 hours ago
0 replies
J
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Sum of groups
MithsApprentice   14
N 3 hours ago by Bread10
Source: USAMO 1996
For any nonempty set $S$ of real numbers, let $\sigma(S)$ denote the sum of the elements of $S$. Given a set $A$ of $n$ positive integers, consider the collection of all distinct sums $\sigma(S)$ as $S$ ranges over the nonempty subsets of $A$. Prove that this collection of sums can be partitioned into $n$ classes so that in each class, the ratio of the largest sum to the smallest sum does not exceed 2.
14 replies
MithsApprentice
Oct 22, 2005
Bread10
3 hours ago
J
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Lower bound for magnitudes of angles
old_csk_mo   0
3 hours ago
Source: Czech and Slovak Olympiad 2025, National Round, Problem 2
Let $n$ be a positive integer. Consider five distinct points in plane such that two of them are inner points of a (non-degenerate) triangle given by the other three. Determine the greatest $n$ for which there is an angle $\varphi$ given by three of these points such that $n^\circ<\varphi\le 180^\circ.$
0 replies
old_csk_mo
3 hours ago
0 replies
J
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J
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euler function
mathsearcher   0
May 16, 2025
Prove that there exists infinitely many positive integers n such that
ϕ(n) | n+1
0 replies
mathsearcher
May 16, 2025
0 replies
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euler function
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mathsearcher
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mathsearcher
#1 h  May 16, 2025, 4:41 AM
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Prove that there exists infinitely many positive integers n such that
ϕ(n) | n+1
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