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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
and train with the best! Please note that early bird pricing ends August 19th!
Are you tired of the heat and thinking about Fall? 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.

Our full course list for upcoming classes is below:
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0 replies
jwelsh
Jul 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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Problem 1
SpectralS   148
N 15 minutes ago by happypi31415
Given triangle $ABC$ the point $J$ is the centre of the excircle opposite the vertex $A.$ This excircle is tangent to the side $BC$ at $M$, and to the lines $AB$ and $AC$ at $K$ and $L$, respectively. The lines $LM$ and $BJ$ meet at $F$, and the lines $KM$ and $CJ$ meet at $G.$ Let $S$ be the point of intersection of the lines $AF$ and $BC$, and let $T$ be the point of intersection of the lines $AG$ and $BC.$ Prove that $M$ is the midpoint of $ST.$

(The excircle of $ABC$ opposite the vertex $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$.)

Proposed by Evangelos Psychas, Greece
148 replies
SpectralS
Jul 10, 2012
happypi31415
15 minutes ago
J
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An interesting number theory problem.It seems easy but it's not.
Imanamiri   0
35 minutes ago
Source: Old Russian number theory.
Let \( a \) and \( b \) be natural numbers. Prove that \( a^2 - 2 \) is not divisible by \( 2b^2 + 3 \).
0 replies
Imanamiri
35 minutes ago
0 replies
J
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Orthocentre is collinear with two tangent points
vladimir92   43
N 36 minutes ago by HamstPan38825
Source: Chinese MO 1996
Let $\triangle{ABC}$ be a triangle with orthocentre $H$. The tangent lines from $A$ to the circle with diameter $BC$ touch this circle at $P$ and $Q$. Prove that $H,P$ and $Q$ are collinear.
43 replies
vladimir92
Jul 29, 2010
HamstPan38825
36 minutes ago
J
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orthocenters lie on a fixed circle, starting with another fixed circle
parmenides51   3
N an hour ago by Schintalpati
Source: 2018 Oral Moscow Geometry Olympiad grades 10-11 p3
A circle is fixed, point $A$ is on it and point $K$ outside the circle. The secant passing through $K$ intersects circle at points $P$ and $Q$. Prove that the orthocenters of the triangle $APQ$ lie on a fixed circle.
3 replies
parmenides51
Jul 25, 2019
Schintalpati
an hour ago
J
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Bonza functions
KevinYang2.71   65
N an hour ago by numbertheory97
Source: 2025 IMO P3
Let $\mathbb{N}$ denote the set of positive integers. A function $f\colon\mathbb{N}\to\mathbb{N}$ is said to be bonza if
\[ f(a)~~\text{divides}~~b^a-f(b)^{f(a)} \]for all positive integers $a$ and $b$.

Determine the smallest real constant $c$ such that $f(n)\leqslant cn$ for all bonza functions $f$ and all positive integers $n$.

Proposed by Lorenzo Sarria, Colombia
65 replies
KevinYang2.71
Jul 15, 2025
numbertheory97
an hour ago
J
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IMO ShortList 1998, algebra problem 1
orl   41
N an hour ago by lpieleanu
Source: IMO ShortList 1998, algebra problem 1
Let $a_{1},a_{2},\ldots ,a_{n}$ be positive real numbers such that $a_{1}+a_{2}+\cdots +a_{n}<1$. Prove that

\[ \frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_{1})(1 - a_{2}) \cdots (1 - a_{n})} \leq \frac{1}{ n^{n+1}}. \]
41 replies
orl
Oct 22, 2004
lpieleanu
an hour ago
J
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Isogonal lines
Giahuytls2326   5
N an hour ago by Giahuytls2326
Source: my cute teacher
Given a triangle \( ABC \) inscribed in a circle \( (O) \), let the altitudes \( AD, BE, CF \) concur at the orthocenter \( H \). Let \( X \) be the circumcenter of triangle \( DEF \) , and let \( Y \) be the circumcenter of triangle \( BOC \). Let \( U \) and \( V \) be the two intersection points of the circles centered at \( X \) and \( Y \). Prove that \( AU \) and \( AV \) are isogonal lines with respect to \( \angle BAC \).
5 replies
Giahuytls2326
Today at 1:33 PM
Giahuytls2326
an hour ago
J
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IMO Shortlist 2012, Combinatorics 1
lyukhson   77
N an hour ago by happypi31415
Source: IMO Shortlist 2012, Combinatorics 1
Several positive integers are written in a row. Iteratively, Alice chooses two adjacent numbers $x$ and $y$ such that $x>y$ and $x$ is to the left of $y$, and replaces the pair $(x,y)$ by either $(y+1,x)$ or $(x-1,x)$. Prove that she can perform only finitely many such iterations.

Proposed by Warut Suksompong, Thailand
77 replies
lyukhson
Jul 29, 2013
happypi31415
an hour ago
J
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Cyclic sum of 1/((3-c)(4-c))
v_Enhance   25
N an hour ago by lpieleanu
Source: ELMO Shortlist 2013: Problem A6, by David Stoner
Let $a, b, c$ be positive reals such that $a+b+c=3$. Prove that \[18\sum_{\text{cyc}}\frac{1}{(3-c)(4-c)}+2(ab+bc+ca)\ge 15. \]Proposed by David Stoner
25 replies
v_Enhance
Jul 23, 2013
lpieleanu
an hour ago
J
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Junior Balkan Mathematical Olympiad 2024- P2
Lukaluce   19
N an hour ago by lendsarctix280
Source: JBMO 2024
Let $ABC$ be a triangle such that $AB < AC$. Let the excircle opposite to A be tangent to the lines $AB, AC$, and $BC$ at points $D, E$, and $F$, respectively, and let $J$ be its centre. Let $P$ be a point on the side $BC$. The circumcircles of the triangles $BDP$ and $CEP$ intersect for the second time at $Q$. Let $R$ be the foot of the perpendicular from $A$ to the line $FJ$. Prove that the points $P, Q$, and $R$ are collinear.

(The excircle of a triangle $ABC$ opposite to $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$.)

Proposed by Bozhidar Dimitrov, Bulgaria
19 replies
Lukaluce
Jun 27, 2024
lendsarctix280
an hour ago
J
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Inequality
SunnyEvan   5
N an hour ago by JARP091
Let $ a,b,c \in R $ such that: $ abc>0 $ and $a^2+b^2+c^2=4(ab+bc+ca)$
Prove that: $$\frac{abc(a^3+b^3+c^3+ab(a+b)+bc(b+c)+ca(c+a))}{730a^2b^2c^2+k((a-b)(b-c)(c-a))^2} \leq \frac{2}{\sqrt{\frac{730}{3}k-9k^2}-3k} $$Where $ k\in(0,\frac{365(2-\sqrt2)}{54}]. $


$$ \frac{abc(a^3+b^3+c^3+ab(a+b)+bc(b+c)+ca(c+a))}{730a^2b^2c^2+k((a-b)(b-c)(c-a))^2} \leq \frac{18(\sqrt2+1)}{365} $$Where $ k\in[\frac{365(2-\sqrt2)}{54}, +\infty). $


$$ \frac{abc(a^3+b^3+c^3+ab(a+b)+bc(b+c)+ca(c+a))}{730a^2b^2c^2+k((a-b)(b-c)(c-a))^2} \geq \frac{18(\sqrt2+1)}{365} $$Where $ k\in(-\infty ,0). $
5 replies
SunnyEvan
Jul 1, 2025
JARP091
an hour ago
J
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Number theory
truongngochieu   1
N an hour ago by truongngochieu
Prove that \[\tau\big(\varphi(n)\big) \geq \varphi\big(\tau(n)\big)\]with all integer n.
1 reply
truongngochieu
Yesterday at 5:06 AM
truongngochieu
an hour ago
J
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sumcif..
teomihai   1
N an hour ago by Royal_mhyasd
Let $a=123456789^{123456789}$ ,$a_{1}=sumcif\{a\}$ ,$a_{2}=sumcifa\{1\}$...
Find number $a_{k}$ with one digit.
1 reply
teomihai
2 hours ago
Royal_mhyasd
an hour ago
J
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2024 SL C5
Twoisaprime   7
N an hour ago by NTguy
Source: 2024 IMO Shortlist C5
Let $N$ be a positive integer. Geoff and Ceri play a game in which they start by writing the numbers $1, 2, \dots, N$ on a board. They then take turns to make a move, starting with Geoff. Each move consists of choosing a pair of integers $(k, n)$, where $k \geq 0$ and $n$ is one of the integers on the board, and then erasing every integer $s$ on the board such that $2^k \mid n - s$. The game continues until the board is empty. The player who erases the last integer on the board loses.

Determine all values of $N$ for which Geoff can ensure that he wins, no matter how Ceri plays.
7 replies
Twoisaprime
Jul 16, 2025
NTguy
an hour ago
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J
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FE i created on bijective function with x≠y
benjaminchew13   9
N Jun 7, 2025 by benjaminchew13
Source: own (probably)
Find all bijective functions $f:\mathbb{R}\to \mathbb{R}$ such that $$(x-y)f(x+f(f(y)))=xf(x)+f(y)^{2}$$for all $x,y\in \mathbb{R}$ such that $x\neq y$.
9 replies
benjaminchew13
Jun 1, 2025
benjaminchew13
Jun 7, 2025
J
FE i created on bijective function with x≠y
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Reveal topic tags
function
algebra
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G H BBookmark kLocked kLocked NReply
Source: own (probably)
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benjaminchew13
26 posts
benjaminchew13
#1 h  Jun 1, 2025, 11:22 AM
Y by
Find all bijective functions $f:\mathbb{R}\to \mathbb{R}$ such that $$(x-y)f(x+f(f(y)))=xf(x)+f(y)^{2}$$for all $x,y\in \mathbb{R}$ such that $x\neq y$.
Z K Y
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benjaminchew13
26 posts
benjaminchew13
#2 h  Jun 1, 2025, 11:24 AM
Y by
Let $P(x,y)$ denote the above assertion.
Define $u$ such that $f(u)=0$.
$P(x,u):u=0$
So $f(0)=0$.
$P(0,x):f(f(f(x)))=-\frac{f(x)^{2}}{x}$ for all nonzero $x$.
Z K Y
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benjaminchew13
26 posts
benjaminchew13
#3 h  Jun 1, 2025, 11:25 AM
Y by
$f(x)=-x$ is a solution, but idk if it is the only one
Z K Y
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benjaminchew13
26 posts
benjaminchew13
#4 h  Jun 1, 2025, 11:33 AM
Y by
fakesolve
Click to reveal hidden text
This post has been edited 1 time. Last edited by benjaminchew13, Jun 1, 2025, 11:33 AM
Reason: realized that f(x)=-x is a sol
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benjaminchew13
26 posts
benjaminchew13
#5 h  Jun 1, 2025, 12:07 PM
Y by
$\text{bump}$
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ethan2011
407 posts
ethan2011
#6 h  Jun 1, 2025, 12:37 PM
Y by
Does $f(x)=-x$ not work?
My Solution
Z K Y
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benjaminchew13
26 posts
benjaminchew13
#7 h  Jun 1, 2025, 12:39 PM
Y by
notice x ≠ y
Z K Y
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benjaminchew13
26 posts
benjaminchew13
#8 h  Jun 1, 2025, 12:48 PM
Y by
bump again
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benjaminchew13
26 posts
benjaminchew13
#9 h  Jun 1, 2025, 1:02 PM
Y by
current progress, how should i continue
Click to reveal hidden text
This post has been edited 1 time. Last edited by benjaminchew13, Jun 1, 2025, 1:02 PM
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benjaminchew13
26 posts
benjaminchew13
#12 h  Jun 7, 2025, 11:56 AM
Y by
$$\text{rev}$$
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N Quick Reply
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