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k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
Jun 2, 2025
Congratulations to all the mathletes who competed at National MATHCOUNTS! If you missed the exciting Countdown Round, you can watch the video at this link. Are you interested in training for MATHCOUNTS or AMC 10 contests? How would you like to train for these math competitions in half the time? We have accelerated sections which meet twice per week instead of once starting on July 8th (7:30pm ET). These sections fill quickly so enroll today!

[list][*]MATHCOUNTS/AMC 8 Basics
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[*]AMC 10 Problem Series[/list]
For those interested in Olympiad level training in math, computer science, physics, and chemistry, be sure to enroll in our WOOT courses before August 19th to take advantage of early bird pricing!

Summer camps are starting this 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 a transformative 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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0 replies
1 viewing
jlacosta
Jun 2, 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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students in a classroom sit in a round table, possible to split into 3 groups
parmenides51   1
N 2 minutes ago by Magnetoninja
Source: Dutch IMO TST 2018 day 2 p4
In the classroom of at least four students the following holds: no matter which four of them take seats around a round table, there is always someone who either knows both of his neighbours, or does not know either of his neighbours. Prove that it is possible to divide the students into two groups such that in one of them, all students know one another, and in the other, none of the students know each other.

(Note: if student A knows student B, then student B knows student A as well.)
1 reply
parmenides51
Aug 30, 2019
Magnetoninja
2 minutes ago
J
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Sharing is a nontrivial task
bjump   11
N 16 minutes ago by HamstPan38825
Source: USA TST 2024 P5
Suppose $a_{1} < a_{2}< \cdots < a_{2024}$ is an arithmetic sequence of positive integers, and $b_{1} <b_{2} < \cdots <b_{2024}$ is a geometric sequence of positive integers. Find the maximum possible number of integers that could appear in both sequences, over all possible choices of the two sequences.

Ray Li
11 replies
bjump
Jan 15, 2024
HamstPan38825
16 minutes ago
J
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Non-classical FE
M11100111001Y1R   0
26 minutes ago
Source: Iran TST 2025 Test 2 Problem 3
Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that for all $x, y >0$ we have:
$$f(f(f(xy))+x^2)=f(y)(f(x)-f(x+y))$$
0 replies
M11100111001Y1R
26 minutes ago
0 replies
J
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My Unsolved Problem
ZeltaQN2008   4
N 37 minutes ago by ZeltaQN2008
Source: IDK
Let \( ABC \) be an acute triangle inscribed in its circumcircle \( (O) \), and let \( (I) \) be its incircle. Let \( K \) be the point where the $A-mixtilinear$ incircle of triangle $ABC$ touches \((O)\). Suppose line \( OI \) intersects segment \( AK \) at \( P \), and intersects line \( BC \) at \( Q \). Let the line through \( I \) perpendicular to \( BC \) intersect line \( KQ \) at \( A' \). Prove that: \[AI \parallel PA'.\]
4 replies
ZeltaQN2008
Yesterday at 1:23 PM
ZeltaQN2008
37 minutes ago
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Most accurate rounding
popcorn1   24
N 42 minutes ago by HamstPan38825
Source: USA Team Selection Test for IMO 2024, Problem 1
Find the smallest constant $C > 1$ such that the following statement holds: for every integer $n \geq 2$ and sequence of non-integer positive real numbers $a_1, a_2, \dots, a_n$ satisfying $$\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n} = 1,$$it's possible to choose positive integers $b_i$ such that
(i) for each $i = 1, 2, \dots, n$, either $b_i = \lfloor a_i \rfloor$ or $b_i = \lfloor a_i \rfloor + 1$, and
(ii) we have $$1 < \frac{1}{b_1} + \frac{1}{b_2} + \cdots + \frac{1}{b_n} \leq C.$$(Here $\lfloor \bullet \rfloor$ denotes the floor function, as usual.)

Merlijn Staps
24 replies
popcorn1
Dec 11, 2023
HamstPan38825
42 minutes ago
J
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Concurrence
LiamChen   1
N 44 minutes ago by phonghatemath
Source: MOP1998
Problem:
1 reply
LiamChen
3 hours ago
phonghatemath
44 minutes ago
J
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Merlijn Has 100 Coins
tastymath75025   40
N 44 minutes ago by peace09
Source: USA TSTST 2019 Problem 4
Consider coins with positive real denominations not exceeding 1. Find the smallest $C>0$ such that the following holds: if we have any $100$ such coins with total value $50$, then we can always split them into two stacks of $50$ coins each such that the absolute difference between the total values of the two stacks is at most $C$.

Merlijn Staps
40 replies
1 viewing
tastymath75025
Jun 25, 2019
peace09
44 minutes ago
J
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IMO ShortList 1998, algebra problem 1
orl   39
N an hour ago by Siddharthmaybe
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

\[ \frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_{1})(1 - a_{2}) \cdots (1 - a_{n})} \leq \frac{1}{ n^{n+1}}. \]
39 replies
orl
Oct 22, 2004
Siddharthmaybe
an hour ago
J
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2013 Japan MO Finals P5
parkjungmin   0
an hour ago
2013 Japan MO Finals
0 replies
parkjungmin
an hour ago
0 replies
J
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Might be slightly generalizable
Rijul saini   6
N an hour ago by WLOGQED1729
Source: India IMOTC Day 3 Problem 1
Let $ABC$ be an acute angled triangle with orthocenter $H$ and $AB<AC$. Let $T(\ne B,C, H)$ be any other point on the arc $\stackrel{\LARGE\frown}{BHC}$ of the circumcircle of $BHC$ and let line $BT$ intersect line $AC$ at $E(\ne A)$ and let line $CT$ intersect line $AB$ at $F(\ne A)$. Let the circumcircles of $AEF$ and $ABC$ intersect again at $X$ ($\ne A$). Let the lines $XE,XF,XT$ intersect the circumcircle of $(ABC)$ again at $P,Q,R$ ($\ne X$). Prove that the lines $AR,BC,PQ$ concur.
6 replies
Rijul saini
Yesterday at 6:39 PM
WLOGQED1729
an hour ago
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Prove that circumcircle of the triangle TEF tangent to (O), (K), (L)
centos6   8
N an hour ago by ihategeo_1969
Let $(O)$ be the circumcircle of the triangle $\triangle ABC$. $A’$ be the antipode of $A$ in $(O)$. Angle bisector of angle $\angle A$ meets $BC$ and $A-Mixtilinear$ at $D$ and $E$. Let $N$ be the midpoind of the arc $BAC$. $ T = A’E \cap (O), T \neq A’, F = AD \cap NT$. $(K)$ and $(L)$ be the Thebault circles of the cevian $AD$. Prove that circumcircle of the triangle $\triangle TEF$ tangent to $(O)$, $(K)$ and $(L)$.

IMAGE
8 replies
centos6
Nov 30, 2018
ihategeo_1969
an hour ago
J
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What the isogonal conjugate on IO
reni_wee   1
N an hour ago by Funcshun840
Source: buratinogigle
Given a triangle $ABC$ with incircle $(I)$ tangent to $BC, CA, AB$ at points $D, E, F$, respectively. Let $P$ be a point such that its isogonal conjugate lies on the line $OI$ (where $O$ is the circumcenter and $I$ the incenter of $ABC$). The line $PA$ intersects segments $DE$ and $DF$ at points $M_a$ and $N_a$, respectively, such that the circle with diameter $M_a N_a$ meets $BC$ at points $P_a$ and $Q_a$.

1) Prove that the circle $(AP_a Q_a)$ is tangent to the incircle $(I)$ at some point $X$.

2) Similarly define points $Y, Z$ corresponding to vertices $B, C$. Prove that the lines $AX, BY, CZ$ are concurrent.
1 reply
reni_wee
3 hours ago
Funcshun840
an hour ago
J
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Maybe a old inequality?
pxchg1200   5
N an hour ago by teomihai
Source: Tran Quoc Anh
Let $ a,b,c>0$ with $ a+b+c=3 $ prove that:
\[ (a^2+1)(b^2+1)(c^2+1)\geq (a+1)(b+1)(c+1) \]



Click to reveal hidden text
5 replies
pxchg1200
Mar 6, 2012
teomihai
an hour ago
J
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My Unsolved Problem
ZeltaQN2008   2
N 2 hours ago by ladder123
Source: IDK
Let \( P(x) = x^{2024} + a_{2023}x^{2023} + \cdots + a_1x + a_0 \) be a polynomial with real coefficients.

(a) Suppose that \( 2023a_{2023}^2 - 4048a_{2022} < 0 \). Prove that the polynomial \( P(x) \) cannot have 2024 real roots.

(b) Suppose that \( a_0 = 1 \) and \( 2023(a_1^2 + a_2^2 + \cdots + a_{2023}^2) \leq 4 \). Prove that \( P(x) \geq 0 \) for all real numbers \( x \).
2 replies
ZeltaQN2008
May 29, 2025
ladder123
2 hours ago
J
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J
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NT algebra
UlanKZ   4
N May 24, 2018 by quangminh1173
Find all integers, can be written of form $$a^3+b^3+c^3-3abc$$, where $a,b,c$ are integers
4 replies
UlanKZ
May 24, 2018
quangminh1173
May 24, 2018
J
NT algebra
G H J
Reveal topic tags
number theory
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UlanKZ
38 posts
UlanKZ
#1 h  May 24, 2018, 12:51 PM • 2 Y
Y by Adventure10, Mango247
Find all integers, can be written of form $$a^3+b^3+c^3-3abc$$, where $a,b,c$ are integers
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TuZo
19351 posts
TuZo
#2 h  May 24, 2018, 2:23 PM • 2 Y
Y by Adventure10, Mango247
It is impossible to find all, because if you write any number instead of $a,b,c$, you get an integer number, we have an infinite many integer numbers!
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ythomashu
6322 posts
ythomashu
#3 h  May 24, 2018, 2:25 PM • 1 Y
Y by Adventure10
how about all integers that cannot be written of this form?
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test20
988 posts
test20
#4 h  May 24, 2018, 2:28 PM • 1 Y
Y by Adventure10
This problem is closely related to:
https://artofproblemsolving.com/community/c6h1640540
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quangminh1173
727 posts
quangminh1173
#5 h  May 24, 2018, 2:44 PM • 2 Y
Y by Adventure10, Mango247
An integer $n$ can be written of the form $a^3+b^3+c^3-3abc$ iff $n$ is not divisible by $3$ or $n$ is divisible by $9.$
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