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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 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
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Benelux fe
ErTeeEs06   8
N 13 minutes ago by NicoN9
Source: BxMO 2025 P1
Does there exist a function $f:\mathbb{R}\to \mathbb{R}$ such that $$f(x^2+f(y))=f(x)^2-y$$for all $x, y\in \mathbb{R}$?
8 replies
ErTeeEs06
Yesterday at 11:05 AM
NicoN9
13 minutes ago
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weird symmetric equation
giangtruong13   1
N 16 minutes ago by pooh123
Solve the equation: $$8x^2-11x+1=(1-x)\sqrt{4x^2-6x+5}$$
1 reply
giangtruong13
4 hours ago
pooh123
16 minutes ago
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P(x) | P(x^2-2)
GreenTea2593   5
N 44 minutes ago by amogususususus
Source: Valentio Iverson
Let $P(x)$ be a monic polynomial with complex coefficients such that there exist a polynomial $Q(x)$ with complex coefficients for which \[P(x^2-2)=P(x)Q(x).\]Determine all complex numbers that could be the root of $P(x)$.

Proposed by Valentio Iverson, Indonesia
5 replies
GreenTea2593
Apr 22, 2025
amogususususus
44 minutes ago
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Interesting inequalities
sqing   3
N an hour ago by sqing
Source: Own
Let $ a,b\geq 0 $ and $ a+b+ab=3. $ Prove that
$$ab^2( b +1) \leq 4$$$$ab( b +1) \leq \frac{9}{4} $$$$a^2b ( a+b^2 ) \leq \frac{76}{27}$$$$a^2b( b +1 ) \leq \frac{3(69-11\sqrt{33})}{8} $$$$a^2b^2( b +1 ) \leq \frac{2(73\sqrt{73}-595)}{27} $$
3 replies
sqing
5 hours ago
sqing
an hour ago
J
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Interesting inequalities
sqing   8
N an hour ago by sqing
Source: Own
Let $ a,b\geq 0 $ and $ a+b=2. $ Prove that
$$ ab( a^2+ b^2)^2 \leq \frac{128}{27}$$$$ ab( a^2-ab+ b^2)^2 \leq \frac{256}{81}$$$$ ab\sqrt{ab}( a^2+ b^2)^2 \leq \frac{1536}{343}\sqrt{\frac{6}{7}}$$$$ ab\sqrt{ab}( a^2-ab+ b^2)^2 \leq \frac{2048}{343\sqrt{7}}$$
8 replies
sqing
4 hours ago
sqing
an hour ago
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Functional xf(x+f(y))=(y-x)f(f(x)) for all reals x,y
cretanman   58
N an hour ago by GreekIdiot
Source: BMO 2023 Problem 1
Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$,
\[xf(x+f(y))=(y-x)f(f(x)).\]
Proposed by Nikola Velov, Macedonia
58 replies
cretanman
May 10, 2023
GreekIdiot
an hour ago
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easy functional
B1t   9
N an hour ago by GreekIdiot
Source: Mongolian TST 2025 P1.
Denote the set of real numbers by $\mathbb{R}$. Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that for all $x, y, z \in \mathbb{R}$,
\[ f(xf(x+y)+z) = f(z) + f(x)y + f(xf(x)). \]
9 replies
B1t
Yesterday at 6:45 AM
GreekIdiot
an hour ago
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IMO Shortlist 2012, Number Theory 1
lyukhson   45
N 2 hours ago by sansgankrsngupta
Source: IMO Shortlist 2012, Number Theory 1
Call admissible a set $A$ of integers that has the following property:
If $x,y \in A$ (possibly $x=y$) then $x^2+kxy+y^2 \in A$ for every integer $k$.
Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers.

Proposed by Warut Suksompong, Thailand
45 replies
lyukhson
Jul 29, 2013
sansgankrsngupta
2 hours ago
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Easy OH Problem in Taiwan TST
chengbilly   2
N 2 hours ago by CrazyInMath
Source: 2025 Taiwan TST Round 3 Independent Study 1-G
Let $O, H$ be the circumcenter and orthocenter of triangle $ABC$, respectively. Let $\odot(ABC)$ and $\odot(AOH)$ intersect at point $S\neq A$. If $T$ is an arbitrary point on
$\odot(AOH)$, and $X$ and $Y$ be the intersection of $OH $with $BC$ and $AT$ , respectively.
Prove that $S, T , X, Y$ are concyclic.

proposed by chengbilly
2 replies
chengbilly
Yesterday at 1:54 PM
CrazyInMath
2 hours ago
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nice geo
Melid   3
N 2 hours ago by Melid
Source: 2025 Japan Junior MO preliminary P9
Let ABCD be a cyclic quadrilateral, which is AB=7 and BC=6. Let E be a point on segment CD so that BE=9. Line BE and AD intersect at F. Suppose that A, D, and F lie in order. If AF=11 and DF:DE=7:6, find the length of segment CD.
3 replies
Melid
Apr 23, 2025
Melid
2 hours ago
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external bisector in 2 angle
crocodilepradita   5
N 3 hours ago by ihategeo_1969
Given a $\triangle ABC$ with incenter $I$. Line $BI$ and $CI$ intersects $CA$ and $AB$ at $E$ and $F$, respectively. Let $M$ and $N$ be the midpoints of $BI$ and $CI$. Line $FM$ meets the external bisector of angle $B$ at $K$ and line $EN$ meets the external bisector of angle $C$ at $L$. Prove that $K, B, L, C$ are concylic.
5 replies
crocodilepradita
Aug 22, 2024
ihategeo_1969
3 hours ago
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Difficult combinatorics problem about distinct sums under shifts
CBMaster   0
3 hours ago
Source: Korea
Problem. Let $a_1, ..., a_n$ be the nonnegative integers in $\{0, 1, ..., m\}$ where $m=\left\lceil \frac{n^{2/3}}{4} \right\rceil $. Define $A=\{a_i+a_j+(j-i)|1\leq i<j\leq n\}$. Prove that $|A|\geq m$.

Bonus problem (Open). Can we prove a tighter result than the one above? That is, is there a function $f(n)$ such that $f(n)=O(n^\alpha)$ where $\alpha>\frac{2}{3}$, and the statement is still true when $m=f(n)$?
Or, is there a function $f(n)$ such that $f(n)\geq C \cdot n^{2/3}$ where $C>\frac{1}{4}$, and the statement is still true when $m=f(n)$?.
0 replies
CBMaster
3 hours ago
0 replies
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4 variables with quadrilateral sides
mihaig   5
N 3 hours ago by mihaig
Source: VL
Let $a,b,c,d\geq0$ satisfying
$$\frac1{a+1}+\frac1{b+1}+\frac1{c+1}+\frac1{d+1}=2.$$Prove
$$4\left(abc+abd+acd+bcd\right)\geq3\left(a+b+c+d\right)+4.$$
5 replies
mihaig
Apr 25, 2025
mihaig
3 hours ago
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CGMO5: Carlos Shine's Fact 5
v_Enhance   60
N 3 hours ago by Sedro
Source: 2012 China Girl's Mathematical Olympiad
As shown in the figure below, the in-circle of $ABC$ is tangent to sides $AB$ and $AC$ at $D$ and $E$ respectively, and $O$ is the circumcenter of $BCI$. Prove that $\angle ODB = \angle OEC$.
IMAGE
60 replies
v_Enhance
Aug 13, 2012
Sedro
3 hours ago
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Inequalities with centroid
indybar   1
N Nov 21, 2006 by niyo
Source: JBMO Selection Test 2005, Problem 9
Let $ABC$ be a triangle with $BC>CA>AB$ and let $G$ be the centroid of the triangle. Prove that \[ \angle GCA+\angle GBC<\angle BAC<\angle GAC+\angle GBA . \]
Dinu Serbanescu
1 reply
indybar
Mar 13, 2006
niyo
Nov 21, 2006
J
Inequalities with centroid
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Reveal topic tags
inequalities
geometry unsolved
geometry
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Source: JBMO Selection Test 2005, Problem 9
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indybar
398 posts
indybar
#1 h  Mar 13, 2006, 9:17 AM • 2 Y
Y by Adventure10, Mango247
Let $ABC$ be a triangle with $BC>CA>AB$ and let $G$ be the centroid of the triangle. Prove that \[ \angle GCA+\angle GBC<\angle BAC<\angle GAC+\angle GBA . \]
Dinu Serbanescu
Z K Y
The post below has been deleted. Click to close.
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niyo
7 posts
niyo
#2 h  Nov 21, 2006, 9:59 AM • 2 Y
Y by Adventure10, Mango247
indybar wrote:
Let $ABC$ be a triangle with $BC>CA>AB$ and let $G$ be the centroid of the triangle. Prove that
\[\angle GCA+\angle GBC<\angle BAC<\angle GAC+\angle GBA . \]
Dinu Serbanescu

it is false?? :?:
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