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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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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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abc(a+b+c)=3, show that prod(a+b)>=8 [Indian RMO 2012(b) Q4]
Potla   27
N 6 minutes ago by sqing
Let $a,b,c$ be positive real numbers such that $abc(a+b+c)=3.$ Prove that we have
\[(a+b)(b+c)(c+a)\geq 8.\]
Also determine the case of equality.
27 replies
2 viewing
Potla
Dec 2, 2012
sqing
6 minutes ago
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Inequality
giangtruong13   1
N 9 minutes ago by arqady
Let $a,b,c >0$ such that: $a^2+b^2+c^2=3$. Prove that: $$\frac{b^2}{a}+\frac{c^2}{b}+\frac{a^2}{c}+abc \geq 4$$
1 reply
giangtruong13
4 hours ago
arqady
9 minutes ago
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hard problem....
Cobedangiu   1
N 16 minutes ago by arqady
let $a,b,c$ be the lengths of the sides of the triangle. Prove that:
$(a+b+c)(\dfrac{3a-b}{a^2+ab}+\dfrac{3b-c}{b^2+bc}+\dfrac{3c-a}{c^2+ac})\le 9$
1 reply
Cobedangiu
an hour ago
arqady
16 minutes ago
J
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Prove the inequality with the condition (a+1)(b+1)(c+1)=8
hlminh   1
N 18 minutes ago by quacksaysduck
Let $a,b,c>0$ such that $(a+1)(b+1)(c+1)=8.$ Prove that $abc(a+b+c)\leq 3.$
1 reply
hlminh
2 hours ago
quacksaysduck
18 minutes ago
J
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integer functional equation
ABCDE   147
N 22 minutes ago by Adywastaken
Source: 2015 IMO Shortlist A2
Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\]holds for all $x,y\in\mathbb{Z}$.
147 replies
ABCDE
Jul 7, 2016
Adywastaken
22 minutes ago
J
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number theory FE
pomodor_ap   0
39 minutes ago
Source: Own, PDC002-P7
Let $f : \mathbb{Z}^+ \to \mathbb{Z}^+$ be a function such that
$$f(m) + mn + n^2 \mid f(m)^2 + m^2 f(n) + f(n)^2$$for all $m, n \in \mathbb{Z}^+$. Find all such functions $f$.
0 replies
pomodor_ap
39 minutes ago
0 replies
J
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real+ FE
pomodor_ap   0
41 minutes ago
Source: Own, PDC001-P7
Let $f : \mathbb{R}^+ \to \mathbb{R}^+$ be a function such that
$$f(x)f(x^2 + y f(y)) = f(x)f(y^2) + x^3$$for all $x, y \in \mathbb{R}^+$. Determine all such functions $f$.
0 replies
pomodor_ap
41 minutes ago
0 replies
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Is this FE solvable?
ItzsleepyXD   2
N an hour ago by ItzsleepyXD
Source: Original
Let $c_1,c_2 \in \mathbb{R^+}$. Find all $f : \mathbb{R^+} \rightarrow \mathbb{R^+}$ such that for all $x,y \in \mathbb{R^+}$ $$f(x+c_1f(y))=f(x)+c_2f(y)$$
2 replies
ItzsleepyXD
Today at 3:02 AM
ItzsleepyXD
an hour ago
J
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AM-GM FE ineq
navi_09220114   2
N an hour ago by navi_09220114
Source: Own. Malaysian IMO TST 2025 P3
Let $\mathbb R$ be the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ where there exist a real constant $c\ge 0$ such that $$x^3+y^2f(y)+zf(z^2)\ge cf(xyz)$$holds for all reals $x$, $y$, $z$ that satisfy $x+y+z\ge 0$.

Proposed by Ivan Chan Kai Chin
2 replies
navi_09220114
Mar 22, 2025
navi_09220114
an hour ago
J
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Strange Geometry
Itoz   2
N an hour ago by hectorraul
Source: Own
Given a fixed circle $\omega$ with its center $O$. There are two fixed points $B, C$ and one moving point $A$ on $\omega$. The midpoint of the line segment $BC$ is $M$. $R$ is a fixed point on $\omega$. Line $AO$ intersects$\odot(AMR)$ at $P(\ne A)$, and line $BP$ intersects $\odot(BOC)$ at $Q(\ne B)$.

Find all the fixed points $R$ such that $\omega$ is always tangent to $\odot (OPQ)$ when $A$ varies.
Hint
2 replies
Itoz
Yesterday at 2:00 PM
hectorraul
an hour ago
J
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From Recreatii Matematice 1/2025
mihaig   0
an hour ago
Source: Own
Given a non-degenerate $\Delta ABC,$
find $x,y,z\geq0$ such that
$$x+y+z+\sqrt{\sum_{\text{cyc}}{x^2}-2\sum_{\text{cyc}}{yz\cos A}}=\sum_{\text{cyc}}{\sqrt{y^2-2yz\cos A+z^2}}.$$
0 replies
mihaig
an hour ago
0 replies
J
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Medium geometry with AH diameter circle
v_Enhance   93
N an hour ago by waterbottle432
Source: USA TSTST 2016 Problem 2, by Evan Chen
Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Denote by $M$, $N$ the midpoints of $\overline{AH}$, $\overline{BC}$. Suppose the circle $\gamma$ with diameter $\overline{AH}$ meets the circumcircle of $ABC$ at $G \neq A$, and meets line $AN$ at a point $Q \neq A$. The tangent to $\gamma$ at $G$ meets line $OM$ at $P$. Show that the circumcircles of $\triangle GNQ$ and $\triangle MBC$ intersect at a point $T$ on $\overline{PN}$.

Proposed by Evan Chen
93 replies
v_Enhance
Jun 28, 2016
waterbottle432
an hour ago
J
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International FE olympiad P3
Functional_equation   21
N 2 hours ago by ItzsleepyXD
Source: IFEO Day 1 P3
Find all functions $f:\mathbb R^+\rightarrow \mathbb R^+$ such that$$f(f(x)f(f(x))+y)=xf(x)+f(y)$$for all $x,y\in \mathbb R^+$

$\textit{Proposed by Functional\_equation, Mr.C and TLP.39}$
21 replies
Functional_equation
Feb 6, 2021
ItzsleepyXD
2 hours ago
J
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HANDOUT!! On the Angle Bisector Miquel Point
cursed_tangent1434   9
N 2 hours ago by quantam13
Source: Neat Configuration
Hi! This is a handout on the Configuration of the Angle Bisector Miquel Point, which originated from a series of notes made by Om245 for a lecture conducted by him for (Unofficial) INMO Training Camp.

Many thanks to stillwater_25 (for group-solving the key problem in the second section and finding a majority of it's key claims) and Takumi Higashida (for discovering most properties in relation to $\overline{WI}$) for all their time and support. We received immense help from TestX01 for the proof of claim 2.19 and it's associated lemma.

The point(s) that the handout deals with are very rich and there are numerous properties that we discovered. There are precious few contest problems related to this configuration and it remains relatively unknown among most of the community. However, we feel there is much more to this configuration to be explored and we hope that it may be as popular as other contemporary configurations in the future.

Due to the AoPS file sharing size restrictions, we have replaced the PDF with a google drive link.

Dive In!
9 replies
cursed_tangent1434
Mar 1, 2025
quantam13
2 hours ago
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Nice problem - Prove that we can dissect into lozenges
Amir Hossein   1
N Sep 20, 2010 by oneplusone
Source: IMO LongList 1979 - P1
Prove that in the Euclidean plane every regular polygon having an even number of sides can be dissected into lozenges. (A lozenge is a quadrilateral whose four sides are all of equal length).
1 reply
Amir Hossein
Sep 15, 2010
oneplusone
Sep 20, 2010
J
Nice problem - Prove that we can dissect into lozenges
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Reveal topic tags
geometry
polygon
dissection
rhombus
IMO Shortlist
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Source: IMO LongList 1979 - P1
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Amir Hossein
5452 posts
Amir Hossein
#1 h  Sep 15, 2010, 9:32 PM • 2 Y
Y by Adventure10, Mango247
Prove that in the Euclidean plane every regular polygon having an even number of sides can be dissected into lozenges. (A lozenge is a quadrilateral whose four sides are all of equal length).
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oneplusone
1459 posts
oneplusone
#2 h  Sep 20, 2010, 10:21 AM • 1 Y
Y by Adventure10
We claim that any even sided polygon with all equal lengths and parallel opposite sides (we call it a nice polygon) can be dissected into lozenges (rhombus). Clearly quadrilaterals works, now suppose it works until it has $2n-2$ sides with $n\geq 3$. Now consider a nice $(2n)$-gon $A_1...A_{2n}$. Now let $B_2,B_3,...,B_{n-1}$ inside the polygon such that the line segments $A_2B_2,...,A_{n-1}B_{n-1}$ are equal in length and parallel to $A_{2n}A_1$. Then the quadrilaterals $A_{2n}A_1A_2B_2,B_2A_2A_3B_3,B_3A_3A_4B_4,...,B_{n-1}A_{n-1}A_nA_{n+1}$ are all rhombus. Dissect them away and the remaining figure is a nice $(2n-2)$-gon. Thus by induction all nice polygon can be dissected into rhombus.
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