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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 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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Nice one
Blacklord   10
N a minute ago by Pal702004
Source: ....
Find all integers numbers (a,b,c) such that
$a/b + b/c + c/a =3$
10 replies
Blacklord
Jan 26, 2017
Pal702004
a minute ago
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Four points are concyclic
DreamTeam   9
N 7 minutes ago by AylyGayypow009
Source: Moldova IMO-BMO TST 2003, day 1, problem 3
Let $ ABCD$ be a quadrilateral inscribed in a circle of center $ O$. Let M and N be the midpoints of diagonals $ AC$ and $ BD$, respectively and let $ P$ be the intersection point of the diagonals $ AC$ and $ BD$ of the given quadrilateral .It is known that the points $ O,M,Np$ are distinct. Prove that the points $ O,N,A,C$ are concyclic if and only if the points $ O,M,B,D$ are concyclic.

Proposer: Dorian Croitoru
9 replies
DreamTeam
Aug 14, 2008
AylyGayypow009
7 minutes ago
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cubefree divisibility
DottedCaculator   63
N 11 minutes ago by Assassino9931
Source: 2021 ISL N1
Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$
63 replies
DottedCaculator
Jul 12, 2022
Assassino9931
11 minutes ago
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$n$ with $2000$ divisors divides $2^n+1$ (IMO 2000)
Valentin Vornicu   67
N 23 minutes ago by alexanderchew
Source: IMO 2000, Problem 5, IMO Shortlist 2000, Problem N3
Does there exist a positive integer $ n$ such that $ n$ has exactly 2000 prime divisors and $ n$ divides $ 2^n + 1$?
67 replies
Valentin Vornicu
Oct 24, 2005
alexanderchew
23 minutes ago
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most bruh geo you can imagineeeeeeeeeeeeee
ItzsleepyXD   0
40 minutes ago
Source: bruhhhhhh
Let $ABC$ be triangle with $AC>AB$ and $B'$ on the segment $AC$ such that $AB=AB'$ . Consider point $E,F$ on $AB,AC$ such that $EF \parallel BB'$ . Point $T$ is the intersection of tangent line through point $E,F$ to circle $(EBC),(FBC)$ respectively . If the tangent through point $B'$ to $(BB'C)$ intersect $AB$ at $K$ . Line $KT$ intersect $BC$ at $D$ . Prove that $AD$ bisect $\angle BAC$ .
0 replies
ItzsleepyXD
40 minutes ago
0 replies
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Less than or equal to 30°
orl   11
N an hour ago by Twan
Source: IMO 1991, Day 2, Problem 4, IMO ShortList 1991, Problem 24 (FRA 2)
Let $ \,ABC\,$ be a triangle and $ \,P\,$ an interior point of $ \,ABC\,$. Show that at least one of the angles $ \,\angle PAB,\;\angle PBC,\;\angle PCA\,$ is less than or equal to $ 30^{\circ }$.
11 replies
orl
Nov 11, 2005
Twan
an hour ago
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Simple Geometry
AbdulWaheed   4
N an hour ago by shanelin-sigma
Source: EGMO
Try to avoid Directed angles
Let ABC be an acute triangle inscribed in circle $\Omega$. Let $X$ be the midpoint of the arc $\overarc{BC}$ not containing $A$ and define $Y, Z$ similarly. Show that the orthocenter of $XYZ$ is the incenter $I$ of $ABC$.
4 replies
AbdulWaheed
Yesterday at 5:15 AM
shanelin-sigma
an hour ago
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2021 EGMO P4: Reflection of A over EF lies on BC
anser   48
N 2 hours ago by cursed_tangent1434
Source: EGMO 2021 P4
Let $ABC$ be a triangle with incenter $I$ and let $D$ be an arbitrary point on the side $BC$. Let the line through $D$ perpendicular to $BI$ intersect $CI$ at $E$. Let the line through $D$ perpendicular to $CI$ intersect $BI$ at $F$. Prove that the reflection of $A$ across the line $EF$ lies on the line $BC$.
48 replies
anser
Apr 13, 2021
cursed_tangent1434
2 hours ago
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Serbian selection contest for the IMO 2025 - P2
OgnjenTesic   10
N 2 hours ago by Mahdi_Mashayekhi
Source: Serbian selection contest for the IMO 2025
Let $ABC$ be an acute triangle. Let $A'$ be the reflection of point $A$ over the line $BC$. Let $O$ and $H$ be the circumcenter and the orthocenter of triangle $ABC$, respectively, and let $E$ be the midpoint of segment $OH$. Let $D$ and $L$ be the points where the reflection of line $AA'$ with respect to line $OA'$ intersects the circumcircle of triangle $ABC$, where point $D$ lies on the arc $BC$ not containing $A$. If \( M \) is a point on the line \( BC \) such that \( OM \perp AD \), prove that \( \angle MAD = \angle EAL \).

Proposed by Strahinja Gvozdić
10 replies
OgnjenTesic
Thursday at 4:02 PM
Mahdi_Mashayekhi
2 hours ago
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AT // BC wanted
parmenides51   105
N 2 hours ago by Adywastaken
Source: IMO 2019 SL G1
Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.

(Nigeria)
105 replies
parmenides51
Sep 22, 2020
Adywastaken
2 hours ago
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Two lines concur on (ABC)
amar_04   19
N 3 hours ago by Giant_PT
Source: XVII Sharygin Corespondnce Round P13
In triangle $ABC$ with circumcircle $\Omega$ and incenter $I$, point $M$ bisects arc $BAC$ and line $\overline{AI}$ meets $\Omega$ at $N\ne A$. The excircle opposite to $A$ touches $\overline{BC}$ at point $E$. Point $Q\ne I$ on the circumcircle of $\triangle MIN$ is such that $\overline{QI}\parallel\overline{BC}$. Prove that the lines $\overline{AE}$ and $\overline{QN}$ meet on $\Omega$.
19 replies
amar_04
Mar 2, 2021
Giant_PT
3 hours ago
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Nice problem of concurrency
deraxenrovalo   0
3 hours ago
Let $(I)$ be an inscribed circle of $\triangle$$ABC$ and touching $BC$, $CA$, $AB$ at $D$, $E$, $F$ respectively. Let $EE'$ and $FF'$ be diameters of $(I)$. Let $X$ and $Y$ be the pole of $DE'$ and $DF'$ with respect to $(I)$, respectively. $BE$ cuts $(I)$ again at $K$. $CF$ cuts $(I)$ again at $L$. The tangent at $K$ of $(I)$ cuts $AX$ at $M$. The tangent at $L$ of $(I)$ cuts $AY$ at $N$. Let $U$ and $V$ be midpoint of $IM$ and $IN$, respectively.

Show that : $UV$, $E'F'$ and perpendicular bisector of $ID$ are concurrent.
0 replies
deraxenrovalo
3 hours ago
0 replies
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SL 2015 G1: Prove that IJ=AH
Problem_Penetrator   137
N 4 hours ago by heheman
Source: IMO 2015 Shortlist, G1
Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.
137 replies
Problem_Penetrator
Jul 7, 2016
heheman
4 hours ago
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IMO Shortlist 2011, G2
WakeUp   30
N 4 hours ago by ezpotd
Source: IMO Shortlist 2011, G2
Let $A_1A_2A_3A_4$ be a non-cyclic quadrilateral. Let $O_1$ and $r_1$ be the circumcentre and the circumradius of the triangle $A_2A_3A_4$. Define $O_2,O_3,O_4$ and $r_2,r_3,r_4$ in a similar way. Prove that
\[\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.\]

Proposed by Alexey Gladkich, Israel
30 replies
WakeUp
Jul 13, 2012
ezpotd
4 hours ago
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An fe based off of another trivial problem
benjaminchew13   1
N Apr 25, 2025 by Mathzeus1024
Source: Own
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all real numbers $x$ and $y$, $f(x+y-f(x))f(f(x+y)-y)=f(xy)$
1 reply
benjaminchew13
Apr 25, 2025
Mathzeus1024
Apr 25, 2025
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An fe based off of another trivial problem
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Reveal topic tags
function
algebra
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Source: Own
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benjaminchew13
11 posts
benjaminchew13
#1 h  Apr 25, 2025, 7:10 AM
Y by
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all real numbers $x$ and $y$, $f(x+y-f(x))f(f(x+y)-y)=f(xy)$
This post has been edited 1 time. Last edited by benjaminchew13, Apr 25, 2025, 7:10 AM
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Mathzeus1024
919 posts
Mathzeus1024
#2 h  Apr 25, 2025, 9:52 AM
Y by
It works for $\textcolor{red}{f(x)=0,1,x}$.
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