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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
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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 nt
HopefullyMcNats2025   0
17 minutes ago
How many positive integers from 1-100000 fit the conditions, they are not divisble by 3, 5, or 8 and that they are palindromes,
0 replies
HopefullyMcNats2025
17 minutes ago
0 replies
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Two Orthocenters and an Invariant Point
Mathdreams   2
N 39 minutes ago by hukilau17
Source: 2025 Nepal Mock TST Day 1 Problem 3
Let $\triangle{ABC}$ be a triangle, and let $P$ be an arbitrary point on line $AO$, where $O$ is the circumcenter of $\triangle{ABC}$. Define $H_1$ and $H_2$ as the orthocenters of triangles $\triangle{APB}$ and $\triangle{APC}$. Prove that $H_1H_2$ passes through a fixed point which is independent of the choice of $P$.

(Kritesh Dhakal, Nepal)
2 replies
Mathdreams
Today at 1:30 PM
hukilau17
39 minutes ago
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Intermediate Counting
RenheMiResembleRice   4
N an hour ago by Apple_maths60
A coin is flipped, a 6-sided die numbered 1 through 6 is rolled, and a 10-sided die numbered 0
through 9 is rolled. What is the probability that the coin comes up heads and the sum of the
numbers that show on the dice is 8?
4 replies
RenheMiResembleRice
Today at 7:46 AM
Apple_maths60
an hour ago
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Cute inequality in equilateral triangle
Miquel-point   0
an hour ago
Source: Romanian IMO TST 1981, Day 3 P5
Let $ABC$ be an equilateral triangle, $M$ be a point inside it, and $A',B',C'$ be the intersections of $AM,\; BM,\; CM$ with the sides of $ABC$. If $A'',\; B'',\; C''$ are the midpoints of $BC$, $CA$, $AB$, show that there is a triangle with sides $A'A''$, $B'B''$ and $C'C''$.

Laurențiu Panaitopol
0 replies
Miquel-point
an hour ago
0 replies
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perpendicularity involving ex and incenter
Erken   19
N an hour ago by Primeniyazidayi
Source: Kazakhstan NO 2008 problem 2
Suppose that $ B_1$ is the midpoint of the arc $ AC$, containing $ B$, in the circumcircle of $ \triangle ABC$, and let $ I_b$ be the $ B$-excircle's center. Assume that the external angle bisector of $ \angle ABC$ intersects $ AC$ at $ B_2$. Prove that $ B_2I$ is perpendicular to $ B_1I_B$, where $ I$ is the incenter of $ \triangle ABC$.
19 replies
Erken
Dec 24, 2008
Primeniyazidayi
an hour ago
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Locus of sphere cutting three spheres along great circles
Miquel-point   0
an hour ago
Source: Romanian IMO TST 1981, Day 2 P3
Consider three fixed spheres $S_1, S_2, S_3$ with pairwise disjoint interiors. Determine the locus of the centre of the sphere intersecting each $S_i$ along a great circle of $S_i$.

Stere Ianuș
0 replies
Miquel-point
an hour ago
0 replies
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Locus problem with circles in space
Miquel-point   0
2 hours ago
Source: RNMO 1979 10.4
Consider two circles $\mathcal C_1$ and $\mathcal C_2$ lying in parallel planes. Describe the locus of the midpoint of $M_1M_2$ when $M_i$ varies along $\mathcal C_i$ for $i=1,2$.

Ioan Tomescu
0 replies
Miquel-point
2 hours ago
0 replies
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Geometry
youochange   3
N 2 hours ago by Double07
m:}
Let $\triangle ABC$ be a triangle inscribed in a circle, where the tangents to the circle at points $B$ and $C$ intersect at the point $P$. Let $M$ be a point on the arc $AC$ (not containing $B$) such that $M \neq A$ and $M \neq C$. Let the lines $BC$ and $AM$ intersect at point $K$. Let $P'$ be the reflection of $P$ with respect to the line $AM$. The lines $AP'$ and $PM$ intersect at point $Q$, and $PM$ intersects the circumcircle of $\triangle ABC$ again at point $N$.

Prove that the point $Q$ lies on the circumcircle of $\triangle ANK$.
3 replies
youochange
Today at 11:27 AM
Double07
2 hours ago
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Sequence of projections is convergent
Filipjack   0
3 hours ago
Source: Romanian National Olympiad 1997 - Grade 10 - Problem 3
A point $A_0$ and two lines $d_1$ and $d_2$ are given in the space. For each nonnegative integer $n$ we denote by $B_n$ the projection of $A_n$ on $d_2,$ and by $A_{n+1}$ the projection of $B_n$ on $d_1.$ Prove that there exist two segments $[A'A''] \subset d_1$ and $[B'B''] \subset d_2$ of length $0.001$ and a nonnegative integer $N$ such that $A_n \in [A'A'']$ and $B_n \in [B'B'']$ for any $n \ge N.$
0 replies
Filipjack
3 hours ago
0 replies
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Right-angled triangle if circumcentre is on circle
liberator   76
N 4 hours ago by numbertheory97
Source: IMO 2013 Problem 3
Let the excircle of triangle $ABC$ opposite the vertex $A$ be tangent to the side $BC$ at the point $A_1$. Define the points $B_1$ on $CA$ and $C_1$ on $AB$ analogously, using the excircles opposite $B$ and $C$, respectively. Suppose that the circumcentre of triangle $A_1B_1C_1$ lies on the circumcircle of triangle $ABC$. Prove that triangle $ABC$ is right-angled.

Proposed by Alexander A. Polyansky, Russia
76 replies
liberator
Jan 4, 2016
numbertheory97
4 hours ago
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APMO 2016: Great triangle
shinichiman   26
N 4 hours ago by ray66
Source: APMO 2016, problem 1
We say that a triangle $ABC$ is great if the following holds: for any point $D$ on the side $BC$, if $P$ and $Q$ are the feet of the perpendiculars from $D$ to the lines $AB$ and $AC$, respectively, then the reflection of $D$ in the line $PQ$ lies on the circumcircle of the triangle $ABC$. Prove that triangle $ABC$ is great if and only if $\angle A = 90^{\circ}$ and $AB = AC$.

Senior Problems Committee of the Australian Mathematical Olympiad Committee
26 replies
shinichiman
May 16, 2016
ray66
4 hours ago
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IMO ShortList 2001, geometry problem 2
orl   48
N 5 hours ago by legogubbe
Source: IMO ShortList 2001, geometry problem 2
Consider an acute-angled triangle $ABC$. Let $P$ be the foot of the altitude of triangle $ABC$ issuing from the vertex $A$, and let $O$ be the circumcenter of triangle $ABC$. Assume that $\angle C \geq \angle B+30^{\circ}$. Prove that $\angle A+\angle COP < 90^{\circ}$.
48 replies
orl
Sep 30, 2004
legogubbe
5 hours ago
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Excalibur Identity
jjsunpu   12
N 5 hours ago by SomeonecoolLovesMaths
proof is below
12 replies
1 viewing
jjsunpu
Apr 3, 2025
SomeonecoolLovesMaths
5 hours ago
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Inequalities
nhathhuyyp5c   3
N 5 hours ago by mathprodigy2011
Prove that for all positive real numbers \( a, b, c \), the following inequality holds:

\[ \sqrt{a + b} + \sqrt{b + c} + \sqrt{c + a} \geq \frac{4(ab + bc + ca)}{\sqrt{(a + b)(b + c)(c + a)}} \]
3 replies
nhathhuyyp5c
Today at 4:45 AM
mathprodigy2011
5 hours ago
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k help me solve this problem
aothatday   1
N Mar 31, 2025 by jasperE3
find f: R+ —> R+
f(2f(x)+f(y)+xy)=xy+2x+y with x,y>0
1 reply
aothatday
Mar 31, 2025
jasperE3
Mar 31, 2025
J
help me solve this problem
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Reveal topic tags
High School
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aothatday
2 posts
aothatday
#1 h  Mar 31, 2025, 2:57 PM
Y by
find f: R+ —> R+
f(2f(x)+f(y)+xy)=xy+2x+y with x,y>0
Z Y
The post below has been deleted. Click to close.
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jasperE3
11168 posts
jasperE3
#2 h  Mar 31, 2025, 3:08 PM • 1 Y
Y by aothatday
This problem gets reposted 2 or 3 times in the last month by new accounts, weird
https://artofproblemsolving.com/community/c4h3536833p34392946
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