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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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Advanced topics in Inequalities
va2010   22
N a minute ago by Primeniyazidayi
So a while ago, I compiled some tricks on inequalities. You are welcome to post solutions below!
22 replies
va2010
Mar 7, 2015
Primeniyazidayi
a minute ago
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Funny easy transcendental geo
qwerty123456asdfgzxcvb   0
an hour ago
Let $\mathcal{S}$ be a logarithmic spiral centered at the origin (ie curve satisfying for any point $X$ on it, line $OX$ makes a fixed angle with the tangent to $\mathcal{S}$ at $X$). Let $\mathcal{H}$ be a rectangular hyperbola centered at the origin, scaled such that it is tangent to the logarithmic spiral at some point.

Prove that for a point $P$ on the spiral, the polar of $P$ wrt. $\mathcal{H}$ is tangent to the spiral.
0 replies
+2 w
qwerty123456asdfgzxcvb
an hour ago
0 replies
J
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Nice problem about a trapezoid
manlio   1
N an hour ago by kiyoras_2001
Have you a nice solution for this problem?
Thank you very much
1 reply
2 viewing
manlio
Apr 19, 2025
kiyoras_2001
an hour ago
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product of the first n terms
FireBreathers   5
N an hour ago by ihategeo_1969
Does there exist an infinite sequence of positive integers $a_i$ such that every positive integer appears exactly once and the product of the first $n$ terms is a perfect $n - th$ power ?
5 replies
FireBreathers
Dec 16, 2024
ihategeo_1969
an hour ago
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Concurrency with 10 lines
oVlad   1
N 3 hours ago by kokcio
Source: Romania EGMO TST 2017 Day 1 P1
Consider five points on a circle. For every three of them, we draw the perpendicular from the centroid of the triangle they determine to the line through the remaining two points. Prove that the ten lines thus formed are concurrent.
1 reply
oVlad
Today at 1:31 PM
kokcio
3 hours ago
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Concurrence, Isogonality
Wictro   40
N 4 hours ago by CatinoBarbaraCombinatoric
Source: BMO 2019, Problem 3
Let $ABC$ be an acute scalene triangle. Let $X$ and $Y$ be two distinct interior points of the segment $BC$ such that $\angle{CAX} = \angle{YAB}$. Suppose that:
$1)$ $K$ and $S$ are the feet of the perpendiculars from from $B$ to the lines $AX$ and $AY$ respectively.
$2)$ $T$ and $L$ are the feet of the perpendiculars from $C$ to the lines $AX$ and $AY$ respectively.
Prove that $KL$ and $ST$ intersect on the line $BC$.
40 replies
Wictro
May 2, 2019
CatinoBarbaraCombinatoric
4 hours ago
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Prove excircle is tangent to circumcircle
sarjinius   7
N 4 hours ago by markam
Source: Philippine Mathematical Olympiad 2025 P4
Let $ABC$ be a triangle with incenter $I$, and let $D$ be a point on side $BC$. Points $X$ and $Y$ are chosen on lines $BI$ and $CI$ respectively such that $DXIY$ is a parallelogram. Points $E$ and $F$ are chosen on side $BC$ such that $AX$ and $AY$ are the angle bisectors of angles $\angle BAE$ and $\angle CAF$ respectively. Let $\omega$ be the circle tangent to segment $EF$, the extension of $AE$ past $E$, and the extension of $AF$ past $F$. Prove that $\omega$ is tangent to the circumcircle of triangle $ABC$.
7 replies
sarjinius
Mar 9, 2025
markam
4 hours ago
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Easy geo
oVlad   3
N 5 hours ago by Primeniyazidayi
Source: Romania EGMO TST 2019 Day 1 P1
A line through the vertex $A{}$ of the triangle $ABC{}$ which doesn't coincide with $AB{}$ or $AC{}$ intersectes the altitudes from $B{}$ and $C{}$ at $D{}$ and $E{}$ respectively. Let $F{}$ be the reflection of $D{}$ in $AB{}$ and $G{}$ be the reflection of $E{}$ in $AC{}.$ Prove that the circles $ABF{}$ and $ACG{}$ are tangent.
3 replies
oVlad
Today at 1:45 PM
Primeniyazidayi
5 hours ago
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abc(a+b+c)=3, show that prod(a+b)>=8 [Indian RMO 2012(b) Q4]
Potla   28
N 5 hours ago by mihaig
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.
28 replies
Potla
Dec 2, 2012
mihaig
5 hours ago
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In triangle \( PQR \), points \( A, B, C, D, E, F \) are constructed as follows:
Jackson0423   0
5 hours ago
In triangle \( PQR \), points \( A, B, C, D, E, F \) are constructed as follows: Points \( A \) and \( B \) lie on the extension of side \( QR \) such that \( AP = BP = QR \). Points \( C \) and \( D \) lie on the extension of side \( PQ \) such that \( CR = DR = PQ \). Points \( E \) and \( F \) lie on the extension of side \( RP \) such that \( EQ = FQ = RP \).

The points are placed in a clockwise order around triangle \( PQR \).

Prove that:
\[ \angle ACE + \angle FBD + \angle EAC = 180^\circ. \]
0 replies
Jackson0423
5 hours ago
0 replies
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Similar triangles formed by angular condition
Mahdi_Mashayekhi   5
N Today at 2:02 PM by sami1618
Source: Iran 2025 second round P3
Point $P$ lies inside of scalene triangle $ABC$ with incenter $I$ such that $:$
$$ 2\angle ABP = \angle BCA , 2\angle ACP = \angle CBA $$Lines $PB$ and $PC$ intersect line $AI$ respectively at $B'$ and $C'$. Line through $B'$ parallel to $AB$ intersects $BI$ at $X$ and line through $C'$ parallel to $AC$ intersects $CI$ at $Y$. Prove that triangles $PXY$ and $ABC$ are similar.
5 replies
Mahdi_Mashayekhi
Apr 19, 2025
sami1618
Today at 2:02 PM
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Easy Geometry
TheOverlord   33
N Today at 1:37 PM by math.mh
Source: Iran TST 2015, exam 1, day 1 problem 2
$I_b$ is the $B$-excenter of the triangle $ABC$ and $\omega$ is the circumcircle of this triangle. $M$ is the middle of arc $BC$ of $\omega$ which doesn't contain $A$. $MI_b$ meets $\omega$ at $T\not =M$. Prove that
$$ TB\cdot TC=TI_b^2.$$
33 replies
TheOverlord
May 10, 2015
math.mh
Today at 1:37 PM
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Existence of a circle tangent to four lines
egxa   3
N Today at 1:36 PM by mathuz
Source: All Russian 2025 10.2
Inside triangle \(ABC\), point \(P\) is marked. Point \(Q\) is on segment \(AB\), and point \(R\) is on segment \(AC\) such that the circumcircles of triangles \(BPQ\) and \(CPR\) are tangent to line \(AP\). Lines are drawn through points \(B\) and \(C\) passing through the center of the circumcircle of triangle \(BPC\), and through points \(Q\) and \(R\) passing through the center of the circumcircle of triangle \(PQR\). Prove that there exists a circle tangent to all four drawn lines.
3 replies
egxa
Apr 18, 2025
mathuz
Today at 1:36 PM
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Two very hard parallel
jayme   0
Today at 12:46 PM
Source: own inspired by EGMO
Dear Mathlinkers,

1. ABC a triangle
2. D, E two point on the segment BC so that BD = DE= EC
3. M, N the midpoint of ED, AE
4. H the orthocenter of the acutangle triangle ADE
5. 1, 2 the circumcircle of the triangle DHM, EHN
6. P, Q the second point of intersection of 1 and BM, 2 and CN
7. U, V the second points of intersection of 2 and MN, PQ.

Prove : UV is parallel to PM.

Sincerely
Jean-Louis
0 replies
jayme
Today at 12:46 PM
0 replies
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J
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Regular tetrahedron can be an obtuse-angled triangle
orl   0
Sep 18, 2008
Source: IMO Shortlist 1989, Problem 21, ILL 67
Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than $ 120^{\circ}.$
0 replies
orl
Sep 18, 2008
0 replies
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Regular tetrahedron can be an obtuse-angled triangle
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Reveal topic tags
geometry
3D geometry
tetrahedron
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geometric inequality
IMO Shortlist
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Source: IMO Shortlist 1989, Problem 21, ILL 67
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orl
3647 posts
orl
#1 h  Sep 18, 2008, 3:42 PM • 2 Y
Y by Adventure10, Mango247
Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than $ 120^{\circ}.$
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