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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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Perpendicular Bisectors and Point Distributions
mojyla222   1
N 4 minutes ago by Quantum-Phantom
Source: IDMC 2025 P3
Given $n\geq 6$ points in the plane such that all pairwise distances between them are distinct, prove that there exist two points $A$,$B$ among them such that the perpendicular bisector of segment $AB$ has at least two points on each of its sides (i.e., each side contains at least one point other than $A$ and $B$).
1 reply
mojyla222
4 hours ago
Quantum-Phantom
4 minutes ago
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Inspired by Bet667
sqing   3
N 39 minutes ago by sqing
Source: Own
Let $x,y\ge 0$ such that $k(x+y)=1+xy. $ Prove that $$x+y+\frac{1}{x}+\frac{1}{y}\geq 4k $$Where $k\geq 1. $
3 replies
+1 w
sqing
Today at 2:34 AM
sqing
39 minutes ago
J
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Equal Distances in an Isosceles Setting
mojyla222   2
N 41 minutes ago by Mahdi_Mashayekhi
Source: IDMC 2025 P4
Let $ABC$ be an isosceles triangle with $AB=AC$. The circle $\omega_1$, passing through $B$ and $C$, intersects segment $AB$ at $K\neq B$. The circle $\omega_2$ is tangent to $BC$ at $B$ and passes through $K$. Let $M$ and $N$ be the midpoints of segments $AB$ and $AC$, respectively. The line $MN$ intersects $\omega_1$ and $\omega_2$ at points $P$ and $Q$, respectively, where $P$ and $Q$ are the intersections closer to $M$. Prove that $MP=MQ$.

Proposed by Hooman Fattahi
2 replies
mojyla222
4 hours ago
Mahdi_Mashayekhi
41 minutes ago
J
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Prove that the line $MN$ is tangent to the inscribed circle
janssv.200603   9
N 43 minutes ago by Captainscrubz
Source: Peru TST
Let $I$ be the incenter of the $ABC$ triangle. The circumference that passes through $I$ and has center
in $A$ intersects the circumscribed circumference of the $ABC$ triangle at points $M$ and
$N$. Prove that the line $MN$ is tangent to the inscribed circle of the $ABC$ triangle.
9 replies
janssv.200603
Feb 3, 2019
Captainscrubz
43 minutes ago
J
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nice fe with non-linear function being the answer
jjkim0336   3
N an hour ago by Lufin
Source: own
f:R+ -> R+

f(xf(y)+y) = y f(y^2 +x)
3 replies
jjkim0336
Apr 8, 2025
Lufin
an hour ago
J
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Equal distances between pairs of orthocenters in cyclic quad
Shu   2
N an hour ago by Nari_Tom
Source: XVII Tuymaada Mathematical Olympiad (2010), Senior Level
In a cyclic quadrilateral $ABCD$, the extensions of sides $AB$ and $CD$ meet at point $P$, and the extensions of sides $AD$ and $BC$ meet at point $Q$. Prove that the distance between the orthocenters of triangles $APD$ and $AQB$ is equal to the distance between the orthocenters of triangles $CQD$ and $BPC$.
2 replies
Shu
Jul 31, 2011
Nari_Tom
an hour ago
J
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a^2b(a-b) + b^2c(b-c) + c^2a(c-a) >= 0 for a triangle
ashwath.rabindranath   58
N an hour ago by Tony_stark0094
Source: IMO 1983, Day 2, Problem 6, proposed by Klamkin; E. Catalan, Educational Times N.S. 10, 57 (1906)
Let $ a$, $ b$ and $ c$ be the lengths of the sides of a triangle. Prove that
\[ a^{2}b(a - b) + b^{2}c(b - c) + c^{2}a(c - a)\ge 0. \]
Determine when equality occurs.
58 replies
ashwath.rabindranath
Oct 9, 2005
Tony_stark0094
an hour ago
J
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a_k+a_n/1+a_ka_n is constant when k+n is constant
gghx   2
N 2 hours ago by lightsynth123
Source: SMO senior 2024 Q5
Let $a_1,a_2,\dots$ be a sequence of positive numbers satisfying, for any positive integers $k,l,m,n$ such that $k+n=m+l$, $$\frac{a_k+a_n}{1+a_ka_n}=\frac{a_m+a_l}{1+a_ma_l}.$$Show that there exist positive numbers $b,c$ so that $b\le a_n\le c$ for any positive integer $n$.
2 replies
gghx
Aug 3, 2024
lightsynth123
2 hours ago
J
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Euler line of incircle touching points /Reposted/
Eagle116   5
N 2 hours ago by Tsikaloudakis
Let $ABC$ be a triangle with incentre $I$ and circumcentre $O$. Let $D,E,F$ be the touchpoints of the incircle with $BC$, $CA$, $AB$ respectively. Prove that $OI$ is the Euler line of $\vartriangle DEF$.
5 replies
Eagle116
Yesterday at 2:48 PM
Tsikaloudakis
2 hours ago
J
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Bunch of midpoints
Retemoeg   0
2 hours ago
Source: Own
Let $ABC$ be a scalene triangle with orthocenter $H$ and medial triangle $MNP$. Let $F$ be a point on $AC$ such that $\angle HMF = 90^{\circ}$. If $L$ is the midpoint of segment $BF$, show that $\triangle NLP$ is isoceles.
0 replies
Retemoeg
2 hours ago
0 replies
J
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Floor sequence
va2010   86
N 2 hours ago by math-olympiad-clown
Source: 2015 ISL N1
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \]contains at least one integer term.
86 replies
va2010
Jul 7, 2016
math-olympiad-clown
2 hours ago
J
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Inequality from my inequality training.
Orkhan-Ashraf_2002   3
N 2 hours ago by SunnyEvan
Let $a,b,c$ non-negative real numbers,but $ab+bc+ca\not=$0.Prove that
\[1\leq \frac{a+b}{a+4b+c}+\frac{b+c}{b+4c+a}+\frac{c+a}{c+4a+b}\leq \frac{4}{3}\]
3 replies
Orkhan-Ashraf_2002
Aug 21, 2016
SunnyEvan
2 hours ago
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SL 2015 G1: Prove that IJ=AH
Problem_Penetrator   135
N 3 hours ago by math-olympiad-clown
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 $IJ = AH$.
135 replies
Problem_Penetrator
Jul 7, 2016
math-olympiad-clown
3 hours ago
J
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Easy diophantine
gghx   3
N 3 hours ago by lightsynth123
Source: SMO senior 2024 Q2 / SMO junior 2024 Q5
Find all integer solutions of the equation $$y^2+2y=x^4+20x^3+104x^2+40x+2003.$$
Note: has appeared many times before, see here
3 replies
gghx
Aug 3, 2024
lightsynth123
3 hours ago
J
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J
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In triangle ABC prove that:
khanhsy   0
Mar 29, 2025
$$\dfrac{1}{cC+aB+bA}+\dfrac{1}{bB+aC+cA}+\dfrac{1}{aA+bC+cB}\ge \dfrac{1}{aA+bB+cC}+\dfrac{4}{a(B+C)+b(C+A)+c(A+B)}.$$
0 replies
khanhsy
Mar 29, 2025
0 replies
J
In triangle ABC prove that:
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khanhsy
129 posts
khanhsy
#1 h  Mar 29, 2025, 11:12 AM • 1 Y
Y by arqady
$$\dfrac{1}{cC+aB+bA}+\dfrac{1}{bB+aC+cA}+\dfrac{1}{aA+bC+cB}\ge \dfrac{1}{aA+bB+cC}+\dfrac{4}{a(B+C)+b(C+A)+c(A+B)}.$$
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