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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 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
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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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P>2D
gwen01   3
N 5 minutes ago by AshAuktober
Source: Baltic Way 1992 #18
Show that in a non-obtuse triangle the perimeter of the triangle is always greater than two times the diameter of the circumcircle.
3 replies
gwen01
Feb 18, 2009
AshAuktober
5 minutes ago
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inequality with interesting conditions
Cobedangiu   2
N 21 minutes ago by musava_ribica
Let $x,y,z>0$:
2 replies
Cobedangiu
an hour ago
musava_ribica
21 minutes ago
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Symmedian
Pomegranat   0
42 minutes ago
Source: Russian geometry olympiad
In triangle $ABC$, point $M$ is the midpoint of $BC$, $P$ the point of intersection of the tangents at points $B$ and $C$ of the circumscribed circle of $ABC$, $N$ is the midpoint of the segment $MP$. The segment $AN$ meets the circumcircle $ABC$ at the point $Q$. Prove that $\angle PMQ = \angle MAQ$.
0 replies
1 viewing
Pomegranat
42 minutes ago
0 replies
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Hard geometry proof
radhoan_rikto-   1
N an hour ago by GreekIdiot
Source: BDMO 2025
Let ABC be an acute triangle and D the foot of the altitude from A onto BC. A semicircle with diameter BC intersects segments AB,AC and AD in the points F,E and X respectively.The circumcircles of the triangles DEX and DFX intersect BC in L and N respectively, other than D. Prove that BN=LC.
1 reply
2 viewing
radhoan_rikto-
Apr 25, 2025
GreekIdiot
an hour ago
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Inspired by JK1603JK
sqing   0
an hour ago
Source: Own
Let $ a,b,c $ be reals such that $ abc\neq 0$ and $ a+b+c=0. $ Prove that
$$\left|\frac{a-b}{c}\right|+k\left|\frac{b-c}{a} \right|+k^2\left|\frac{c-a}{b} \right|\ge 3(k+1)$$Where $ k>0.$
$$\left|\frac{a-b}{c}\right|+2\left|\frac{b-c}{a} \right|+4\left|\frac{c-a}{b} \right|\ge 9$$
0 replies
sqing
an hour ago
0 replies
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problem interesting
Cobedangiu   9
N an hour ago by Cobedangiu
Let $a=3k^2+3k+1 (a,k \in N)$
$i)$ Prove that: $a^2$ is the sum of $3$ square numbers
$ii)$ Let $b \vdots a$ and $b$ is the sum of $3$ square numbers. Prove that: $b^n$ is the sum of $3$ square numbers
9 replies
Cobedangiu
Yesterday at 5:06 AM
Cobedangiu
an hour ago
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4-var inequality
RainbowNeos   0
an hour ago
Given $a,b,c,d>0$, show that
\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]
0 replies
RainbowNeos
an hour ago
0 replies
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Find all integer pairs (m,n) such that 2^n! + 1 | 2^m! + 19
Goblik   0
an hour ago
Find all positive integer pairs $(m,n)$ such that $2^{n!} + 1 | 2^{m!} + 19$
0 replies
Goblik
an hour ago
0 replies
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Junior Balkan Mathematical Olympiad 2024- P3
Lukaluce   15
N an hour ago by MATHS_ENTUSIAST
Source: JBMO 2024
Find all triples of positive integers $(x, y, z)$ that satisfy the equation

$$2020^x + 2^y = 2024^z.$$
Proposed by Ognjen Tešić, Serbia
15 replies
Lukaluce
Jun 27, 2024
MATHS_ENTUSIAST
an hour ago
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AD is Euler line of triangle IKL
VicKmath7   16
N 2 hours ago by ErTeeEs06
Source: IGO 2021 Advanced P5
Given a triangle $ABC$ with incenter $I$. The incircle of triangle $ABC$ is tangent to $BC$ at $D$. Let $P$ and $Q$ be points on the side BC such that $\angle PAB = \angle BCA$ and $\angle QAC = \angle ABC$, respectively. Let $K$ and $L$ be the incenter of triangles $ABP$ and $ACQ$, respectively. Prove that $AD$ is the Euler line of triangle $IKL$.

Proposed by Le Viet An, Vietnam
16 replies
VicKmath7
Dec 30, 2021
ErTeeEs06
2 hours ago
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Twin Prime Diophantine
awesomeming327.   22
N 2 hours ago by MATHS_ENTUSIAST
Source: CMO 2025
Determine all positive integers $a$, $b$, $c$, $p$, where $p$ and $p+2$ are odd primes and
\[2^ap^b=(p+2)^c-1.\]
22 replies
awesomeming327.
Mar 7, 2025
MATHS_ENTUSIAST
2 hours ago
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Inequality with 3 variables and a special condition
Nuran2010   3
N 2 hours ago by sqing
Source: Azerbaijan Al-Khwarizmi IJMO TST 2024
For positive real numbers $a,b,c$ we have $3abc \geq ab+bc+ca$.
Prove that:

$\frac{1}{a^3+b^3+c}+\frac{1}{b^3+c^3+a}+\frac{1}{c^3+a^3+b} \leq \frac{3}{a+b+c}$.

Determine the equality case.
3 replies
Nuran2010
Tuesday at 5:06 PM
sqing
2 hours ago
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Inspired by JK1603JK and arqady
sqing   1
N 2 hours ago by sqing
Source: Own
Let $ a,b,c $ be reals such that $ abc\neq 0$ and $ a+b+c=0. $ Prove that
$$\left|\frac{a-2b}{c}\right|+\left|\frac{b-2c}{a} \right|+\left|\frac{c-2a}{b} \right|\ge \frac{1+3\sqrt{13+16\sqrt{2}}}{2}$$$$\left|\frac{a-3b}{c}\right|+\left|\frac{b-3c}{a}\right|+\left|\frac{c-3a}{b}\right|\ge 1+2\sqrt{13+16\sqrt{2}} $$
1 reply
sqing
2 hours ago
sqing
2 hours ago
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An easiest problem ever
Asilbek777   0
2 hours ago
Simplify
0 replies
Asilbek777
2 hours ago
0 replies
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<BDC when AC\perp BD, <BCA = 10^o,<BDA = 20^o, <BAC = 40^o
parmenides51   0
Apr 7, 2020
Source: 2011 Oral Moscow Geometry Olympiad grades 10-11 p5
In a convex quadrilateral $ABCD, AC\perp BD, \angle BCA = 10^o,\angle BDA = 20^o, \angle BAC = 40^o$. Find $\angle BDC$.
0 replies
parmenides51
Apr 7, 2020
0 replies
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<BDC when AC\perp BD, <BCA = 10^o,<BDA = 20^o, <BAC = 40^o
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Reveal topic tags
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Source: 2011 Oral Moscow Geometry Olympiad grades 10-11 p5
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parmenides51
30650 posts
parmenides51
#1 h  Apr 7, 2020, 11:15 PM • 2 Y
Y by Mango247, Mango247
In a convex quadrilateral $ABCD, AC\perp BD, \angle BCA = 10^o,\angle BDA = 20^o, \angle BAC = 40^o$. Find $\angle BDC$.
This post has been edited 1 time. Last edited by parmenides51, Apr 7, 2020, 11:16 PM
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