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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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A point on BC
jayme   5
N 9 minutes ago by Boulets
Source: Own ?
Dear Mathlinkers,

1. ABC a triangle
2. 0 the circumcircle
3. D the pole of BC wrt 0
4. B', C' the symmetrics of B, C wrt AC, AB
5. 1b, 1c the circumcircles of the triangles BB'D, CC'D
6. T the second point of intersection of the tangent to 1c at D with 1b.

Prove : B, C and T are collinear.

Sincerely
Jean-Louis
5 replies
jayme
5 hours ago
Boulets
9 minutes ago
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power of a point
BekzodMarupov   2
N 22 minutes ago by BekzodMarupov
Source: lemmas in olympiad geometry
Epsilon 1.3. Let ABC be a triangle and let D, E, F be the feet of the altitudes, with D on BC, E on CA, and F on AB. Let the parallel through D to EF meet AB at X and AC at Y. Let T be the intersection of EF with BC and let M be the midpoint of side BC. Prove that the points T, M, X, Y are concyclic.
2 replies
BekzodMarupov
Yesterday at 5:41 AM
BekzodMarupov
22 minutes ago
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Insspired by Shandong 2025
sqing   3
N 24 minutes ago by JARP091
Source: Own
Let $ a,b,c>0,abc>1$. Prove that$$ \frac {abc(a+b+c+ab+bc+ca+3)}{ abc-1}\geq \frac {81}{4}$$$$ \frac {abc(a+b+c+ab+bc+ca+abc+2)}{ abc-1}\geq 12+8\sqrt{2}$$
3 replies
sqing
2 hours ago
JARP091
24 minutes ago
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Find the minimum
sqing   4
N an hour ago by JARP091
Source: China Shandong High School Mathematics Competition 2025 Q4
Let $ a,b,c>0,abc>1$. Find the minimum value of $ \frac {abc(a+b+c+8)}{abc-1}. $
4 replies
sqing
2 hours ago
JARP091
an hour ago
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sequence positive
malinger   38
N 4 hours ago by ezpotd
Source: ISL 2006, A2, VAIMO 2007, P4, Poland 2007
The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$.

Proposed by Mariusz Skalba, Poland
38 replies
malinger
Apr 22, 2007
ezpotd
4 hours ago
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A sharp one with 3 var
mihaig   4
N Today at 5:21 AM by arqady
Source: Own
Let $a,b,c\geq0$ satisfying
$$\left(a+b+c-2\right)^2+8\leq3\left(ab+bc+ca\right).$$Prove
$$ab+bc+ca+abc\geq4.$$
4 replies
mihaig
May 13, 2025
arqady
Today at 5:21 AM
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Inequality on APMO P5
Jalil_Huseynov   41
N Today at 4:16 AM by Mathandski
Source: APMO 2022 P5
Let $a,b,c,d$ be real numbers such that $a^2+b^2+c^2+d^2=1$. Determine the minimum value of $(a-b)(b-c)(c-d)(d-a)$ and determine all values of $(a,b,c,d)$ such that the minimum value is achived.
41 replies
Jalil_Huseynov
May 17, 2022
Mathandski
Today at 4:16 AM
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Again
heartwork   11
N Today at 12:18 AM by Mathandski
Source: Vietnam MO 2002, Problem 5
Determine for which $ n$ positive integer the equation: $ a + b + c + d = n \sqrt {abcd}$ has positive integer solutions.
11 replies
heartwork
Dec 16, 2004
Mathandski
Today at 12:18 AM
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Sequence inequality
hxtung   20
N Yesterday at 11:44 PM by awesomeming327.
Source: IMO ShortList 2003, algebra problem 6
Let $n$ be a positive integer and let $(x_1,\ldots,x_n)$, $(y_1,\ldots,y_n)$ be two sequences of positive real numbers. Suppose $(z_2,\ldots,z_{2n})$ is a sequence of positive real numbers such that $z_{i+j}^2 \geq x_iy_j$ for all $1\le i,j \leq n$.

Let $M=\max\{z_2,\ldots,z_{2n}\}$. Prove that \[ \left( \frac{M+z_2+\dots+z_{2n}}{2n} \right)^2 \ge \left( \frac{x_1+\dots+x_n}{n} \right) \left( \frac{y_1+\dots+y_n}{n} \right). \]

comment

Proposed by Reid Barton, USA
20 replies
hxtung
Jun 9, 2004
awesomeming327.
Yesterday at 11:44 PM
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A little problem
TNKT   3
N Yesterday at 4:42 PM by Pengu14
Source: Tran Ngoc Khuong Trang
Problem. Let a,b,c be three positive real numbers with a+b+c=3. Prove that \color{blue}{\frac{1}{4a^{2}+9}+\frac{1}{4b^{2}+9}+\frac{1}{4c^{2}+9}\le \frac{3}{abc+12}.}
When does equality hold?
P/s: Could someone please convert it to latex help me? Thank you!
See also MSE: https://math.stackexchange.com/questions/5065499/prove-that-frac14a29-frac14b29-frac14c29-le-frac3
3 replies
TNKT
Thursday at 1:17 PM
Pengu14
Yesterday at 4:42 PM
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inequalities
Ducksohappi   1
N Yesterday at 3:22 PM by Nguyenhuyen_AG
let a,b,c be non-negative numbers such that ab+bc+ca>0. Prove:
$ \sum_{cyc} \frac{b+c}{2a^2+bc}\ge \frac{6}{a+b+c}$
P/s: I have analysed:$ S_a=\frac{b^2+c^2+3bc-ab-ac}{(2b^2+ac)(2c^2+2ab)}$, similarly to $S_b, S_c$, by SOS
1 reply
Ducksohappi
Yesterday at 12:57 PM
Nguyenhuyen_AG
Yesterday at 3:22 PM
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Interesting inequalities
sqing   3
N Yesterday at 12:55 PM by sqing
Source: Own
Let $a,b,c \geq 0 $ and $ab+bc+ca- abc =3.$ Show that
$$a+k(b+c)\geq 2\sqrt{3 k}$$Where $ k\geq 1. $
Let $a,b,c \geq 0 $ and $2(ab+bc+ca)- abc =31.$ Show that
$$a+k(b+c)\geq \sqrt{62k}$$Where $ k\geq 1. $
3 replies
sqing
Yesterday at 4:34 AM
sqing
Yesterday at 12:55 PM
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INMO 2007 Problem 6
Sathej   32
N Yesterday at 11:50 AM by math_genie
Source: Inequality
If $ x$, $ y$, $ z$ are positive real numbers, prove that
\[ \left(x + y + z\right)^2 \left(yz + zx + xy\right)^2 \leq 3\left(y^2 + yz + z^2\right)\left(z^2 + zx + x^2\right)\left(x^2 + xy + y^2\right) .\]
32 replies
Sathej
Feb 4, 2007
math_genie
Yesterday at 11:50 AM
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inequality
mathematical-forest   2
N Yesterday at 10:14 AM by mathematical-forest
For positive real intengers $x_{1} ,x_{2} ,\cdots,x_{n} $, such that $\prod_{i=1}^{n} x_{i} =1$
proof:
$$\sum_{i=1}^{n} \frac{1}{1+\sum _{j\ne i}x_{j} } \le 1$$
2 replies
mathematical-forest
Thursday at 12:40 PM
mathematical-forest
Yesterday at 10:14 AM
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Find the smallest of sum of elements
hlminh   0
Apr 24, 2025
Let $S=\{1,2,...,2014\}$ and $X=\{a_1,a_2,...,a_{30}\}$ is a subset of $S$ such that if $a,b\in X,a+b\leq 2014$ then $a+b\in X.$ Find the smallest of $\dfrac{a_1+a_2+\cdots+a_{30}}{30}.$
0 replies
hlminh
Apr 24, 2025
0 replies
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Find the smallest of sum of elements
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hlminh
#1 h  Apr 24, 2025, 3:37 AM
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Let $S=\{1,2,...,2014\}$ and $X=\{a_1,a_2,...,a_{30}\}$ is a subset of $S$ such that if $a,b\in X,a+b\leq 2014$ then $a+b\in X.$ Find the smallest of $\dfrac{a_1+a_2+\cdots+a_{30}}{30}.$
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