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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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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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Inequalities
sqing   0
10 minutes ago
Let $ a,b> 0 ,\frac{a}{2b+1}+\frac{b}{3}+\frac{1}{2a+1} \leq 1.$ Prove that
$$ a^2+b^2 -ab\leq 1$$$$ a^2+b^2 +ab \leq3$$
0 replies
1 viewing
sqing
10 minutes ago
0 replies
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Inequalities
sqing   19
N 31 minutes ago by sqing
Let $ a,b>0 $ . Prove that
$$ \frac{a}{a^2+a +2b+1}+ \frac{b}{b^2+2a +b+1} \leq \frac{2}{5} $$$$ \frac{a}{a^2+2a +b+1}+ \frac{b}{b^2+a +2b+1} \leq \frac{2}{5} $$
19 replies
1 viewing
sqing
May 13, 2025
sqing
31 minutes ago
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inequality
luckvoltia.112   9
N an hour ago by Shan3t
Given that \( a, b, c, d \) are nonzero real numbers, find the minimum value of the expression
\[ P = \left| \frac{b + c + d}{a} \right| + \left| \frac{c + d + a}{b} \right| + \left| \frac{d + a + b}{c} \right| + \left| \frac{a + b + c}{d} \right|. \]
9 replies
luckvoltia.112
3 hours ago
Shan3t
an hour ago
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Inequality for Sequences
steven_zhang123   0
2 hours ago
Given a positive number \( t \) and integers \( m, n \geq 2 \), prove that for any \( n \) positive numbers \( a_1, a_2, \ldots, a_n \) satisfying \( a_j - a_{j-1} \leq t \) for \( j = 1, 2, \ldots, n \) (with the convention \( a_0 = 0 \)), the following inequality holds:
\[ \sum_{j=1}^n a_j^{2m-1} \leq \frac{mt}{2} \left( \sum_{j=1}^n a_j^{m-1} \right)^2. \]
0 replies
steven_zhang123
2 hours ago
0 replies
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prove that
luckvoltia.112   0
3 hours ago
Let \( a, b, c \) be non-negative real numbers such that \( a + b + c > 0 \) and
\[ \frac{25a + 36b + 49c}{5a + 6b + 7c} + \frac{25b + 36c + 49a}{5b + 6c + 7a} + \frac{25c + 36a + 49b}{5c + 6a + 7b} = 18. \]Prove that exactly two of the numbers \( a, b, c \) are equal to 0.
0 replies
luckvoltia.112
3 hours ago
0 replies
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Circumcircle of ADM
v_Enhance   70
N 3 hours ago by Shan3t
Source: USA TSTST 2012, Problem 7
Triangle $ABC$ is inscribed in circle $\Omega$. The interior angle bisector of angle $A$ intersects side $BC$ and $\Omega$ at $D$ and $L$ (other than $A$), respectively. Let $M$ be the midpoint of side $BC$. The circumcircle of triangle $ADM$ intersects sides $AB$ and $AC$ again at $Q$ and $P$ (other than $A$), respectively. Let $N$ be the midpoint of segment $PQ$, and let $H$ be the foot of the perpendicular from $L$ to line $ND$. Prove that line $ML$ is tangent to the circumcircle of triangle $HMN$.
70 replies
v_Enhance
Jul 19, 2012
Shan3t
3 hours ago
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Prove this relation in triangle ABC
Entrepreneur   1
N 3 hours ago by MathIQ.
Source: Conjectured by me
In $\Delta ABC,$ prove that$$\color{blue}{2\angle A=3\angle B\implies c^3a^3(c+a)^2=b^2(c^2+a^2-b^2+2ca)(c^2+a^2-b^2)^2.}$$
1 reply
Entrepreneur
Aug 1, 2024
MathIQ.
3 hours ago
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45 degrees everywhere
Rijul saini   12
N 3 hours ago by guptaamitu1
Source: India IMOTC 2024 Day 2 Problem 2
Let $ABC$ be an acute angled triangle with $AC>AB$ and incircle $\omega$. Let $\omega$ touch the sides $BC, CA,$ and $AB$ at $D, E,$ and $F$ respectively. Let $X$ and $Y$ be points outside $\triangle ABC$ satisfying \[\angle BDX = \angle XEA = \angle YDC = \angle AFY = 45^{\circ}.\]Prove that the circumcircles of $\triangle AXY, \triangle AEF$ and $\triangle ABC$ meet at a point $Z\ne A$.

Proposed by Atul Shatavart Nadig and Shantanu Nene
12 replies
Rijul saini
May 31, 2024
guptaamitu1
3 hours ago
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equation 2025
mohamed-adam   1
N 3 hours ago by MathIQ.
Source: own
Find all positive integers $a,b$ such that $$a^8-2b^5=2025b$$
1 reply
mohamed-adam
Yesterday at 7:49 PM
MathIQ.
3 hours ago
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Graph Theory
ABCD1728   1
N 3 hours ago by ABCD1728
Can anyone provide the PDF version of "Graphs: an introduction" by Radu Bumbacea (XYZ press), thanks!
1 reply
ABCD1728
Yesterday at 5:10 AM
ABCD1728
3 hours ago
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Rectangles of grid cells
tapir1729   11
N 3 hours ago by Mathandski
Source: TSTST 2024, problem 9
Let $n \ge 2$ be a fixed integer. The cells of an $n \times n$ table are filled with the integers from $1$ to $n^2$ with each number appearing exactly once. Let $N$ be the number of unordered quadruples of cells on this board which form an axis-aligned rectangle, with the two smaller integers being on opposite vertices of this rectangle. Find the largest possible value of $N$.

Anonymous
11 replies
tapir1729
Jun 24, 2024
Mathandski
3 hours ago
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Interesting problem from a friend
v4913   11
N 4 hours ago by YaoAOPS
Source: I'm not sure...
Let the incircle $(I)$ of $\triangle{ABC}$ touch $BC$ at $D$, $ID \cap (I) = K$, let $\ell$ denote the line tangent to $(I)$ through $K$. Define $E, F \in \ell$ such that $\angle{EIF} = 90^{\circ}, EI, FI \cap (AEF) = E', F'$. Prove that the circumcenter $O$ of $\triangle{ABC}$ lies on $E'F'$.
11 replies
v4913
Nov 25, 2023
YaoAOPS
4 hours ago
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Curious inequality
produit   2
N 5 hours ago by RainbowNeos
Positive real numbers x_1, x_2, . . . x_n satisfy x_1 + x_2 + . . . + x_n = 1.
Prove that
1/(1 −√x_1)+1/(1 −√x_2)+ . . . +1/(1 −√x_n)⩾ n + 4.
2 replies
produit
May 10, 2025
RainbowNeos
5 hours ago
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Inspired by an old problem
RainbowNeos   0
5 hours ago
Given $n\geq 3$ and $a_i \geq 0, i=1,2,...,n$. Show that
\[\sum_{i=1}^n (a_i-a_{i+1})^3\leq \frac{1}{2\sqrt{3}} (\sum_{i=1}^n a_i )^3,\]where $a_{n+1}=a_1$.
0 replies
RainbowNeos
5 hours ago
0 replies
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J
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Three variables inequality
Headhunter   6
N Apr 30, 2025 by lbh_qys
$\forall a\in R$ ,$~\forall b\in R$ ,$~\forall c \in R$
Prove that at least one of $(a-b)^{2}$, $(b-c)^{2}$, $(c-a)^{2}$ is not greater than $\frac{a^{2}+b^{2}+c^{2}}{2}$.

I assume that all are greater than it, but can't go more.
6 replies
Headhunter
Apr 20, 2025
lbh_qys
Apr 30, 2025
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Three variables inequality
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Reveal topic tags
inequalities
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Headhunter
1963 posts
Headhunter
#1 h  Apr 20, 2025, 6:58 AM
Y by
$\forall a\in R$ ,$~\forall b\in R$ ,$~\forall c \in R$
Prove that at least one of $(a-b)^{2}$, $(b-c)^{2}$, $(c-a)^{2}$ is not greater than $\frac{a^{2}+b^{2}+c^{2}}{2}$.

I assume that all are greater than it, but can't go more.
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programjames1
3046 posts
programjames1
#2 h  Apr 20, 2025, 7:20 AM
Y by
Can you give the reference? I think this is from a USAJMO contest in the 2010s.

EDIT: I was thinking of USA(J)MO 2018 #1 (#2) which can be rearranged to a similar inequality.
This post has been edited 2 times. Last edited by programjames1, Apr 20, 2025, 7:23 AM
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Headhunter
1963 posts
Headhunter
#3 h  Apr 20, 2025, 9:01 AM
Y by
I guess that this problem is from chinese materials at 1990~2004. but I'm not sure. Thanks.
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lbh_qys
599 posts
lbh_qys
#4 h  Apr 21, 2025, 3:04 AM
Y by
Hint
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lbh_qys
599 posts
lbh_qys
#5 h  Apr 21, 2025, 3:18 AM • 2 Y
Y by programjames1, spy27
another solution
This post has been edited 1 time. Last edited by lbh_qys, Apr 21, 2025, 3:18 AM
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spy27
8 posts
spy27
#6 h  Apr 30, 2025, 5:38 AM
Y by
lbh_qys wrote:
another solution

Can you explain the \( a + b + c \neq 0 \) case in some detail?
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lbh_qys
599 posts
lbh_qys
#7 h  Apr 30, 2025, 6:08 AM
Y by
spy27 wrote:
lbh_qys wrote:
another solution

Can you explain the \( a + b + c \neq 0 \) case in some detail?

$f(x) = (a+x)^2 + (b+x)^2 + (c+x)^2 $ get minimum at $x=0$ iff $a+b+c=0$.
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