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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   1
N an hour ago by sqing
Let $ a,b> 0 $ and $2a+2b+ab=5. $ Prove that
$$ \frac{a^4}{b^4}+\frac{1}{a^4}+42ab-a^4\geq 43$$$$ \frac{a^5}{b^5}+\frac{1}{a^5}+64ab-a^5\geq 65$$$$ \frac{a^6}{b^6}+\frac{1}{a^6}+90ab-a^6\geq 91$$$$ \frac{a^7}{b^7}+\frac{1}{a^7}+121ab-a^7\geq 122$$
1 reply
sqing
Yesterday at 2:31 PM
sqing
an hour ago
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Every subset of size k has sum at most N/2
orl   50
N 2 hours ago by de-Kirschbaum
Source: USAMO 2006, Problem 2, proposed by Dick Gibbs
For a given positive integer $k$ find, in terms of $k$, the minimum value of $N$ for which there is a set of $2k + 1$ distinct positive integers that has sum greater than $N$ but every subset of size $k$ has sum at most $\tfrac{N}{2}.$
50 replies
orl
Apr 20, 2006
de-Kirschbaum
2 hours ago
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Inspired by a9opsow_
sqing   2
N 2 hours ago by sqing
Source: Own
Let $ a,b > 0 .$ Prove that
$$ \frac{(ka^2 - kab-b)^2 + (kb^2 - kab-a)^2 + (ab-ka-kb )^2}{ (ka+b)^2 + (kb+a)^2+(a - b)^2 }\geq \frac {1}{(k+1)^2}$$Where $ k\geq 0.37088 .$
$$\frac{(a^2 - ab-b)^2 + (b^2 - ab-a)^2 + ( ab-a-b)^2}{a^2 +b^2+(a - b)^2 } \geq 1$$$$ \frac{(2a^2 - 2ab-b)^2 + (2b^2 - 2ab-a)^2 + (ab-2a-2b )^2}{ (2a+b)^2 + (2b+a)^2+(a - b)^2 }\geq \frac 19$$
2 replies
sqing
3 hours ago
sqing
2 hours ago
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Cute NT Problem
M11100111001Y1R   5
N 3 hours ago by compoly2010
Source: Iran TST 2025 Test 4 Problem 1
A number \( n \) is called lucky if it has at least two distinct prime divisors and can be written in the form:
\[ n = p_1^{\alpha_1} + \cdots + p_k^{\alpha_k} \]where \( p_1, \dots, p_k \) are distinct prime numbers that divide \( n \). (Note: it is possible that \( n \) has other prime divisors not among \( p_1, \dots, p_k \).) Prove that for every prime number \( p \), there exists a lucky number \( n \) such that \( p \mid n \).
5 replies
M11100111001Y1R
Tuesday at 7:20 AM
compoly2010
3 hours ago
J
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Can anyone help me with this inequality problem
a9opsow_   3
N 3 hours ago by sqing
Let a, b, c be nonnegative real numbers, not all equal. Prove that

\frac{(a - bc)^2 + (b - ca)^2 + (c - ab)^2}{(a - b)^2 + (b - c)^2 + (c - a)^2} \geq \frac{1}{2}.
3 replies
a9opsow_
Yesterday at 11:50 AM
sqing
3 hours ago
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3 var inequality
SunnyEvan   13
N 3 hours ago by Nguyenhuyen_AG
Let $ a,b,c \in R $ ,such that $ a^2+b^2+c^2=4(ab+bc+ca)$Prove that :$$ \frac{7-2\sqrt{14}}{48} \leq \frac{a^3b+b^3c+c^3a}{(a^2+b^2+c^2)^2} \leq \frac{7+2\sqrt{14}}{48} $$
13 replies
SunnyEvan
May 17, 2025
Nguyenhuyen_AG
3 hours ago
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trigonometric inequality
MATH1945   13
N 4 hours ago by sqing
Source: ?
In triangle $ABC$, prove that $$sin^2(A)+sin^2(B)+sin^2(C) \leq \frac{9}{4}$$
13 replies
MATH1945
May 26, 2016
sqing
4 hours ago
J
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Inequalities
sqing   21
N 4 hours ago by sqing
Let $ a,b,c\geq 0 ,a+b+c\leq 3. $ Prove that
$$a^2+b^2+c^2+ab +2ca+2bc + abc \leq \frac{251}{27}$$$$ a^2+b^2+c^2+ab+2ca+2bc + \frac{2}{5}abc \leq \frac{4861}{540}$$$$ a^2+b^2+c^2+ab+2ca+2bc + \frac{7}{20}abc \leq \frac{2381411}{26460}$$
21 replies
sqing
May 21, 2025
sqing
4 hours ago
J
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Iran TST Starter
M11100111001Y1R   2
N 4 hours ago by sami1618
Source: Iran TST 2025 Test 1 Problem 1
Let \( a_n \) be a sequence of positive real numbers such that for every \( n > 2025 \), we have:
\[ a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i} \]Prove that there exists a natural number \( M \) such that for all \( n > M \), the following holds:
\[ a_n < \frac{1}{1404} \]
2 replies
M11100111001Y1R
Tuesday at 7:36 AM
sami1618
4 hours ago
J
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Polar Coordinates
pingpongmerrily   4
N 4 hours ago by K124659
Convert the equation $r=\tan(\theta)$ into rectangular form.
4 replies
pingpongmerrily
4 hours ago
K124659
4 hours ago
J
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Twin Prime Diophantine
awesomeming327.   23
N 5 hours ago by HDavisWashu
Source: CMO 2025
Determine all positive integers $a$, $b$, $c$, $p$, where $p$ and $p+2$ are odd primes and
\[2^ap^b=(p+2)^c-1.\]
23 replies
awesomeming327.
Mar 7, 2025
HDavisWashu
5 hours ago
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Troublesome median in a difficult inequality
JG666   2
N 5 hours ago by navid
Source: 2022 Spring NSMO Day 2 Problem 3
Determine the minimum value of $\lambda\in\mathbb{R}$, such that for any positive integer $n$ and non-negative reals $x_1, x_2, \cdots, x_n$, the following inequality always holds:
$$\sum_{i=1}^n(m_i-a_i)^2\leqslant \lambda\cdot\sum_{i=1}^nx_i^2,$$Here $m_i$ and $a_i$ denote the median and arithmetic mean of $x_1, x_2, \cdots, x_i$, respectively.

Duanyang ZHANG, High School Affiliated to Renmin University of China
2 replies
1 viewing
JG666
May 22, 2022
navid
5 hours ago
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Concurrent lines, angle bisectors
legogubbe   0
5 hours ago
Source: ???
Hi AoPS!

Let $ABC$ be an isosceles triangle with $AB=AC$, and $M$ an arbitrary point on side $BC$. The internal angle bisector of $\angle MAB$ meets the circumcircle of $\triangle ABC$ again at $P \neq A$, and the internal angle bisector of $\angle CAM$ meets it again at $Q \neq A$. Show that lines $AM$, $BQ$ and $CP$ are concurrent.
0 replies
legogubbe
5 hours ago
0 replies
J
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Fractional Inequality
sqing   33
N 5 hours ago by Learning11
Source: Chinese Girls Mathematical Olympiad 2012, Problem 1
Let $ a_1, a_2,\ldots, a_n$ be non-negative real numbers. Prove that
$\frac{1}{1+ a_1}+\frac{ a_1}{(1+ a_1)(1+ a_2)}+\frac{ a_1 a_2}{(1+ a_1)(1+ a_2)(1+ a_3)}+$ $\cdots+\frac{ a_1 a_2\cdots a_{n-1}}{(1+ a_1)(1+ a_2)\cdots (1+ a_n)} \le 1.$
33 replies
sqing
Aug 10, 2012
Learning11
5 hours ago
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a point P such that AP +BP+CP is minimised
Iris Aliaj   3
N Jul 22, 2004 by beta
Triangle ABC has a right angle at A. Among all the points P on the perimeter of the triangle, find the position of P such that
AP + BP + CP
is minimised
3 replies
Iris Aliaj
Jul 22, 2004
beta
Jul 22, 2004
J
a point P such that AP +BP+CP is minimised
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Reveal topic tags
geometry
perimeter
quadratics
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Iris Aliaj
165 posts
Iris Aliaj
#1 h  Jul 22, 2004, 12:15 PM • 2 Y
Y by Adventure10, Mango247
Triangle ABC has a right angle at A. Among all the points P on the perimeter of the triangle, find the position of P such that
AP + BP + CP
is minimised
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beta
3001 posts
beta
#2 h  Jul 22, 2004, 8:02 PM • 2 Y
Y by Adventure10, Mango247
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This post has been edited 2 times. Last edited by beta, Jul 28, 2004, 8:26 PM
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Pork_Chop8
179 posts
Pork_Chop8
#3 h  Jul 22, 2004, 9:08 PM • 2 Y
Y by Adventure10, Mango247
Edit:
The minimum of a quadratic $f(x) = ax^{2}+bx+c$ with $a>0$, occurs when $x=\displaystyle-\frac{b}{2a}$, which is not, of course, the same as when $x=\displaystyle-\frac{b}{a}$. So, in conclusion, thank you beta for the mistake check. Our methods were exactly the same.
This post has been edited 1 time. Last edited by Pork_Chop8, Jul 23, 2004, 3:06 AM
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beta
#4 h  Jul 22, 2004, 10:21 PM • 2 Y
Y by Adventure10, Mango247
Pork_Chop8 wrote:
$\displaystyle \frac{\sqrt{(c^{2}+1)k^{2}-2kc+c^{2}}}{c}$ which is minimized when $\displaystyle k=-\frac{-2c}{c^{2}+1}$ and plugging that in for k, we get $\frac{\sqrt{c^{2}}}{c}=1$. So the minimum would be $BC + 1$ or $BC + AB$.
I am pretty sure if you minimize $\displaystyle \frac{\sqrt{(c^{2}+1)k^{2}-2kc+c^{2}}}{c}$ right, it is $\frac{c}{\sqrt{c^2+1}}$, not 1, because if AP is minimized, then AP is perpendicular to BC, which according to your definition, is $\frac{c}{\sqrt{c^2+1}}$, there's no way AB is perpendicular to BC..
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