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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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Inequality em981
oldbeginner   21
N 26 minutes ago by sqing
Source: Own
Let $a, b, c>0, a+b+c=3$. Prove that
\[\sqrt{a+\frac{9}{b+2c}}+\sqrt{b+\frac{9}{c+2a}}+\sqrt{c+\frac{9}{a+2b}}+\frac{2(ab+bc+ca)}{9}\ge\frac{20}{3}\]
21 replies
+1 w
oldbeginner
Sep 22, 2016
sqing
26 minutes ago
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Find the minimum
sqing   29
N 28 minutes ago by sqing
Source: Zhangyanzong
Let $a,b$ be positive real numbers such that $a^2b^2+\frac{4a}{a+b}=4.$ Find the minimum value of $a+2b.$
29 replies
1 viewing
sqing
Sep 4, 2018
sqing
28 minutes ago
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Interesting inequality
sqing   3
N 31 minutes ago by sqing
Source: Own
Let $ (a+b)^2+(a-b)^2=1. $ Prove that
$$0\geq (a+b-1)(a-b+1)\geq -\frac{3}{2}-\sqrt 2$$$$ -\frac{9}{2}+2\sqrt 2\geq (a+b-2)(a-b+2)\geq -\frac{9}{2}-2\sqrt 2$$
3 replies
sqing
an hour ago
sqing
31 minutes ago
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Simple triangle geometry [a fixed point]
darij grinberg   50
N 32 minutes ago by ezpotd
Source: German TST 2004, IMO ShortList 2003, geometry problem 2
Three distinct points $A$, $B$, and $C$ are fixed on a line in this order. Let $\Gamma$ be a circle passing through $A$ and $C$ whose center does not lie on the line $AC$. Denote by $P$ the intersection of the tangents to $\Gamma$ at $A$ and $C$. Suppose $\Gamma$ meets the segment $PB$ at $Q$. Prove that the intersection of the bisector of $\angle AQC$ and the line $AC$ does not depend on the choice of $\Gamma$.
50 replies
darij grinberg
May 18, 2004
ezpotd
32 minutes ago
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Inspired by RMO 2006
sqing   6
N 32 minutes ago by sqing
Source: Own
Let $ a,b >0 . $ Prove that
$$ \frac {a^{2}+1}{b+k}+\frac { b^{2}+1}{ka+1}+\frac {2}{a+kb} \geq \frac {6}{k+1} $$Where $k\geq 0.03 $
$$ \frac {a^{2}+1}{b+1}+\frac { b^{2}+1}{a+1}+\frac {2}{a+b} \geq 3 $$
6 replies
sqing
Saturday at 3:24 PM
sqing
32 minutes ago
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IMO 2009, Problem 2
orl   143
N 37 minutes ago by ezpotd
Source: IMO 2009, Problem 2
Let $ ABC$ be a triangle with circumcentre $ O$. The points $ P$ and $ Q$ are interior points of the sides $ CA$ and $ AB$ respectively. Let $ K,L$ and $ M$ be the midpoints of the segments $ BP,CQ$ and $ PQ$. respectively, and let $ \Gamma$ be the circle passing through $ K,L$ and $ M$. Suppose that the line $ PQ$ is tangent to the circle $ \Gamma$. Prove that $ OP = OQ.$

Proposed by Sergei Berlov, Russia
143 replies
orl
Jul 15, 2009
ezpotd
37 minutes ago
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Number Theory problem
StoicFTW   0
an hour ago
Determine the number of solutions $(i, j, k)$ which satisfy the simultaneous equations
$$ i j=6 i+6 j+6 k $$and
$$ k^2=i^2+j^2 $$where $i, j$ and $k$ are integers such that $k^2>i^2$ and $k^2>j^2$.
0 replies
StoicFTW
an hour ago
0 replies
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Inequalities from SXTX
sqing   24
N an hour ago by sqing
T702. Let $ a,b,c>0 $ and $ a+2b+3c=\sqrt{13}. $ Prove that $$ \sqrt{a^2+1} +2\sqrt{b^2+1} +3\sqrt{c^2+1} \geq 7$$S
T703. Let $ a,b $ be real numbers such that $ a+b\neq 0. $. Find the minimum of $ a^2+b^2+(\frac{1-ab}{a+b} )^2.$
T704. Let $ a,b,c>0 $ and $ a+b+c=3. $ Prove that $$ \frac{a^2+7}{(c+a)(a+b)} + \frac{b^2+7}{(a+b)(b+c)} +\frac{c^2+7}{(b+c)(c+a)} \geq 6$$S
24 replies
sqing
Feb 18, 2025
sqing
an hour ago
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A sharp one with 3 var (2)
mihaig   0
an hour ago
Source: Own
Let $a,b,c\geq0$ satisfying
$$\left(a+b+c-2\right)^2+8\leq3\left(ab+bc+ca\right).$$Prove
$$a+b+c+\sqrt{abc}\geq4.$$
0 replies
mihaig
an hour ago
0 replies
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f(1)f(2)...f(n) has at most n prime factors
MarkBcc168   39
N an hour ago by cursed_tangent1434
Source: 2020 Cyberspace Mathematical Competition P2
Let $f(x) = 3x^2 + 1$. Prove that for any given positive integer $n$, the product
$$f(1)\cdot f(2)\cdot\dots\cdot f(n)$$has at most $n$ distinct prime divisors.

Proposed by Géza Kós
39 replies
MarkBcc168
Jul 15, 2020
cursed_tangent1434
an hour ago
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a tst 2013 test
Math2030   9
N 2 hours ago by Shan3t
Given the sequence $(a_n): a_1=1, a_2=11$ and $a_{n+2}=a_{n+1}+5a_{n}, n \geq 1$
. Prove that $a_n $not is a perfect square for all $n > 3$.
9 replies
Math2030
Saturday at 5:26 AM
Shan3t
2 hours ago
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Challenge Problem in Similar Triangles
yes45   1
N 2 hours ago by LilKirb
\(\triangle ABC\) has a right angle at \(B\). \(AB = 9\) and \(AC = 15\). Point \(D\) lies on \(AB\) such that \(AD = 2\) and point \(E\) lies on \(AC\) such that \(AE\) = \(\frac{12}{5}\). Find the perimeter of \(BCED\).
Answer Confirmation
Solution
1 reply
yes45
Yesterday at 5:26 PM
LilKirb
2 hours ago
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Inspired by 2025 Beijing
sqing   11
N 2 hours ago by sqing
Source: Own
Let $ a,b,c,d >0 $ and $ (a^2+b^2+c^2)(b^2+c^2+d^2)=36. $ Prove that
$$ab^2c^2d \leq 8$$$$a^2bcd^2 \leq 16$$$$ ab^3c^3d \leq \frac{2187}{128}$$$$ a^3bcd^3 \leq \frac{2187}{32}$$
11 replies
sqing
Saturday at 4:56 PM
sqing
2 hours ago
J
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A functional equation
super1978   1
N 2 hours ago by CheerfulZebra68
Source: Somewhere
Find all functions $f: \mathbb R \to \mathbb R$ such that:$$ f(f(x)(y+f(y)))=xf(y)+f(xy) $$for all $x,y \in \mathbb R$
1 reply
super1978
3 hours ago
CheerfulZebra68
2 hours ago
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that statement is true
pennypc123456789   3
N Apr 5, 2025 by sqing
we have $a^3+b^3 = 2$ and $3(a^4+b^4)+2a^4b^4 \le 8 $ , then we can deduce $a^2+b^2$ \le 2 $ ?
3 replies
pennypc123456789
Mar 23, 2025
sqing
Apr 5, 2025
J
that statement is true
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Reveal topic tags
inqualities
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pennypc123456789
38 posts
pennypc123456789
#1 h  Mar 23, 2025, 9:27 AM
Y by
we have $a^3+b^3 = 2$ and $3(a^4+b^4)+2a^4b^4 \le 8 $ , then we can deduce $a^2+b^2$ \le 2 $ ?
This post has been edited 1 time. Last edited by pennypc123456789, Mar 23, 2025, 10:10 AM
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no_room_for_error
337 posts
no_room_for_error
#2 h  Mar 23, 2025, 11:15 AM • 2 Y
Y by ehuseyinyigit, pennypc123456789
pennypc123456789 wrote:
we have $a^3+b^3 = 2$ and $3(a^4+b^4)+2a^4b^4 \le 8 $ , then we can deduce $a^2+b^2 \le 2 $ ?

Just $3(a^4+b^4)+2a^4b^4\leq 8$ is enough. It follows from Cauchy-Schwarz

$$25 \geq (2a^4+3)(2b^4+3)\geq (2a^2+2b^2+1)^2$$
Z K Y
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pennypc123456789
38 posts
pennypc123456789
#3 h  Mar 23, 2025, 2:44 PM
Y by
wow thanks
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sqing
42414 posts
sqing
#4 h  Apr 5, 2025, 12:32 PM
Y by
Let $ a,b \geq 0 $ and $ a^3 +b^3=2. $ Prove that
$$3(a^4+b^4)+2a^4b^4\leq 8$$
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