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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 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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Right tetrahedron of fixed volume and min perimeter
Miquel-point   0
a few seconds ago
Source: Romanian IMO TST 1981, Day 4 P3
Determine the lengths of the edges of a right tetrahedron of volume $a^3$ so that the sum of its edges' lengths is minumum.

0 replies
Miquel-point
a few seconds ago
0 replies
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Every subset of size k has sum at most N/2
orl   49
N a minute ago by Marcus_Zhang
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 $\tfrac{N}{2}.$
49 replies
orl
Apr 20, 2006
Marcus_Zhang
a minute ago
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V \le RS/2 in tetrahderon with equil base
Miquel-point   0
3 minutes ago
Source: Romanian IMO TST 1981, Day 4 P2
Consider a tetrahedron $OABC$ with $ABC$ equilateral. Let $S$ be the area of the triangle of sides $OA$, $OB$ and $OC$. Show that $V\leqslant \dfrac12 RS$ where $R$ is the circumradius and $V$ is the volume of the tetrahedron.

Stere Ianuș
0 replies
Miquel-point
3 minutes ago
0 replies
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Beautiful problem
luutrongphuc   21
N 4 minutes ago by hukilau17
Let triangle $ABC$ be circumscribed about circle $(I)$, and let $H$ be the orthocenter of $\triangle ABC$. The circle $(I)$ touches line $BC$ at $D$. The tangent to the circle $(BHC)$ at $H$ meets $BC$ at $S$. Let $J$ be the midpoint of $HI$, and let the line $DJ$ meet $(I)$ again at $X$. The tangent to $(I)$ parallel to $BC$ meets the line $AX$ at $T$. Prove that $ST$ is tangent to $(I)$.
21 replies
luutrongphuc
Apr 4, 2025
hukilau17
4 minutes ago
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Cute inequality in equilateral triangle
Miquel-point   0
an hour ago
Source: Romanian IMO TST 1981, Day 3 P5
Let $ABC$ be an equilateral triangle, $M$ be a point inside it, and $A',B',C'$ be the intersections of $AM,\; BM,\; CM$ with the sides of $ABC$. If $A'',\; B'',\; C''$ are the midpoints of $BC$, $CA$, $AB$, show that there is a triangle with sides $A'A''$, $B'B''$ and $C'C''$.

Laurențiu Panaitopol
0 replies
Miquel-point
an hour ago
0 replies
J
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Conditional upper bound for binomial coeff
Miquel-point   0
an hour ago
Source: Romanian IMO TST 1981, Day 3 P3
Let $n>r\geqslant 3$ be two integers and $d$ be a positive integer such that $nd\geqslant \dbinom{n+r}{r+1}$. Show that \[(n-t)(d-t)>\dbinom{n-t+r}{r+1},\]for $t=1,2,\ldots,n-1$

Vasile Brânzănescu
0 replies
Miquel-point
an hour ago
0 replies
J
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Simple cube root inequality [Taiwan 2014 Quizzes]
v_Enhance   43
N 3 hours ago by arqady
Prove that for positive reals $a$, $b$, $c$ we have \[ 3(a+b+c) \ge 8\sqrt[3]{abc} + \sqrt[3]{\frac{a^3+b^3+c^3}{3}}. \]
43 replies
v_Enhance
Jul 18, 2014
arqady
3 hours ago
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hard problem
Cobedangiu   16
N 3 hours ago by InftyByond
problem
16 replies
Cobedangiu
Mar 27, 2025
InftyByond
3 hours ago
J
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A very nice inequality
KhuongTrang   4
N 3 hours ago by arqady
Source: own
Problem. Let $a,b,c\in \mathbb{R}:\ a+b+c=3.$ Prove that $$\color{black}{\sqrt{5a^{2}-ab+5b^{2}}+\sqrt{5b^{2}-bc+5c^{2}}+\sqrt{5c^{2}-ca+5a^{2}}\le 2(a^2+b^2+c^2)+ab+bc+ca.}$$When does equality hold?
4 replies
KhuongTrang
Today at 1:35 PM
arqady
3 hours ago
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Conditional maximum
giangtruong13   3
N 4 hours ago by imnotgoodatmathsorry
Source: Specialized Math
Let $a,b$ satisfy that: $1 \leq a \leq2$ and $1 \leq b \leq 2$. Find the maximum: $$A=(a+b^2+\frac{4}{a^2}+\frac{2}{b})(b+a^2+\frac{4}{b^2}+\frac{2}{a})$$
3 replies
giangtruong13
Mar 22, 2025
imnotgoodatmathsorry
4 hours ago
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3 var inquality
sqing   0
Today at 1:11 PM
Source: Own
Let $ a,b,c>0 $ and $ \dfrac{a}{bc}+\dfrac{2b}{ca}+\dfrac{5c}{ab}\leq 12.$ Prove that$$ a^2+b^2+c^2\geq 1$$
0 replies
sqing
Today at 1:11 PM
0 replies
J
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inequality
pennypc123456789   6
N Today at 1:03 PM by sqing
Let \( x, y \) be positive real numbers satisfying \( x + y = 2 \). Prove that

\[ 3(x^{\frac{2}{3}} + y^{\frac{2}{3}}) \geq 4 + 2x^{\frac{1}{3}}y^{\frac{1}{3}}. \]
6 replies
pennypc123456789
Mar 24, 2025
sqing
Today at 1:03 PM
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2018 PAMO Shortlist: Inequality with condition $a^3 + b^3 + c^3 = 5abc$
DylanN   24
N Today at 1:01 PM by sqing
Source: 2018 Pan-African Shortlist - A6
Let $a, b, c$ be positive real numbers such that $a^3 + b^3 + c^3 = 5abc$.

Show that
\[ \left( \frac{a + b}{c} \right) \left( \frac{b + c}{a} \right) \left( \frac{c + a}{b} \right) \geq 9. \]
24 replies
DylanN
May 6, 2019
sqing
Today at 1:01 PM
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inequality ( 4 var
SunnyEvan   5
N Today at 12:53 PM by sqing
Let $ a,b,c,d \in R $ , such that $ a+b+c+d=4 . $ Prove that :
$$ a^4+b^4+c^4+d^4+3 \geq \frac{7}{4}(a^3+b^3+c^3+d^3) $$$$ a^4+b^4+c^4+d^4+ \frac{252}{25} \geq \frac{88}{25}(a^3+b^3+c^3+d^3) $$equality cases : ?
5 replies
SunnyEvan
Apr 4, 2025
sqing
Today at 12:53 PM
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Burak0609
Burak0609   0
Apr 3, 2025
So $b=x+y+z$ $x^3+a=-3y-3z$ $P(t)=t^3-3t+a-3b\implies x+y+z=0$ from vieta $x^3+a=-3y-3z,y^3+a=-3x-3z\implies x^2+xy+y^2=3\implies y^2+xy+(x^2-3)=0\implies \Delta\ge 0 \implies 12\ge 3x^2 \implies x\in [-2,2]$. Notice in similar fashion we get $y\in [-2,2] , z\in [-2,2]$, also its easy to observe that none of $\{x,y,z\} \in \{-2,2,0\}$. So we have$\boxed{a \in (-2,2)-\{0\}}$
0 replies
Burak0609
Apr 3, 2025
0 replies
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Burak0609
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Burak0609
6 posts
Burak0609
#1 h  Apr 3, 2025, 3:18 PM
Y by
So $b=x+y+z$ $x^3+a=-3y-3z$ $P(t)=t^3-3t+a-3b\implies x+y+z=0$ from vieta $x^3+a=-3y-3z,y^3+a=-3x-3z\implies x^2+xy+y^2=3\implies y^2+xy+(x^2-3)=0\implies \Delta\ge 0 \implies 12\ge 3x^2 \implies x\in [-2,2]$. Notice in similar fashion we get $y\in [-2,2] , z\in [-2,2]$, also its easy to observe that none of $\{x,y,z\} \in \{-2,2,0\}$. So we have$\boxed{a \in (-2,2)-\{0\}}$
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