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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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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
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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Regarding Maaths olympiad prepration
omega2007   3
N 2 minutes ago by MathleteMystic
<Hey Everyone'>
I'm 10 grader student and Im starting prepration for maths olympiad..>>> From scratch (not 2+2=4 )

Do you haves compiled resources of Handouts,
PDF,
Links,
List of books topic wise
which are shared on AOPS (and from your perspective) for maths olympiad and any useful thing, which will help me in boosting Maths olympiad prepration.
3 replies
omega2007
Yesterday at 3:13 PM
MathleteMystic
2 minutes ago
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square root problem that involves geometry
kjhgyuio   2
N 32 minutes ago by ND_
If x is a nonnegative real number , find the minimum value of √x^2+4 + √x^2 -24x +153

2 replies
kjhgyuio
2 hours ago
ND_
32 minutes ago
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inquequality
ngocthi0101   9
N 2 hours ago by sqing
let $a,b,c > 0$ prove that
$\frac{a}{b} + \sqrt {\frac{b}{c}} + \sqrt[3]{{\frac{c}{a}}} > \frac{5}{2}$
9 replies
ngocthi0101
Sep 26, 2014
sqing
2 hours ago
J
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Assisted perpendicular chasing
sarjinius   5
N 2 hours ago by hukilau17
Source: Philippine Mathematical Olympiad 2025 P7
In acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$, let $D$ be an arbitrary point on the circumcircle of triangle $ABC$ such that $D$ does not lie on line $OB$ and that line $OD$ is not parallel to line $BC$. Let $E$ be the point on the circumcircle of triangle $ABC$ such that $DE$ is perpendicular to $BC$, and let $F$ be the point on line $AC$ such that $FA = FE$. Let $P$ and $R$ be the points on the circumcircle of triangle $ABC$ such that $PE$ is a diameter, and $BH$ and $DR$ are parallel. Let $M$ be the midpoint of $DH$.
(a) Show that $AP$ and $BR$ are perpendicular.
(b) Show that $FM$ and $BM$ are perpendicular.
5 replies
sarjinius
Mar 9, 2025
hukilau17
2 hours ago
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Tangent.
steven_zhang123   2
N 3 hours ago by AshAuktober
Source: China TST 2001 Quiz 6 P1
In \( \triangle ABC \) with \( AB > BC \), a tangent to the circumcircle of \( \triangle ABC \) at point \( B \) intersects the extension of \( AC \) at point \( D \). \( E \) is the midpoint of \( BD \), and \( AE \) intersects the circumcircle of \( \triangle ABC \) at \( F \). Prove that \( \angle CBF = \angle BDF \).
2 replies
steven_zhang123
Mar 23, 2025
AshAuktober
3 hours ago
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IMO ShortList 1998, algebra problem 1
orl   37
N 3 hours ago by Marcus_Zhang
Source: IMO ShortList 1998, algebra problem 1
Let $a_{1},a_{2},\ldots ,a_{n}$ be positive real numbers such that $a_{1}+a_{2}+\cdots +a_{n}<1$. Prove that

\[ \frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_{1})(1 - a_{2}) \cdots (1 - a_{n})} \leq \frac{1}{ n^{n+1}}. \]
37 replies
orl
Oct 22, 2004
Marcus_Zhang
3 hours ago
J
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Integer Coefficient Polynomial with order
MNJ2357   9
N 3 hours ago by v_Enhance
Source: 2019 Korea Winter Program Practice Test 1 Problem 3
Find all polynomials $P(x)$ with integer coefficients such that for all positive number $n$ and prime $p$ satisfying $p\nmid nP(n)$, we have $ord_p(n)\ge ord_p(P(n))$.
9 replies
MNJ2357
Jan 12, 2019
v_Enhance
3 hours ago
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Inspired by bamboozled
sqing   0
3 hours ago
Source: Own
Let $ a,b,c $ be reals such that $(a^2+1)(b^2+1)(c^2+1) = 27. $Prove that $$1-3\sqrt 3\leq ab + bc + ca\leq 6$$
0 replies
sqing
3 hours ago
0 replies
J
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Range of ab + bc + ca
bamboozled   1
N 3 hours ago by sqing
Let $(a^2+1)(b^2+1)(c^2+1) = 9$, where $a, b, c \in R$, then the number of integers in the range of $ab + bc + ca$ is __
1 reply
bamboozled
4 hours ago
sqing
3 hours ago
J
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Functional Equation
AnhQuang_67   4
N 3 hours ago by AnhQuang_67
Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $$2\cdot f\Big(\dfrac{-xy}{2}+f(x+y)\Big)=xf(y)+yf(x), \forall x, y \in \mathbb{R} $$
4 replies
AnhQuang_67
Yesterday at 4:50 PM
AnhQuang_67
3 hours ago
J
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Inradius and ex-radii
bamboozled   0
3 hours ago
Let $ABC$ be a triangle and $r, r_1, r_2, r_3$ denote its inradius and ex-radii opposite to the vertices $A, B, C$ respectively. If $a> r_1, b > r_2$ and $c > r_3$, then which of the following is/are true?
(A) $\angle{B}$ is obtuse
(B) $\angle{A}$ is acute
(C) $3r > s$, where $s$ is semi perimeter
(D) $3r < s$, where $s$ is semi perimeter
0 replies
bamboozled
3 hours ago
0 replies
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Inspired by giangtruong13
sqing   1
N 4 hours ago by sqing
Source: Own
Let $ a,b\in[\frac{1}{2},1] $. Prove that$$ 64\leq (a+b^2+\frac{4}{a^2}+\frac{2}{b})(b+a^2+\frac{4}{b^2}+\frac{2}{a})\leq\frac{6889}{16} $$Let $ a,b\in[\frac{1}{2},2] $. Prove that$$ 8(3+2\sqrt 2)\leq (a+b^2+\frac{4}{a^2}+\frac{2}{b})(b+a^2+\frac{4}{b^2}+\frac{2}{a})\leq\frac{6889}{16} $$
1 reply
sqing
4 hours ago
sqing
4 hours ago
J
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Conditional maximum
giangtruong13   2
N 4 hours ago by sqing
Source: Specialized Math
Let $a,b$ satisfy that: $1 \leq a \leq2$ and $1 \leq b \leq 2$. Find the maximum: $$A=(a+b^2+\frac{4}{a^2}+\frac{2}{b})(b+a^2+\frac{4}{b^2}+\frac{2}{a})$$
2 replies
giangtruong13
Mar 22, 2025
sqing
4 hours ago
J
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Inspired by JK1603JK
sqing   16
N 4 hours ago by sqing
Source: Own
Let $ a,b,c\geq 0 $ and $ab+bc+ca=1.$ Prove that$$\frac{abc-2}{abc-1}\ge \frac{4(a^2b+b^2c+c^2a)}{a^3b+b^3c+c^3a+1} $$
16 replies
sqing
Yesterday at 3:31 AM
sqing
4 hours ago
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J
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a bulgarian beauty, regular hexagon inscribed in a polygon
pohoatza   4
N Feb 7, 2020 by Why-
Source: bmo 1991
A regular hexagon of area $H$ is inscribed in a convex polygon of area $P$. Show that $P \leq \frac{3}{2}H$. When does the equality occur?
4 replies
pohoatza
Apr 23, 2007
Why-
Feb 7, 2020
J
a bulgarian beauty, regular hexagon inscribed in a polygon
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Reveal topic tags
geometry
geometry proposed
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Source: bmo 1991
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pohoatza
1145 posts
pohoatza
#1 h  Apr 23, 2007, 8:31 PM • 2 Y
Y by Adventure10, Mango247
A regular hexagon of area $H$ is inscribed in a convex polygon of area $P$. Show that $P \leq \frac{3}{2}H$. When does the equality occur?
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andreica
18 posts
andreica
#2 h  Apr 28, 2009, 2:02 PM • 2 Y
Y by Adventure10, Mango247
in the figure below $ S[A_1B_2B_1] \leq S[B_2'QB_2] , B_2B_1'B_1 and B_2B_3B_2'$ are equilateral triangles same with $ B_iB_{i+1}O$ with area $ \frac{H}{6}\Rightarrow$
$ S[B_1A_1B_2] + S[B_2A_2B_3] \leq \frac{H}{6}$
$ S[B_3A_3B_4] + S[B_4B_4B_5] \leq \frac{H}{6}$
$ S[B_5A_5B_6] + S[B_6A_6B_1] \leq \frac{H}{6}$
summing this three and H we get $ P \leq \frac{3}{2}H$
equality holds when $ A_1$ coresponds with the center of $ B_1B_2B_2'$ same for the other $ A_i$
Attachments:
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andreica
18 posts
andreica
#3 h  May 4, 2009, 6:06 AM • 2 Y
Y by Adventure10, Mango247
I'm wrong about the equality case and i do not know obtained
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silvermoon
1 post
silvermoon
#4 h  Apr 26, 2010, 3:36 AM • 2 Y
Y by Adventure10, Mango247
can anyone post more precise solution,please
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Why-
27 posts
Why-
#5 h  Feb 7, 2020, 12:10 PM • 2 Y
Y by Adventure10, Mango247
equality holds only when $p$ is an equilateral triangle
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