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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
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 29 minutes ago by sqing
Let $ a,b,c>0 . $ Prove that
$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq 4\left(\frac{a+b}{b+c}+ \frac{b+c}{a+b}\right)$$$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq \frac{32}{9}\left(\frac{a+b}{b+c}+ \frac{c+a}{a+b}\right)$$$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq \frac{8}{3}\left( \frac{a+b}{b+c}+ \frac{b+c}{c+a}+ \frac{c+a}{a+b}\right)$$$$ \left(1 +\frac{a^2}{b^2}\right)\left(1+\frac{b^2}{c^2}\right)\left(1+\frac{c^2}{a^2}\right )\geq \frac{8}{3}\left( \frac{a^2+bc}{b^2+ca}+\frac{b^2+ca}{c^2+ab}+\frac{c^2+ab}{a^2+bc}\right)$$
1 reply
+1 w
sqing
Today at 12:20 AM
sqing
29 minutes ago
J
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Inequalities
sqing   6
N an hour ago by sqing
Let $a,b,c> 0$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1.$ Prove that
$$ (1-abc) (1-a)(1-b)(1-c) \ge 208 $$$$ (1+abc) (1-a)(1-b)(1-c) \le -224 $$$$(1+a^2b^2c^2) (1-a)(1-b)(1-c) \le -5840 $$
6 replies
sqing
Jul 12, 2024
sqing
an hour ago
J
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A problem with a rectangle
Raul_S_Baz   11
N an hour ago by Raul_S_Baz
On the sides AB and AD of the rectangle ABCD, points M and N are taken such that MB = ND. Let P be the intersection of BN and CD, and Q be the intersection of DM and CB. How can we prove that PQ || MN?
IMAGE
11 replies
Raul_S_Baz
Apr 26, 2025
Raul_S_Baz
an hour ago
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(14n+25)/(2n+1) 'is a perfect square - Portugal OPM 2017 p1
parmenides51   3
N an hour ago by justaguy_69
Determine all integer values of n for which the number $\frac{14n+25}{2n+1}$ 'is a perfect square.
3 replies
parmenides51
May 15, 2024
justaguy_69
an hour ago
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APMO 2015 P1
aditya21   62
N 2 hours ago by Tonne
Source: APMO 2015
Let $ABC$ be a triangle, and let $D$ be a point on side $BC$. A line through $D$ intersects side $AB$ at $X$ and ray $AC$ at $Y$ . The circumcircle of triangle $BXD$ intersects the circumcircle $\omega$ of triangle $ABC$ again at point $Z$ distinct from point $B$. The lines $ZD$ and $ZY$ intersect $\omega$ again at $V$ and $W$ respectively.
Prove that $AB = V W$

Proposed by Warut Suksompong, Thailand
62 replies
aditya21
Mar 30, 2015
Tonne
2 hours ago
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Or statement function
ItzsleepyXD   2
N 2 hours ago by cursed_tangent1434
Source: Own , Mock Thailand Mathematic Olympiad P2
Find all $f: \mathbb{R} \to \mathbb{Z^+}$ such that $$f(x+f(y))=f(x)+f(y)+1\quad\text{ or }\quad f(x)+f(y)-1$$for all real number $x$ and $y$
2 replies
ItzsleepyXD
Yesterday at 9:07 AM
cursed_tangent1434
2 hours ago
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Trivial fun Equilateral
ItzsleepyXD   4
N 3 hours ago by cursed_tangent1434
Source: Own , Mock Thailand Mathematic Olympiad P1
Let $ABC$ be a scalene triangle with point $P$ and $Q$ on the plane such that $\triangle BPC , \triangle CQB$ is an equilateral . Let $AB$ intersect $CP$ and $CQ$ at $X$ and $Z$ respectively and $AC$ intersect $BP$ and $BQ$ at $Y$ and $W$ respectively .
Prove that $XY\parallel ZW$
4 replies
ItzsleepyXD
Yesterday at 9:05 AM
cursed_tangent1434
3 hours ago
J
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Geometry Proof
Jackson0423   2
N 3 hours ago by aidan0626
In triangle \( \triangle ABC \), point \( P \) on \( AB \) satisfies \( DB = BC \) and \( \angle DCA = 30^\circ \).
Let \( X \) be the point where the perpendicular from \( B \) to line \( DC \) meets the angle bisector of \( \angle BCA \).
Then, the relation \( AD \cdot DC = BD \cdot AX \) holds.

Prove that \( \triangle ABC \) is an isosceles triangle.
2 replies
Jackson0423
Yesterday at 4:17 PM
aidan0626
3 hours ago
J
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Do not try to case bash lol
ItzsleepyXD   2
N 3 hours ago by cursed_tangent1434
Source: Own , Mock Thailand Mathematic Olympiad P3
Let $n,d\geqslant 6$ be a positive integer such that $d\mid 6^{n!}+1$ .
Prove that $d>2n+6$ .
2 replies
ItzsleepyXD
Yesterday at 9:08 AM
cursed_tangent1434
3 hours ago
J
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N lines cutting each other in the plane
M.J.Espinas   4
N 3 hours ago by Lemmas
Source: Iranian Math Olympiad(Second Round 2016)
Let $l_1,l_2,l_3,...,L_n$ be lines in the plane such that no two of them are parallel and no three of them are concurrent. Let $A$ be the intersection point of lines $l_i,l_j$. We call $A$ an "Interior Point" if there are points $C,D$ on $l_i$ and $E,F$ on $l_j$ such that $A$ is between $C,D$ and $E,F$. Prove that there are at least $\frac{(n-2)(n-3)}{2}$ Interior points.($n>2$)
note: by point here we mean the points which are intersection point of two of $l_1,l_2,...,l_n$.
4 replies
M.J.Espinas
May 5, 2016
Lemmas
3 hours ago
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Arbitrary point on BC and its relation with orthocenter
falantrng   25
N 4 hours ago by EeEeRUT
Source: Balkan MO 2025 P2
In an acute-angled triangle \(ABC\), \(H\) be the orthocenter of it and \(D\) be any point on the side \(BC\). The points \(E, F\) are on the segments \(AB, AC\), respectively, such that the points \(A, B, D, F\) and \(A, C, D, E\) are cyclic. The segments \(BF\) and \(CE\) intersect at \(P.\) \(L\) is a point on \(HA\) such that \(LC\) is tangent to the circumcircle of triangle \(PBC\) at \(C.\) \(BH\) and \(CP\) intersect at \(X\). Prove that the points \(D, X, \) and \(L\) lie on the same line.

Proposed by Theoklitos Parayiou, Cyprus
25 replies
falantrng
Apr 27, 2025
EeEeRUT
4 hours ago
J
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Hard inequality
JK1603JK   2
N 4 hours ago by arqady
Source: unknown?
Let $a,b,c\in R: abc\neq 0$ and $a+b+c=0$ then prove $$|\frac{a-b}{c}|+|\frac{b-c}{a}|+|\frac{c-a}{b}|\ge 6$$
2 replies
JK1603JK
5 hours ago
arqady
4 hours ago
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BMO 2024 SL A5
MuradSafarli   2
N 5 hours ago by ja.


Let \(\mathbb{R}^+ = (0, \infty)\) be the set of positive real numbers.
Find all non-negative real numbers \(c \geq 0\) such that there exists a function \(f : \mathbb{R}^+ \to \mathbb{R}^+\) with the property:
\[ f(y^2f(x) + y + c) = xf(x+y^2) \]for all \(x, y \in \mathbb{R}^+\).

2 replies
MuradSafarli
Apr 27, 2025
ja.
5 hours ago
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Something nice
KhuongTrang   29
N 5 hours ago by arqady
Source: own
Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that

$$\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\sqrt{a+b+c+abc}.$$
29 replies
KhuongTrang
Nov 1, 2023
arqady
5 hours ago
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2 points M such MA+ BC=MB+AC=MC+AB (1994-95 Savin Competition 6-8 p19)
parmenides51   1
N Jun 4, 2021 by Mathmick51
Think of a triangle $ABC$ for which there are two points $M$ such that $MA + BC = MB + AC = MC + AB.$
1 reply
parmenides51
Jun 3, 2021
Mathmick51
Jun 4, 2021
J
2 points M such MA+ BC=MB+AC=MC+AB (1994-95 Savin Competition 6-8 p19)
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Reveal topic tags
geometry
equal segments
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parmenides51
30650 posts
parmenides51
#1 h  Jun 3, 2021, 5:32 PM • 1 Y
Y by Mango247
Think of a triangle $ABC$ for which there are two points $M$ such that $MA + BC = MB + AC = MC + AB.$
This post has been edited 1 time. Last edited by parmenides51, Jun 3, 2021, 5:57 PM
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Mathmick51
478 posts
Mathmick51
#2 h  Jun 4, 2021, 6:54 AM • 1 Y
Y by Mango247
Ok I am thinking about it.
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