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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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Show that f is bijective on T
Tsukuyomi   34
N 2 minutes ago by lelouchvigeo
Source: IMO Shortlist 2017 A3
Let $S$ be a finite set, and let $\mathcal{A}$ be the set of all functions from $S$ to $S$. Let $f$ be an element of $\mathcal{A}$, and let $T=f(S)$ be the image of $S$ under $f$. Suppose that $f\circ g\circ f\ne g\circ f\circ g$ for every $g$ in $\mathcal{A}$ with $g\ne f$. Show that $f(T)=T$.
34 replies
Tsukuyomi
Jul 10, 2018
lelouchvigeo
2 minutes ago
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Isogonal lines at the intersection of two circles
anantmudgal09   12
N 6 minutes ago by Tony_stark0094
Source: RMO Delhi 2016, P3
Two circles $C_1$ and $C_2$ intersect each other at points $A$ and $B$. Their external common tangent (closer to $B$) touches $C_1$ at $P$ and $C_2$ at $Q$. Let $C$ be the reflection of $B$ in line $PQ$. Prove that $\angle CAP=\angle BAQ$.
12 replies
+1 w
anantmudgal09
Oct 11, 2016
Tony_stark0094
6 minutes ago
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Diophantine equation !
ComplexPhi   10
N 8 minutes ago by Namisgood
Determine all triples $(m , n , p)$ satisfying :
\[n^{2p}=m^2+n^2+p+1\]
where $m$ and $n$ are integers and $p$ is a prime number.
10 replies
ComplexPhi
Feb 4, 2015
Namisgood
8 minutes ago
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Factorial Equation
Alidq   0
11 minutes ago
Solve in $\mathbb{N}$ $$\frac{x!}{(x-y)!} = 10x+2y-29$$
0 replies
Alidq
11 minutes ago
0 replies
J
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Triangular function and quadrilateral
Kunihiko_Chikaya   1
N 16 minutes ago by Mathzeus1024
Source: National Defence Medical college Entrance exam November 2005
Given a convex quadrilateral such that $AB=AD=\sqrt{5},\ CB=CD=2, \tan \angle{DAB=-2,\ \angle{BCD}=2\alpha\ (0<2\alpha <\pi).}$ Find all the integers $n$ satisfying $\cos n\alpha=\cos \alpha.$
1 reply
Kunihiko_Chikaya
Jan 17, 2006
Mathzeus1024
16 minutes ago
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IMO Shortlist 2011, G5
WakeUp   71
N 23 minutes ago by InterLoop
Source: IMO Shortlist 2011, G5
Let $ABC$ be a triangle with incentre $I$ and circumcircle $\omega$. Let $D$ and $E$ be the second intersection points of $\omega$ with $AI$ and $BI$, respectively. The chord $DE$ meets $AC$ at a point $F$, and $BC$ at a point $G$. Let $P$ be the intersection point of the line through $F$ parallel to $AD$ and the line through $G$ parallel to $BE$. Suppose that the tangents to $\omega$ at $A$ and $B$ meet at a point $K$. Prove that the three lines $AE,BD$ and $KP$ are either parallel or concurrent.

Proposed by Irena Majcen and Kris Stopar, Slovenia
71 replies
WakeUp
Jul 13, 2012
InterLoop
23 minutes ago
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Inspired by old results
sqing   2
N 26 minutes ago by sqing
Source: Own
Let $ a,b>0 , a^2+b^2+ab+a+b=5 . $ Prove that
$$ \frac{ 1 }{a+b+ab+1}+\frac{6}{a^2+b^2+ab+1}\geq \frac{7}{4}$$$$ \frac{ 1 }{a+b+ab+1}+\frac{1}{a^2+b^2+ab+1}\geq \frac{1}{2}$$$$ \frac{41}{a+b+2}+\frac{ab}{a^3+b^3+2} \geq \frac{21}{2}$$
2 replies
sqing
an hour ago
sqing
26 minutes ago
J
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If $b^n|a^n-1$ then $a^b >\frac {3^n}{n}$ (China TST 2009)
Fang-jh   15
N 26 minutes ago by ihategeo_1969
Source: Chinese TST 2009 6th P1
Let $ a > b > 1, b$ is an odd number, let $ n$ be a positive integer. If $ b^n|a^n-1,$ then $ a^b > \frac {3^n}{n}.$
15 replies
Fang-jh
Apr 4, 2009
ihategeo_1969
26 minutes ago
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inequality (another entrance exam)
nai0610   1
N an hour ago by sqing
Given positive real numbers $a,b,c$ satisfying
$(a+2)b^2+(b+2)c^2+(c+2)a^2\geq 8+abc$
prove that $2(ab+bc+ca)\leq a^2(a+b)+b^2(b+c)+c^2(c+a)$
1 reply
nai0610
Jun 2, 2024
sqing
an hour ago
J
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Inspired by old results
sqing   6
N an hour ago by sqing
Source: Own
Let $ a,b,c>0 $ and $ a+b+c=3. $ Prove that
$$ \frac{2}{a}+\frac {2}{ab}+\frac{1}{abc}\geq 4$$$$ \frac{1}{a}+\frac {1}{ab}+\frac{2}{abc}\geq 2+\sqrt 3$$$$ \frac{3}{a}+\frac {3}{ab}+\frac{1}{abc}\geq\frac {7+\sqrt {13}}{2}$$$$ \frac{1}{a}+\frac {1}{ab}+\frac{3}{abc}\geq\frac {5+\sqrt {21}}{2}$$$$ \frac{1}{a}+\frac {1}{ab}+\frac{4}{abc}\geq 3+2\sqrt 2$$
6 replies
sqing
Apr 26, 2025
sqing
an hour ago
J
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Integer-Valued FE comes again
lminsl   206
N an hour ago by anudeep
Source: IMO 2019 Problem 1
Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$, $$f(2a)+2f(b)=f(f(a+b)).$$Proposed by Liam Baker, South Africa
206 replies
lminsl
Jul 16, 2019
anudeep
an hour ago
J
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Integer representation
RL_parkgong_0106   2
N an hour ago by maromex
Source: Own
Show that for any positive integer $n$, there exists some positive integer $k$ that makes the following equation have no integer root $(x_1, x_2, x_3, \dots, x_n)$.

$$x_1^{2^1}+x_2^{2^2}+x_3^{2^3}+\dots+x_n^{2^n}=k$$
2 replies
RL_parkgong_0106
Apr 22, 2025
maromex
an hour ago
J
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geometry+algebra(ver beatiful)
ehsan2004   7
N 2 hours ago by NicoN9
Source: Serbia and Montenegro 2004
The side lengths of a triangle are the roots of a cubic polynomial with rational coefficients. Prove that the altitudes of this triangle are roots of a polynomial of sixth degree with rational coefficients.
7 replies
ehsan2004
Aug 11, 2005
NicoN9
2 hours ago
J
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Japan Mathematical Olympiad Preliminary 2007 Problem 3
Kunihiko_Chikaya   1
N 2 hours ago by Mathzeus1024
On a plane given the line segment with length 7. The distance of a point $P$ and the segment is 3. Find the possible minimum value of $AP*BP.$
1 reply
Kunihiko_Chikaya
Jan 18, 2007
Mathzeus1024
2 hours ago
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J
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Bundeswettbewerb Mathematik 1974 Problem 1.1
sqrtX   0
Oct 16, 2022
Source: Bundeswettbewerb Mathematik 1974 Round 1
Find the necessary and sufficient condition that a trapezoid can be formed out of a given four-bar linkage.
0 replies
sqrtX
Oct 16, 2022
0 replies
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Bundeswettbewerb Mathematik 1974 Problem 1.1
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Reveal topic tags
geometry
trapezoid
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Source: Bundeswettbewerb Mathematik 1974 Round 1
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sqrtX
675 posts
sqrtX
#1 h  Oct 16, 2022, 12:32 PM
Y by
Find the necessary and sufficient condition that a trapezoid can be formed out of a given four-bar linkage.
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