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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
J
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An fe based off of another trivial problem
benjaminchew13   0
a few seconds ago
Source: Own
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all real numbers $x$ and $y$, $f(x+y-f(x))f(f(x+y)-y)=f(xy)$
This was based off of a (trivial) fe
$f(x + y - f(x))(f(x + y) - y) = f(xy)$
0 replies
benjaminchew13
a few seconds ago
0 replies
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IMO ShortList 1998, geometry problem 5
nttu   31
N a few seconds ago by Ilikeminecraft
Source: IMO ShortList 1998, geometry problem 5
Let $ABC$ be a triangle, $H$ its orthocenter, $O$ its circumcenter, and $R$ its circumradius. Let $D$ be the reflection of the point $A$ across the line $BC$, let $E$ be the reflection of the point $B$ across the line $CA$, and let $F$ be the reflection of the point $C$ across the line $AB$. Prove that the points $D$, $E$ and $F$ are collinear if and only if $OH=2R$.
31 replies
nttu
Oct 14, 2004
Ilikeminecraft
a few seconds ago
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Line through orthocenter
juckter   13
N a minute ago by Ilikeminecraft
Source: Mexico National Olympiad 2011 Problem 2
Let $ABC$ be an acute triangle and $\Gamma$ its circumcircle. Let $l$ be the line tangent to $\Gamma$ at $A$. Let $D$ and $E$ be the intersections of the circumference with center $B$ and radius $AB$ with lines $l$ and $AC$, respectively. Prove the orthocenter of $ABC$ lies on line $DE$.
13 replies
juckter
Jun 22, 2014
Ilikeminecraft
a minute ago
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ABC is similar to XYZ
Amir Hossein   54
N 2 minutes ago by Ilikeminecraft
Source: China TST 2011 - Quiz 2 - D2 - P1
Let $AA',BB',CC'$ be three diameters of the circumcircle of an acute triangle $ABC$. Let $P$ be an arbitrary point in the interior of $\triangle ABC$, and let $D,E,F$ be the orthogonal projection of $P$ on $BC,CA,AB$, respectively. Let $X$ be the point such that $D$ is the midpoint of $A'X$, let $Y$ be the point such that $E$ is the midpoint of $B'Y$, and similarly let $Z$ be the point such that $F$ is the midpoint of $C'Z$. Prove that triangle $XYZ$ is similar to triangle $ABC$.
54 replies
Amir Hossein
May 20, 2011
Ilikeminecraft
2 minutes ago
J
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Rectangle EFGH in incircle, prove that QIM = 90
v_Enhance   63
N 3 minutes ago by Ilikeminecraft
Source: Taiwan 2014 TST1, Problem 3
Let $ABC$ be a triangle with incenter $I$, and suppose the incircle is tangent to $CA$ and $AB$ at $E$ and $F$. Denote by $G$ and $H$ the reflections of $E$ and $F$ over $I$. Let $Q$ be the intersection of $BC$ with $GH$, and let $M$ be the midpoint of $BC$. Prove that $IQ$ and $IM$ are perpendicular.
63 replies
v_Enhance
Jul 18, 2014
Ilikeminecraft
3 minutes ago
J
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Cyclic points [variations on a Fuhrmann generalization]
shobber   25
N 4 minutes ago by Ilikeminecraft
Source: China TST 2006
$H$ is the orthocentre of $\triangle{ABC}$. $D$, $E$, $F$ are on the circumcircle of $\triangle{ABC}$ such that $AD \parallel BE \parallel CF$. $S$, $T$, $U$ are the semetrical points of $D$, $E$, $F$ with respect to $BC$, $CA$, $AB$. Show that $S, T, U, H$ lie on the same circle.
25 replies
shobber
Jun 18, 2006
Ilikeminecraft
4 minutes ago
J
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20^x+13^y=2013^z
parmenides51   17
N 4 minutes ago by Math_01-person
Source: JBMO Shortlist 2013 NT2
Solve in integers $20^x+13^y=2013^z$.
17 replies
parmenides51
Apr 24, 2019
Math_01-person
4 minutes ago
J
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Quad formed by orthocenters has same area (all 7's!)
v_Enhance   36
N 5 minutes ago by Ilikeminecraft
Source: USA January TST for the 55th IMO 2014
Let $ABCD$ be a cyclic quadrilateral, and let $E$, $F$, $G$, and $H$ be the midpoints of $AB$, $BC$, $CD$, and $DA$ respectively. Let $W$, $X$, $Y$ and $Z$ be the orthocenters of triangles $AHE$, $BEF$, $CFG$ and $DGH$, respectively. Prove that the quadrilaterals $ABCD$ and $WXYZ$ have the same area.
36 replies
v_Enhance
Apr 28, 2014
Ilikeminecraft
5 minutes ago
J
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A geometry problem from the TOT
Invert_DOG_about_centre_O   10
N 5 minutes ago by Ilikeminecraft
Source: Tournament of towns Spring 2018 A-level P4
Let O be the center of the circumscribed circle of the triangle ABC. Let AH be the altitude in this triangle, and let P be the base of the perpendicular drawn from point A to the line CO. Prove that the line HP passes through the midpoint of the side AB. (6 points)

Egor Bakaev
10 replies
Invert_DOG_about_centre_O
Mar 10, 2020
Ilikeminecraft
5 minutes ago
J
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The prime inequality learning problem
orl   138
N 7 minutes ago by Ilikeminecraft
Source: IMO 1995, Problem 2, Day 1, IMO Shortlist 1995, A1
Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc = 1$. Prove that
\[ \frac {1}{a^{3}\left(b + c\right)} + \frac {1}{b^{3}\left(c + a\right)} + \frac {1}{c^{3}\left(a + b\right)}\geq \frac {3}{2}. \]
138 replies
orl
Nov 9, 2005
Ilikeminecraft
7 minutes ago
J
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ab+bc+ca = 1
Tales   119
N 7 minutes ago by Ilikeminecraft
Source: IMO ShortList 2004, algebra problem 5
If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]
119 replies
Tales
Mar 22, 2005
Ilikeminecraft
7 minutes ago
J
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Inequality by Po-Ru Loh
v_Enhance   56
N 8 minutes ago by Ilikeminecraft
Source: ELMO 2003 Problem 4
Let $x,y,z \ge 1$ be real numbers such that \[ \frac{1}{x^2-1} + \frac{1}{y^2-1} + \frac{1}{z^2-1} = 1. \] Prove that \[ \frac{1}{x+1} + \frac{1}{y+1} + \frac{1}{z+1} \le 1. \]
56 replies
v_Enhance
Dec 29, 2012
Ilikeminecraft
8 minutes ago
J
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All Russian Olympiad 2018 Day1 P2
Davrbek   24
N 9 minutes ago by Ilikeminecraft
Source: Grade 11 P2
Let $n\geq 2$ and $x_{1},x_{2},\ldots,x_{n}$ positive real numbers. Prove that
\[\frac{1+x_{1}^2}{1+x_{1}x_{2}}+\frac{1+x_{2}^2}{1+x_{2}x_{3}}+\cdots+\frac{1+x_{n}^2}{1+x_{n}x_{1}}\geq n.\]
24 replies
Davrbek
Apr 28, 2018
Ilikeminecraft
9 minutes ago
J
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Product of three positive reals is less than 1/8
kk108   48
N 9 minutes ago by ohiorizzler1434
Source: RMO Hyderabad 2016 , P2 .
Let $a,b,c$ be positive real numbers such that $$\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=1.$$Prove that $abc \le \frac{1}{8}$.
48 replies
1 viewing
kk108
Oct 12, 2016
ohiorizzler1434
9 minutes ago
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J
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When is this well known sequence periodic?
Assassino9931   2
N Mar 30, 2025 by Assassino9931
Source: Bulgaria Spring Mathematical Competition 2025 12.2
Determine all values of $a_0$ for which the sequence of real numbers with $a_{n+1}=3a_n - 4a_n^3$ for all $n\geq 0$ is periodic from the beginning.
2 replies
Assassino9931
Mar 30, 2025
Assassino9931
Mar 30, 2025
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When is this well known sequence periodic?
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Reveal topic tags
algebra
periodic
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G H BBookmark kLocked kLocked NReply
Source: Bulgaria Spring Mathematical Competition 2025 12.2
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Assassino9931
1252 posts
Assassino9931
#1 h  Mar 30, 2025, 1:17 PM
Y by
Determine all values of $a_0$ for which the sequence of real numbers with $a_{n+1}=3a_n - 4a_n^3$ for all $n\geq 0$ is periodic from the beginning.
Z K Y
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RagvaloD
4909 posts
RagvaloD
#2 h  Mar 30, 2025, 2:28 PM • 1 Y
Y by MihaiT
Let $a_0>1$
Then $a_1 =3a_0-4a_0^3<-a_0$
And $a_2=3a_1-4a_1^3>-a_1>a_0>1$ so $a_0<a_2<...<a_{2k}$ and so sequence is not periodic

Same logic for $a_0<-1$

So $a_0 \in [-1,1]$ and so $a_0= \sin t$
then $a_1=\sin{3t} \in [-1,1]$ so $a_n=\sin {3^n t}$

$a_n=a_0 \to \sin{3^nt}=\sin {t} \to \sin {\frac{3^n+1}{2}t}\cos{\frac{3^n-1}{2}t}=0 \to t = \frac{2\pi m}{3^n+1}$ or $ t =\frac{\pi (2m-1)}{3^n-1}$
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Assassino9931
1252 posts
Assassino9931
#3 h  Mar 30, 2025, 4:56 PM
Y by
For $|a_0| > 1$ one simply has $|a_{n+1}| > |a_n|$ by induction, as $|3 - 4a_n^2| = 4|a_n|^2 - 3 > 1$.
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