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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
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0 replies
jlacosta
Mar 2, 2025
0 replies
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
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
USAMO 2001 Problem 6
MithsApprentice   20
N 30 minutes ago by Ritwin
Each point in the plane is assigned a real number such that, for any triangle, the number at the center of its inscribed circle is equal to the arithmetic mean of the three numbers at its vertices. Prove that all points in the plane are assigned the same number.
20 replies
MithsApprentice
Sep 30, 2005
Ritwin
30 minutes ago
Bijection on the set of integers
talkon   18
N 32 minutes ago by HamstPan38825
Source: InfinityDots MO 2 Problem 2
Determine all bijections $f:\mathbb Z\to\mathbb Z$ satisfying
$$f^{f(m+n)}(mn) = f(m)f(n)$$for all integers $m,n$.

Note: $f^0(n)=n$, and for any positive integer $k$, $f^k(n)$ means $f$ applied $k$ times to $n$, and $f^{-k}(n)$ means $f^{-1}$ applied $k$ times to $n$.

Proposed by talkon
18 replies
+1 w
talkon
Apr 9, 2018
HamstPan38825
32 minutes ago
Inequality
anhduy98   5
N an hour ago by JK1603JK
Source: Own
Given three real numbers $   a,b,c\ge 0   $ satisfying $:   a+b+c=3   $.Prove that:
$$\sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2}+\sqrt{c^2-ca+a^2}\ge 3+\frac{a^2+b^2+c^2-3abc}{3}.$$
5 replies
anhduy98
Oct 28, 2024
JK1603JK
an hour ago
Nice and easy FE on R+
sttsmet   22
N an hour ago by jasperE3
Source: EMC 2024 Problem 4, Seniors
Find all functions $ f: \mathbb{R}^{+} \to \mathbb{R}^{+}$ such that $f(x+yf(x)) = xf(1+y)$
for all x, y positive reals.
22 replies
sttsmet
Dec 23, 2024
jasperE3
an hour ago
Existence of AP of interesting integers
DVDthe1st   33
N an hour ago by tchange7575
Source: 2018 China TST Day 1 Q2
A number $n$ is interesting if 2018 divides $d(n)$ (the number of positive divisors of $n$). Determine all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ whose terms are all interesting.
33 replies
DVDthe1st
Jan 2, 2018
tchange7575
an hour ago
A diophantine equation
crazyfehmy   13
N 2 hours ago by Primeniyazidayi
Source: Turkey Junior National Olympiad 2012 P1
Let $x, y$ be integers and $p$ be a prime for which

\[ x^2-3xy+p^2y^2=12p \]
Find all triples $(x,y,p)$.
13 replies
crazyfehmy
Dec 12, 2012
Primeniyazidayi
2 hours ago
nf(f(n)) = f(n)^2, f : N->N
Zhero   19
N 2 hours ago by HamstPan38825
Source: ELMO Shortlist 2010, A1; also ELMO #4
Determine all strictly increasing functions $f: \mathbb{N}\to\mathbb{N}$ satisfying $nf(f(n))=f(n)^2$ for all positive integers $n$.

Carl Lian and Brian Hamrick.
19 replies
Zhero
Jul 5, 2012
HamstPan38825
2 hours ago
RMM 2019 Problem 2
math90   77
N 2 hours ago by ihatemath123
Source: RMM 2019
Let $ABCD$ be an isosceles trapezoid with $AB\parallel CD$. Let $E$ be the midpoint of $AC$. Denote by $\omega$ and $\Omega$ the circumcircles of the triangles $ABE$ and $CDE$, respectively. Let $P$ be the crossing point of the tangent to $\omega$ at $A$ with the tangent to $\Omega$ at $D$. Prove that $PE$ is tangent to $\Omega$.

Jakob Jurij Snoj, Slovenia
77 replies
math90
Feb 23, 2019
ihatemath123
2 hours ago
Two circles concur on a line
math154   59
N 2 hours ago by Mathandski
Source: ELMO Shortlist 2012, G1; also ELMO #1
In acute triangle $ABC$, let $D,E,F$ denote the feet of the altitudes from $A,B,C$, respectively, and let $\omega$ be the circumcircle of $\triangle AEF$. Let $\omega_1$ and $\omega_2$ be the circles through $D$ tangent to $\omega$ at $E$ and $F$, respectively. Show that $\omega_1$ and $\omega_2$ meet at a point $P$ on $BC$ other than $D$.

Ray Li.
59 replies
math154
Jul 2, 2012
Mathandski
2 hours ago
functional equation
Anni   7
N 2 hours ago by HamstPan38825
Source: albanian TST 2008 bmo
Find all functions $f: \mathbb R \to \mathbb R$ such that
\[ f(x+f(y))=y+f(x+1),\]for all $x,y \in \mathbb R$.
7 replies
Anni
May 24, 2009
HamstPan38825
2 hours ago
Trash Trig Sum
P_Groudon   3
N 4 hours ago by P_Groudon
Find the smallest positive integer $n$ such that $$\sum_{k=0}^{298}\sin(k^2 + 2k + 2)\sin(2k + 2) = \frac{\cos(1) - \cos(n)}{2},$$where degrees are used.
3 replies
P_Groudon
5 hours ago
P_Groudon
4 hours ago
Geometry problem for enthusiasts
Raul_S_Baz   1
N 5 hours ago by Raul_S_Baz
IMAGE
1 reply
Raul_S_Baz
6 hours ago
Raul_S_Baz
5 hours ago
Inequalities
sqing   3
N Today at 4:24 PM by DAVROS
Let $a,b,c\ge \frac{1}{2}$ and $\left(\frac{1}{a}+\frac{1}{b}-\frac{1}{c}\right)\left(\frac{1}{a}-\frac{1}{b}+\frac{1}{c}\right)\le 1. $ Prove that
$$a+b+c\geq 2$$Let $a,b,c\ge \frac{1}{2}$ and $ \left(a+\frac{1}{a}+\frac{1}{b}-\frac{1}{c}\right)\left(a+\frac{1}{a}-\frac{1}{b}+\frac{1}{c}\right)\le \frac{9}{2}. $ Prove that
$$a^2+b^2+c^2\geq 1$$Let $a,b\ge \frac{1}{2}$ and $ \left( \frac{1}{a}-\frac{1}{b}+2\right)\left( \frac{1}{b}-\frac{1}{a}+2\right) \le   \frac{20}{9}. $ Prove that
$$ a+b\geq 2$$Let $a,b\ge \frac{1}{2}$ and $a^2+b^2=1. $ Prove that
$$\left(\frac{2}{a}+\frac{1}{b}-1\right)\left(\frac{2}{a}-\frac{1}{b}+1\right)\ge \frac{13}{3}$$
3 replies
sqing
Mar 15, 2025
DAVROS
Today at 4:24 PM
Classic geometry problem
Raul_S_Baz   2
N Today at 4:10 PM by Raul_S_Baz
IMAGE
2 replies
Raul_S_Baz
Mar 4, 2025
Raul_S_Baz
Today at 4:10 PM
(x^2-3x+2)^2-3(x^2-3x+2)-2-x=0 (Moldova 2000 Grade 9 P5)
jasperE3   12
N Apr 26, 2021 by R-sk
Solve in real numbers the equation
$$\left(x^2-3x-2\right)^2-3\left(x^2-3x-2\right)-2-x=0.$$
12 replies
jasperE3
Apr 26, 2021
R-sk
Apr 26, 2021
(x^2-3x+2)^2-3(x^2-3x+2)-2-x=0 (Moldova 2000 Grade 9 P5)
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jasperE3
11093 posts
#1
Y by
Solve in real numbers the equation
$$\left(x^2-3x-2\right)^2-3\left(x^2-3x-2\right)-2-x=0.$$
This post has been edited 1 time. Last edited by jasperE3, Apr 26, 2021, 1:01 PM
Reason: x^2-3x-2 instead of x^2-3x+2
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Mathematician1010
3261 posts
#2
Y by
Start out by simplifying the left-hand side, and you get
$x^4-6x^3+10x^2-4x-4=0$
After that, I'd recommend using the rational root theorem, and if there are any rational solutions, you can divide out the factor. I haven't done this for this problem yet, but if there are two rational solutions you can reduce down to a quadratic and find any other solutions with the quadratic formula.
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jasperE3
11093 posts
#3
Y by
RRT gives no rational roots, you'd probably have to factor it into two quadratics. Any solutions without undetermined coefficients?
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Lamboreghini
6486 posts
#4
Y by
progress
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jasperE3
11093 posts
#5 • 3 Y
Y by Mango247, Mango247, Mango247
yofro (via PM) wrote:
What happens if x is a root of $x^2-4x-2$?
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Lamboreghini
6486 posts
#6
Y by
Post #5 by jasperE3

@jasperE3 ok

soo... some more progress
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natmath
8219 posts
#7
Y by
jasperE3 wrote:
yofro (via PM) wrote:
What happens if x is a root of $x^2-4x-2$?

That's pretty cool
Click to reveal hidden text

The $x^2-4x-2$ is kind of intuitive because clearly setting $x^2-3x-2=x$ gives a solution to the above equation. We still need the other quadratic factor for 4 roots, but I'm not seeing something that is intuitive. Of course, it's pretty easy now to just factor $x^2-4x-2$ from the equation in #2 and use MUD, but I was wondering if there was a more intuitive approach on that other factor.

@below you really should be thanking yofro for these amazing hints. Do you have a reason to believe the other factor does not have real roots?
This post has been edited 2 times. Last edited by natmath, Apr 26, 2021, 4:55 AM
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Lamboreghini
6486 posts
#8 • 3 Y
Y by Mango247, Mango247, Mango247
Post #7 by natmath

@natmath whoa that's awesome

solutions to $x^2-4x-2$ are $$x=\frac{4\pm\sqrt{16+8}}{2}=\frac{4\pm2\sqrt6}{2}=2\pm\sqrt6.$$Those are real! 2 solutions to the equation already!

If I'm not mistaken, these are the only real solutions and this problem is solved....?
This post has been edited 1 time. Last edited by Lamboreghini, Apr 26, 2021, 4:43 AM
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asbodke
1913 posts
#9
Y by
uh I think you messed up? You're confusing $x^2-4x+2$ and $x^2-4x-2$? Is there a typo in the problem?

If there's a typo in the problem and it is $x^2-3x-2$ instead of $x^2-3x+2$ we can use poly division and get our other answers to be $1\pm \sqrt 5$

and of course using the other roots, we can let $x$ be a root of $x^2-2x-4$, then $x^2-3x-2=-x+2$, and $(-x+2)^2-3(-x+2)-2-x=x^2-4x+4+3x-6-2-x=x^2-2x-4=0$
This post has been edited 4 times. Last edited by asbodke, Apr 26, 2021, 5:33 AM
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Mathematician1010
3261 posts
#10
Y by
@above where are you getting $x^2-2x-4$ from?
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jasperE3
11093 posts
#11
Y by
Very sorry. Yes, the problem was incorrect. It has been edited now.
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natmath
8219 posts
#12 • 1 Y
Y by yofro
Yofro gave me a hint on $x^2-2x-4$. It's not as direct as the first one, but I can try to explain it.

If $f(x)=x^2-3x-2$, then we want to find the solutions to
$$f(f(x))=x$$
Of course, if there existed an $r$ s.t. $f(r)=r$, then this would clearly be a solution to our equation. That $r$ must be either of the roots to
$$x^2-3x-2=x$$$$x^2-4x-2=0$$
Let's say for some constant $k$ there exists an $r$ s.t. $f(r)=k-r$. Now let's say $k-r$ also satisfies the equation $f(x)=k-x$ (i.e. $f(k-r)=k-(k-r)=r$). Then this $r$ would satisfy the original equation.
However, finding this root is not as immediate. We want $r$ and $k-r$ to be the solutions of
$$f(x)=k-x$$$$x^2-3x-2=k-x$$$$x^2-2x-(2+k)=0$$By viete's, the sum of roots is
$$r+k-r=2$$$$k=2$$So the solutions to
$$x^2-2x-4=0$$also satisfy our original equation.

Also I think we know that there are no other solutions aside from those in the form $f(x)=x$ and $f(x)=k-x$ because $x,k-x$ are the only polynomial functions that satisfy $f(f(x))=x$.
This post has been edited 1 time. Last edited by natmath, Apr 26, 2021, 3:29 PM
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R-sk
429 posts
#13
Y by
Take one x in other side and set y=$x^2-3x$ and solve quadratic take one of its root equate it to x then you get a good answer
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