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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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0 replies
jlacosta
May 1, 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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Inequality
lgx57   1
N 32 minutes ago by sqing
$a,b,c \in \mathbb{R}^{+}$,$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=1$. Prove that
$$a^abc+b^bac+c^cab \ge 27(ab+bc+ca)$$
1 reply
lgx57
an hour ago
sqing
32 minutes ago
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Number of real roots
girishpimoli   1
N 43 minutes ago by P0tat0b0y
Number of real roots of

$\displaystyle 2\sin(\theta)\cos(3\theta)\sin(5\theta)=-1$
1 reply
girishpimoli
Yesterday at 5:35 PM
P0tat0b0y
43 minutes ago
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Balkan Mathematical Olympiad
ABCD1728   1
N 43 minutes ago by ABCD1728
Can anyone provide the PDF version of the book "Balkan Mathematical Olympiads" by Mircea Becheanu and Bogdan Enescu (published by XYZ press in 2014), thanks!
1 reply
ABCD1728
May 24, 2025
ABCD1728
43 minutes ago
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Unexpecredly Quick-Solve Inequality
Primeniyazidayi   2
N an hour ago by sqing
Source: German MO 2025,Round 4,Grade 11/12 Day 2 P1
If $a, b, c>0$, prove that $$\frac{a^5}{b^2}+\frac{b}{c}+\frac{c^3}{a^2}>2a$$
2 replies
1 viewing
Primeniyazidayi
3 hours ago
sqing
an hour ago
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cute geo
Royal_mhyasd   0
an hour ago
Source: own(?)
Let $\triangle ABC$ be an acute triangle and $I$ it's incenter. Let $A'$, $B'$ and $C'$ be the projections of $I$ onto $BC$, $AC$ and $AB$ respectively. $BC \cap B'C' = \{K\}$ and $Y$ is the projection of $A'$ onto $KI$. Let $M$ be the middle of the arc $BC$ not containing $A$ and $T$ the second intersection of $A'M$ and the circumcircle of $ABC$. If $N$ is the midpoint of $AI$, $TY \cap IA' = \{P\}$, $BN \cap PC' = \{D\}$ and $CN \cap PB' =\{E\}$, prove that $NEPD$ is cyclic.
PS i'm not sure if this problem is actually original so if it isn't someone please tell me so i can change the source (if that's possible)
0 replies
1 viewing
Royal_mhyasd
an hour ago
0 replies
J
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Nice inequality
TUAN2k8   2
N an hour ago by TUAN2k8
Source: Own
Let $n \ge 2$ be an even integer and let $x_1,x_2,...,x_n$ be real numbers satisfying $x_1^2+x_2^2+...+x_n^2=n$.
Prove that
$\sum_{1 \le i < j \le n} \frac{x_ix_j}{x_i^2+x_j^2+1} \ge \frac{-n}{6}$
2 replies
TUAN2k8
Today at 2:03 AM
TUAN2k8
an hour ago
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Pls help I'm stuck
UtkarshSri   3
N an hour ago by lbh_qys
So I was solving this problem
Cbrt(x+1) - cbrt(x-1) = pow((x²-1),1/6)


Here, the square root is defined in the domain x≤-1 OR x≥1
I substituted a= cbrt(x+1) and b= cbrt(x-1)
And then the equation turned out to be
a-b= sqrt(ab)
Here, since the RHS is +ve, the LHS must be too, so a>b (Done so as to prevent extraneous roots)
Now, I can write the domain in terms of a and b as ab≥0
Squaring, we get a²+b²-3ab=0
On solving we get a/b = (3±sqrt(5))/2
Now, we know that a>b from the above condition, so the only root is a/b= (3+sqrt(5))/2 and on substituting for x, and applying componendo and dividendo after cubing, I get x= sqrt(5)/2

Now, my question is, since the equation is even i.e f(x)=f(-x), it's symmetric about the y-axis, so x= -sqrt(5)/2 must also satisfy the equation, and indeed it does, so where did I made some irreversible step that caused this loss of the root?
when I plugged the equation to wolframalpha, it showed that the "principle root" has the former solⁿ only where's for a "real valued root", it has both.I also found that on solving the equation
b-a= sqrt(ab), I get the latter root.But I mean where did that extra -ve sign come from?
I'm familiar that f(x)=sqrt(g(x)) is equivalent to f²(x)=g(x) AND f(x)≥0 AND g(x)≥0 but the g(x)≥0 condition is not necessary as f²(x) is already ≥0
and also that abs(f(x))=abs(g(x)) is equivalent to f²(x)=g²(x)

PS: Pardon me I don't know latex well, I'm tried to make this as comprehensive as possible and made sure there's no loss of equivalence in any step.
I hope someone clarifies this for me, it's really bothering me for a while.
3 replies
UtkarshSri
Yesterday at 9:57 AM
lbh_qys
an hour ago
J
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An NT for a break
reni_wee   1
N an hour ago by reni_wee
Source: ONTCP 2.4.1
Prove that there are no positive integers $x,k$ and $n \geq 2$ such that $x^2+1 = k(2^n -1)$.
1 reply
reni_wee
an hour ago
reni_wee
an hour ago
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p divides x^x-c
mistakesinsolutions   6
N an hour ago by reni_wee
Show that for integer c and a prime p, $ p |x^x-c $ has a solution
6 replies
mistakesinsolutions
Jun 13, 2023
reni_wee
an hour ago
J
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exponential diophantine in integers
skellyrah   1
N an hour ago by skellyrah
find all integers x,y,z such that $$ 45^x = 5^y + 2000^z $$
1 reply
skellyrah
Yesterday at 7:04 PM
skellyrah
an hour ago
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IMO 2017 Problem 4
Amir Hossein   117
N 2 hours ago by ezpotd
Source: IMO 2017, Day 2, P4
Let $R$ and $S$ be different points on a circle $\Omega$ such that $RS$ is not a diameter. Let $\ell$ be the tangent line to $\Omega$ at $R$. Point $T$ is such that $S$ is the midpoint of the line segment $RT$. Point $J$ is chosen on the shorter arc $RS$ of $\Omega$ so that the circumcircle $\Gamma$ of triangle $JST$ intersects $\ell$ at two distinct points. Let $A$ be the common point of $\Gamma$ and $\ell$ that is closer to $R$. Line $AJ$ meets $\Omega$ again at $K$. Prove that the line $KT$ is tangent to $\Gamma$.

Proposed by Charles Leytem, Luxembourg
117 replies
Amir Hossein
Jul 19, 2017
ezpotd
2 hours ago
J
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Polynomial Equations and Roots
P162008   1
N 2 hours ago by alexheinis
Given that $a,b,c,d \in R$ and $\alpha,\beta,\gamma,\delta$ are distinct complex numbers such that

$f(x) = x^7 - x^3 + x^2 + x + 11$

$g(x) = ax^3 + bx^2 + cx + d$

$p(x) = x^5 + x^4 + x^3 + x^2 + x - 2,$

$p(\alpha) = p(\beta) = p(\gamma) = p(\delta) = 0,$

$f(\alpha) = g(\alpha), f(\beta) = g(\beta), f(\gamma) = g(\gamma)$ and $f(\delta) = g(\delta).$

Find the value of $ac + bd.$
1 reply
P162008
Monday at 11:59 PM
alexheinis
2 hours ago
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x^2+y^2+z^2+xy+yz+zx=6xyz diophantine
parmenides51   7
N 2 hours ago by Assassino9931
Source: Greece Junior Math Olympiad 2024 p4
Prove that there are infinite triples of positive integers $(x,y,z)$ such that
$$x^2+y^2+z^2+xy+yz+zx=6xyz.$$
7 replies
parmenides51
Mar 2, 2024
Assassino9931
2 hours ago
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Turkish JMO 2025?
bitrak   1
N 3 hours ago by blug
Let p and q be prime numbers. Prove that if pq(p+ 1)(q + 1)+ 1 is a perfect square, then pq + 1 is also a perfect square.
1 reply
bitrak
Yesterday at 2:04 PM
blug
3 hours ago
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Geometry Basic
AlexCenteno2007   4
N May 4, 2025 by giratina3
Let $ABC$ be an isosceles triangle such that $AC=BC$. Let $P$ be a dot on the $AC$ side.
The tangent to the circumcircle of $ABP$ at point $P$ intersects the circumcircle of $BCP$ at $D$. Prove that CD$ \parallel$AB
4 replies
AlexCenteno2007
Apr 28, 2025
giratina3
May 4, 2025
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Geometry Basic
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Reveal topic tags
geometry
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AlexCenteno2007
157 posts
AlexCenteno2007
#1 h  Apr 28, 2025, 12:11 AM
Y by
Let $ABC$ be an isosceles triangle such that $AC=BC$. Let $P$ be a dot on the $AC$ side.
The tangent to the circumcircle of $ABP$ at point $P$ intersects the circumcircle of $BCP$ at $D$. Prove that CD$ \parallel$AB
This post has been edited 5 times. Last edited by AlexCenteno2007, Apr 28, 2025, 12:14 AM
Reason: Error
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AlexCenteno2007
157 posts
AlexCenteno2007
#3 h  Apr 28, 2025, 1:44 AM
Y by
Could you give any clues?
This post has been edited 1 time. Last edited by AlexCenteno2007, Apr 28, 2025, 4:16 PM
Reason: Error
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mathafou
421 posts
mathafou
#4 h  Apr 28, 2025, 5:58 PM • 1 Y
Y by AlexCenteno2007
hint
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AlexCenteno2007
157 posts
AlexCenteno2007
#5 h  May 4, 2025, 3:00 AM
Y by
mathafou wrote:
hint

I guess that by power of points it can come out, right?
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giratina3
556 posts
giratina3
#6 h  May 4, 2025, 3:18 AM
Y by
Chasing inscribed angles would definitely be the first step to solving the problem.
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