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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
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Orthocenter config once again
Assassino9931   7
N 25 minutes ago by VicKmath7
Source: Bulgaria National Olympiad 2025, Day 2, Problem 4
Let \( ABC \) be an acute triangle with \( AB < AC \), midpoint $M$ of side $BC$, altitude \( AD \) (\( D \in BC \)), and orthocenter \( H \). A circle passes through points \( B \) and \( D \), is tangent to line \( AB \), and intersects the circumcircle of triangle \( ABC \) at a second point \( Q \). The circumcircle of triangle \( QDH \) intersects line \( BC \) at a second point \( P \). Prove that the lines \( MH \) and \( AP \) are perpendicular.
7 replies
Assassino9931
Tuesday at 1:53 PM
VicKmath7
25 minutes ago
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Angle EBA is equal to Angle DCB
WakeUp   6
N 40 minutes ago by Nari_Tom
Source: Baltic Way 2011
Let $ABCD$ be a convex quadrilateral such that $\angle ADB=\angle BDC$. Suppose that a point $E$ on the side $AD$ satisfies the equality
\[AE\cdot ED + BE^2=CD\cdot AE.\]
Show that $\angle EBA=\angle DCB$.
6 replies
WakeUp
Nov 6, 2011
Nari_Tom
40 minutes ago
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if xy+xz+yz+2xyz+1 prove that...
behdad.math.math   5
N 41 minutes ago by Sadigly
if xy+xz+yz+2xyz+1 prove that x+y+z>=3/2
5 replies
behdad.math.math
Sep 25, 2008
Sadigly
41 minutes ago
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no. of divisors of the form
S_14159   0
an hour ago
Source: idk
If $P=2^5 \cdot 3^6 \cdot 5^4 \cdot 7^3$ then number of positive integral divisors of $``P"$

(A) of form $(2 n+3), n \in \mathbb{N}$, is $=138$
(B) of form $(4 n+1), n \in \mathbb{W}$, is $=70$
(C) of form $(6 n+3), n \in \mathbb{W}$, is $=120$
(D) of form $(4 n+3), n \in \mathbb{W}$, is $=56$

(more than one option may be correct)
0 replies
S_14159
an hour ago
0 replies
J
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max n with n times n square are black
NicoN9   0
an hour ago
Source: Japan Junior MO Preliminary 2022 P5
Find the maximum positive integer $n$ such that for $45\times 45$ grid, no matter how you paint $2022$ unit squares black, there exists $n\times n$ square with all unit square painted black.
0 replies
NicoN9
an hour ago
0 replies
J
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BDE tangent to EF
NicoN9   0
an hour ago
Source: Japan Junior MO Preliminary 2022 P4
Let $ABC$ be a triangle with $AB=5$, $BC=7$, $CA=6$. Let $D, E$, and $F$ be points lying on sides $BC, CA, AB$, respectively. Given that $A, B, D, E$, and $B, C, E, F$ are cyclic respectively, and the circumcircle of $BDE$ are tangent to line $EF$, find the length of segment $AE$.
0 replies
NicoN9
an hour ago
0 replies
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Maximum number of m-tastic numbers
Tsukuyomi   30
N an hour ago by cursed_tangent1434
Source: IMO Shortlist 2017 N4
Call a rational number short if it has finitely many digits in its decimal expansion. For a positive integer $m$, we say that a positive integer $t$ is $m-$tastic if there exists a number $c\in \{1,2,3,\ldots ,2017\}$ such that $\dfrac{10^t-1}{c\cdot m}$ is short, and such that $\dfrac{10^k-1}{c\cdot m}$ is not short for any $1\le k<t$. Let $S(m)$ be the set of $m-$tastic numbers. Consider $S(m)$ for $m=1,2,\ldots{}.$ What is the maximum number of elements in $S(m)$?
30 replies
Tsukuyomi
Jul 10, 2018
cursed_tangent1434
an hour ago
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Inequalities
sqing   7
N an hour ago by sqing
Let $ a,b,c $ be real numbers so that $ a+2b+3c=2 $ and $ 2ab+6bc+3ca =1. $ Show that
$$-\frac{1}{6} \leq ab-bc+ ca\leq \frac{1}{2}$$$$\frac{5-\sqrt{61}}{9} \leq a-b+c\leq \frac{5+\sqrt{61}}{9} $$
7 replies
sqing
Yesterday at 2:40 PM
sqing
an hour ago
J
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interesting ineq
nikiiiita   6
N an hour ago by KhuongTrang
Source: Own
Given $a,b,c$ are positive real numbers satisfied $a^3+b^3+c^3=3$. Prove that:
$$\sqrt{2ab+5c^{2}+2a}+\sqrt{2bc+5a^{2}+2b}+\sqrt{2ac+5b^{2}+2c}\le3\sqrt{3\left(a+b+c\right)}$$
6 replies
nikiiiita
Jan 29, 2025
KhuongTrang
an hour ago
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Prove that x1=x2=....=x2025
Rohit-2006   7
N an hour ago by Fibonacci_math
Source: A mock
The real numbers $x_1,x_2,\cdots,x_{2025}$ satisfy,
$$x_1+x_2=2\bar{x_1}, x_2+x_3=2\bar{x_2},\cdots, x_{2025}+x_1=2\bar{x_{2025}}$$Where {$\bar{x_1},\cdots,\bar{x_{2025}}$} is a permutation of $x_1,x_2,\cdots,x_{2025}$. Prove that $x_1=x_2=\cdots=x_{2025}$
7 replies
Rohit-2006
Yesterday at 5:22 AM
Fibonacci_math
an hour ago
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Inspired by old results
sqing   1
N an hour ago by sqing
Source: Own
Let $a ,b,c \geq 0 $ and $a+b+c=1$. Prove that
$$\frac{1}{2}\leq \frac{ \left(1-a^{2}\right)^2+2\left(1-b^{2}\right) \left(1-c^{2}\right) }{\left(1+a\right)\left(1+b\right)\left(1+c\right)}\leq 1$$$$1 \leq \frac{\left(1-a^{2}\right)^{2}+2\left(1-b^{2}\right) +\left(1-c^{2}\right)^{2}}{\left(1+a\right)\left(1+b\right)\left(1+c\right)}\leq \frac{3}{2}$$
1 reply
sqing
2 hours ago
sqing
an hour ago
J
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Circle and square
Marrelia   1
N 4 hours ago by sunken rock
Given a circle with center $O$, and square $ABCD$. Point $A$ and $B$ are on the circle, and $CD$ is tangent to the circle at point $E$. Let $M$ represent the midpoint of $AD$ and $F$ represent the intersection between $AD$ and circle. Prove that $MF = FD$.
1 reply
Marrelia
Today at 3:00 AM
sunken rock
4 hours ago
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Challenging Trigonometric Sums - AoPS Volume 2 Problem 277
Shiyul   2
N 5 hours ago by sp0rtman00000
Problem #277 (Source: Mu Alpha Theta 1992)

Find $\color[rgb]{0.35,0.35,0.35}\displaystyle\sum_{n=0}^\infty\frac{\sin (nx)}{3^n}$ if $\color[rgb]{0.35,0.35,0.35}\sin x=1/3$ and $\color[rgb]{0.35,0.35,0.35} 0\le x\le \pi/2$.

I know what cosine of x is also positive because of the value of x. I've also tried to see if the value of sin(nx) ever repeats, but it doesn't. Can anyone give me a hint (not the full solution) on how to start on solving this problem? Thank you.
2 replies
Shiyul
6 hours ago
sp0rtman00000
5 hours ago
J
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law of log
Miranda2829   14
N 5 hours ago by sp0rtman00000
5log (5²) + 8 ˡºᵍ₈4 =

is this answer 6?
14 replies
Miranda2829
Today at 2:12 AM
sp0rtman00000
5 hours ago
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J
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Help me please
Hahahsafdk3l   2
N Mar 29, 2025 by Hahahsafdk3l
Find all functions $f: \mathbb{R^+} \to \mathbb{R^+}$ such that $f(2f(x) + f(y) + xyy) = xy + 2x + y, \forall x, y > 0$
2 replies
Hahahsafdk3l
Mar 29, 2025
Hahahsafdk3l
Mar 29, 2025
J
Help me please
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Reveal topic tags
functional equation
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Hahahsafdk3l
32 posts
Hahahsafdk3l
#1 h  Mar 29, 2025, 2:21 AM
Y by
Find all functions $f: \mathbb{R^+} \to \mathbb{R^+}$ such that $f(2f(x) + f(y) + xyy) = xy + 2x + y, \forall x, y > 0$
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jasperE3
11175 posts
jasperE3
#2 h  Mar 29, 2025, 2:40 AM
Y by
Hahahsafdk3l wrote:
Find all functions $f: \mathbb{R^+} \to \mathbb{R^+}$ such that $f(2f(x) + f(y) + xyy) = xy + 2x + y, \forall x, y > 0$

Almost certainly a typo of this question.
https://artofproblemsolving.com/community/c4h3536833p34392946
BTW there is a search function on AoPS that you can use, most problems you've posted have already been posted elsewhere on AoPS so there is no need for the post.
This post has been edited 2 times. Last edited by jasperE3, Mar 29, 2025, 2:53 AM
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Hahahsafdk3l
32 posts
Hahahsafdk3l
#3 h  Mar 29, 2025, 11:43 PM
Y by
thank you
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