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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
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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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Challenging Optimization Problem
Shiyul   4
N 15 minutes ago by vanstraelen
Let $xyz = 1$. Find the minimum and maximum values of $\frac{1}{1 + x + xy}$ + $\frac{1}{1 + y + yz}$ + $\frac{1}{1 + z + zx}$

Can anyone give me a hint? I got that either the minimum or maximum was 1, but I'm sure if I'm correct.
4 replies
Shiyul
Yesterday at 8:20 PM
vanstraelen
15 minutes ago
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Geometry Angle Chasing
Sid-darth-vater   1
N 22 minutes ago by vanstraelen
Is there a way to do this without drawing obscure auxiliary lines? (the auxiliary lines might not be obscure I might just be calling them obscure)

For example I tried rotating triangle MBC 80 degrees around point C (so the BC line segment would now lie on segment AC) but I couldn't get any results. Any help would be appreciated!
1 reply
Sid-darth-vater
Yesterday at 11:50 PM
vanstraelen
22 minutes ago
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Inequalities
sqing   1
N an hour ago by sqing
Let $ a,b,c $ be real numbers such that $ a^2+b^2+c^2=1. $ Prove that$$ |a-b|+|b-2c|+|c-3a|\leq 5$$$$|a-2b|+|b-3c|+|c-4a|\leq \sqrt{42}$$$$ |a-b|+|b-\frac{11}{10}c|+|c-a|\leq \frac{29}{10}$$
1 reply
sqing
5 hours ago
sqing
an hour ago
J
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Incircle of a triangle is tangent to (ABC)
amar_04   11
N an hour ago by Nari_Tom
Source: XVII Sharygin Correspondence Round P18
Let $ABC$ be a scalene triangle, $AM$ be the median through $A$, and $\omega$ be the incircle. Let $\omega$ touch $BC$ at point $T$ and segment $AT$ meet $\omega$ for the second time at point $S$. Let $\delta$ be the triangle formed by lines $AM$ and $BC$ and the tangent to $\omega$ at $S$. Prove that the incircle of triangle $\delta$ is tangent to the circumcircle of triangle $ABC$.
11 replies
+1 w
amar_04
Mar 2, 2021
Nari_Tom
an hour ago
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Inspired by hlminh
sqing   1
N an hour ago by sqing
Source: Own
Let $ a,b,c $ be real numbers such that $ a^2+b^2+c^2=1. $ Prove that$$ |a-kb|+|kb-c|+|c-a|\leq 2\sqrt {k^2+1}$$Where $ k\geq 1.$
$$ |a-kb|+|kb-c|+|c-a|\leq 2\sqrt {2}$$Where $0< k\leq 1.$
1 reply
sqing
2 hours ago
sqing
an hour ago
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Two very hard parallel
jayme   3
N an hour ago by jayme
Source: own inspired by EGMO
Dear Mathlinkers,

1. ABC a triangle
2. D, E two point on the segment BC so that BD = DE= EC
3. M, N the midpoint of ED, AE
4. H the orthocenter of the acutangle triangle ADE
5. 1, 2 the circumcircle of the triangle DHM, EHN
6. P, Q the second point of intersection of 1 and BM, 2 and CN
7. U, V the second points of intersection of 2 and MN, PQ.

Prove : UV is parallel to PM.

Sincerely
Jean-Louis
3 replies
jayme
Yesterday at 12:46 PM
jayme
an hour ago
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Inequality with n-gon sides
mihaig   3
N an hour ago by mihaig
Source: VL
If $a_1,a_2,\ldots, a_n~(n\geq3)$ are are the lengths of the sides of a $n-$gon such that
$$\sum_{i=1}^{n}{a_i}=1,$$then
$$(n-2)\left[\sum_{i=1}^{n}{\frac{a_i^2}{(1-a_i)^2}}-\frac n{(n-1)^2}\right]\geq(2n-1)\left(\sum_{i=1}^{n}{\frac{a_i}{1-a_i}}-\frac n{n-1}\right)^2.$$
When do we have equality?

(V. Cîrtoaje and L. Giugiuc, 2021)
3 replies
mihaig
Feb 25, 2022
mihaig
an hour ago
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Advanced topics in Inequalities
va2010   23
N an hour ago by Novmath
So a while ago, I compiled some tricks on inequalities. You are welcome to post solutions below!
23 replies
va2010
Mar 7, 2015
Novmath
an hour ago
J
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JBMO TST Bosnia and Herzegovina 2022 P3
Motion   7
N an hour ago by cafer2861
Source: JBMO TST Bosnia and Herzegovina 2022
Let $ABC$ be an acute triangle. Tangents on the circumscribed circle of triangle $ABC$ at points $B$ and $C$ intersect at point $T$. Let $D$ and $E$ be a foot of the altitudes from $T$ onto $AB$ and $AC$ and let $M$ be the midpoint of $BC$. Prove:
A) Prove that $M$ is the orthocenter of the triangle $ADE$.
B) Prove that $TM$ cuts $DE$ in half.
7 replies
Motion
May 21, 2022
cafer2861
an hour ago
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hard problem
Cobedangiu   5
N 2 hours ago by arqady
Let $a,b,c>0$ and $a+b+c=3$. Prove that:
$\dfrac{4}{a+b}+\dfrac{4}{b+c}+\dfrac{4}{c+a} \le \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+3$
5 replies
Cobedangiu
Yesterday at 1:51 PM
arqady
2 hours ago
J
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Radical Axes and circles
mathprodigy2011   4
N 2 hours ago by spiderman0
Can someone explain how to do this purely geometrically?
4 replies
mathprodigy2011
Today at 1:58 AM
spiderman0
2 hours ago
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density over modulo M
SomeGuy3335   3
N 2 hours ago by ja.
Let $M$ be a positive integer and let $\alpha$ be an irrational number. Show that for every integer $0\leq a < M$, there exists a positive integer $n$ such that $M \mid \lfloor{n \alpha}\rfloor-a$.
3 replies
SomeGuy3335
Apr 20, 2025
ja.
2 hours ago
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Diophantine equation !
ComplexPhi   5
N 3 hours ago by aops.c.c.
Source: Romania JBMO TST 2015 Day 1 Problem 4
Solve in nonnegative integers the following equation :
$$21^x+4^y=z^2$$
5 replies
ComplexPhi
May 14, 2015
aops.c.c.
3 hours ago
J
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Parity and sets
betongblander   7
N 3 hours ago by ihategeo_1969
Source: Brazil National Olympiad 2020 5 Level 3
Let $n$ and $k$ be positive integers with $k$ $\le$ $n$. In a group of $n$ people, each one or always
speak the truth or always lie. Arnaldo can ask questions for any of these people
provided these questions are of the type: “In set $A$, what is the parity of people who speak to
true? ”, where $A$ is a subset of size $ k$ of the set of $n$ people. The answer can only
be “$even$” or “$odd$”.
a) For which values of $n$ and $k$ is it possible to determine which people speak the truth and
which people always lie?
b) What is the minimum number of questions required to determine which people
speak the truth and which people always lie, when that number is finite?
7 replies
betongblander
Mar 18, 2021
ihategeo_1969
3 hours ago
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.problem.
Cobedangiu   4
N Apr 5, 2025 by Lankou
Find the integer coefficients after expanding Newton's binomial:
$$(\frac{3}{2}-\frac{2}{3}x^2)^n (n \in Z)$$
4 replies
Cobedangiu
Apr 4, 2025
Lankou
Apr 5, 2025
J
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Reveal topic tags
algebra
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Cobedangiu
55 posts
Cobedangiu
#1 h  Apr 4, 2025, 6:20 AM
Y by
Find the integer coefficients after expanding Newton's binomial:
$$(\frac{3}{2}-\frac{2}{3}x^2)^n (n \in Z)$$
This post has been edited 1 time. Last edited by Cobedangiu, Apr 4, 2025, 6:20 AM
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Cobedangiu
55 posts
Cobedangiu
#2 h  Apr 4, 2025, 8:36 AM
Y by
...............
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Lankou
1389 posts
Lankou
#3 h  Apr 4, 2025, 12:06 PM • 1 Y
Y by Cobedangiu
$(\frac{3}{2}-\frac{2}{3}x^2)^n =\sum_{k=0}^n {n\choose k} \cdot \left(\frac{3}{2}\right)^k \left(-\frac{2x^2}{3}\right)^{n-k}$
The coefficient is an integer when $n-k=k$
Coefficient$= {n\choose \frac{n}{2}}$
This post has been edited 1 time. Last edited by Lankou, Apr 4, 2025, 12:11 PM
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Cobedangiu
55 posts
Cobedangiu
#4 h  Apr 5, 2025, 9:45 AM
Y by
Lankou wrote:
$(\frac{3}{2}-\frac{2}{3}x^2)^n =\sum_{k=0}^n {n\choose k} \cdot \left(\frac{3}{2}\right)^k \left(-\frac{2x^2}{3}\right)^{n-k}$
The coefficient is an integer when $n-k=k$
Coefficient$= {n\choose \frac{n}{2}}$

integer coefficients? ${n\choose \frac{n}{2}}$ not integer
This post has been edited 1 time. Last edited by Cobedangiu, Apr 5, 2025, 9:46 AM
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Lankou
1389 posts
Lankou
#5 h  Apr 5, 2025, 11:40 AM • 1 Y
Y by Cobedangiu
${n\choose \frac{n}{2}}$ always an integer
By the way it should be $(-1)^{\frac{n}{2}}{n\choose \frac{n}{2}}$
This post has been edited 3 times. Last edited by Lankou, Apr 5, 2025, 1:29 PM
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