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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
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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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Aime type Geo
ehuseyinyigit   4
N 39 minutes ago by ehuseyinyigit
Source: Turkish First Round 2024
In a scalene triangle $ABC$, let $M$ be the midpoint of side $BC$. Let the line perpendicular to $AC$ at point $C$ intersect $AM$ at $N$. If $(BMN)$ is tangent to $AB$ at $B$, find $AB/MA$.
4 replies
ehuseyinyigit
Monday at 9:04 PM
ehuseyinyigit
39 minutes ago
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n variables with n-gon sides
mihaig   1
N 43 minutes ago by mihaig
Source: Own
Let $n\geq3$ and let $a_1,a_2,\ldots, a_n\geq0$ be reals such that $\sum_{i=1}^{n}{\frac{1}{2a_i+n-2}}=1.$
Prove
$$\frac{24}{(n-1)(n-2)}\cdot\sum_{1\leq i<j<k\leq n}{a_ia_ja_k}\geq3\sum_{i=1}^{n}{a_i}+n.$$
1 reply
mihaig
Apr 25, 2025
mihaig
43 minutes ago
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Geometry
gggzul   5
N an hour ago by nabodorbuco2
In trapezoid $ABCD$ segments $AB$ and $CD$ are parallel. Angle bisectors of $\angle A$ and $\angle C$ meet at $P$. Angle bisectors of $\angle B$ and $\angle D$ meet at $Q$. Prove that $ABPQ$ is cyclic
5 replies
gggzul
Yesterday at 8:22 AM
nabodorbuco2
an hour ago
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Inspired by lgx57
sqing   1
N an hour ago by sqing
Source: Own
Let $ a,b>0, a^4+ab+b^4=60 $. Prove that
$$ 2\sqrt{15} \leq a^2+ab+b^2 \leq \frac{3(\sqrt{481}-1)}{4}$$$$\frac{\sqrt{481}-1}{4}\leq a^2-ab+b^2 \leq 2\sqrt{15} $$Let $ a,b>0, a^4-ab+b^4=60 $. Prove that
$$ 2\sqrt{15} \leq a^2+ab+b^2 \leq \frac{3(\sqrt{481}+1)}{4}$$$$ 5<a^2-ab+b^2 \leq 2\sqrt{15} $$
1 reply
sqing
an hour ago
sqing
an hour ago
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Geometry
Lukariman   4
N an hour ago by lbh_qys
Given circle (O) and point P outside (O). From P draw tangents PA and PB to (O) with contact points A, B. On the opposite ray of ray BP, take point M. The circle circumscribing triangle APM intersects (O) at the second point D. Let H be the projection of B on AM. Prove that $\angle HDM$ = 2∠AMP.
4 replies
Lukariman
Yesterday at 12:43 PM
lbh_qys
an hour ago
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Number Theory
Foxellar   0
2 hours ago
It is known that for all positive integers $k$,
\[ 1^2 + 2^2 + 3^2 + \ldots + k^2 = \frac{k(k + 1)(2k + 1)}{6} \]Find the smallest positive integer $k$ such that $1^2 + 2^2 + 3^2 + \ldots + k^2$ is divisible by 200.
0 replies
Foxellar
2 hours ago
0 replies
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Need help with barycentric
Sadigly   0
2 hours ago
Hi,is there a good handout/book that explains barycentric,other than EGMO?
0 replies
Sadigly
2 hours ago
0 replies
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Find min and max
lgx57   0
3 hours ago
Source: Own
$x_1,x_2, \cdots ,x_n\ge 0$,$\displaystyle\sum_{i=1}^n x_i=m$. $k_1,k_2,\cdots,k_n >0$. Find min and max of
$$\sum_{i=1}^n(k_i\prod_{j=1}^i x_j)$$
0 replies
lgx57
3 hours ago
0 replies
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Find min
lgx57   0
3 hours ago
Source: Own
$a,b>0$, $a^4+ab+b^4=60$. Find min of
$$4a^2-ab+4b^2$$
$a,b>0$, $a^4-ab+b^4=60$. Find min of
$$4a^2-ab+4b^2$$
0 replies
lgx57
3 hours ago
0 replies
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Polynomial Minimization
ReticulatedPython   2
N 3 hours ago by lgx57
Consider the polynomial $$p(x)=x^{n+1}-x^{n}$$, where $x, n \in \mathbb{R+}.$

(a) Prove that the minimum value of $p(x)$ always occurs at $x=\frac{n}{n+1}.$
2 replies
ReticulatedPython
Yesterday at 5:07 PM
lgx57
3 hours ago
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III Lusophon Mathematical Olympiad 2013 - Problem 5
DavidAndrade   2
N 4 hours ago by KTYC
Find all the numbers of $5$ non-zero digits such that deleting consecutively the digit of the left, in each step, we obtain a divisor of the previous number.
2 replies
DavidAndrade
Aug 12, 2013
KTYC
4 hours ago
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Maximum number of terms in the sequence
orl   11
N 4 hours ago by navier3072
Source: IMO LongList, Vietnam 1, IMO 1977, Day 1, Problem 2
In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.
11 replies
orl
Nov 12, 2005
navier3072
4 hours ago
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Compilation of functions problems
Saucepan_man02   0
5 hours ago
Could anyone post some handout/compilation of problems related to functions (difficulty similar to AIME/ARML/HMMT etc)?

Thanks..
0 replies
Saucepan_man02
5 hours ago
0 replies
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Nice Number Theory Question (Inspired by AIME)
MathRook7817   5
N Today at 4:13 AM by jasperE3
Let $n$ be a positive integer. Find the smallest value of $n$ such that:

$2^n + 3^n - n$ is divisible by $216$.

5 replies
MathRook7817
Today at 1:57 AM
jasperE3
Today at 4:13 AM
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BABBAGE'S THEOREM EXTENSION
Mathgloggers   4
N Yesterday at 7:11 AM by Mathgloggers
A few days ago I came across. this interesting result is someone interested in proving this.

$\boxed{\sum_{k=1}^{p-1} \frac{1}{k} \equiv \sum_{k=p+1}^{2p-1} \frac{1}{k} \equiv \sum_{k=2p+1}^{3p-1}\frac{1}{k} \equiv.....\sum_{k=p(p-1)+1}^{p^2-1}\frac{1}{k} \equiv 0(mod p^2)}$
4 replies
Mathgloggers
Apr 29, 2025
Mathgloggers
Yesterday at 7:11 AM
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BABBAGE'S THEOREM EXTENSION
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Reveal topic tags
number theory
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Mathgloggers
82 posts
Mathgloggers
#1 h  Apr 29, 2025, 12:18 PM • 1 Y
Y by orange0707
A few days ago I came across. this interesting result is someone interested in proving this.

$\boxed{\sum_{k=1}^{p-1} \frac{1}{k} \equiv \sum_{k=p+1}^{2p-1} \frac{1}{k} \equiv \sum_{k=2p+1}^{3p-1}\frac{1}{k} \equiv.....\sum_{k=p(p-1)+1}^{p^2-1}\frac{1}{k} \equiv 0(mod p^2)}$
This post has been edited 1 time. Last edited by Mathgloggers, Apr 29, 2025, 12:22 PM
Reason: D
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orange0707
2 posts
orange0707
#2 h  Yesterday at 3:46 AM • 1 Y
Y by Mathgloggers
Is this an extension of Wolstenholme's theorem?
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Mathgloggers
82 posts
Mathgloggers
#3 h  Yesterday at 4:18 AM
Y by
Yes a sort of because that connects with $\frac{1}{k} (mod p^2) $ congruences but this also follows from Babbages theorem which is:
$ \binom{aP}{bP} \equiv 0 (mod p) $ which act a sort of proof for Wolstenholme's theorem.
So when I was working I saw the congruences of of the other part because at $\equiv 0 (mod p) $ modulo inverse does not exist and I came up with this.
This post has been edited 1 time. Last edited by Mathgloggers, Yesterday at 4:19 AM
Reason: m
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orange0707
2 posts
orange0707
#4 h  Yesterday at 6:50 AM • 1 Y
Y by Mathgloggers
This is really an interesting idea. When I saw your conclusion, I happened to be studying the method of constructing polynomials and then proving Wolstenholme's theorem by using Lagrange's theorem and Fermat's little theorem, and found that it could also be used here. Additionally, may I ask when new users can upload images?Can complex mathematical formulas be directly entered using a LaTeX editor?
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Mathgloggers
82 posts
Mathgloggers
#5 h  Yesterday at 7:11 AM
Y by
Yes you can upload images through the pin image icon the extreme right of head bar where you send solution and yes you can upload complex mathematical formula .
I would recommend you the "LATEX FORUMS " on aops you can check that to form your formulas
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