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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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Inequalities
sqing   4
N 21 minutes ago by DAVROS
Let $a,b,c >2 $ and $ ab+bc+ca \leq 75.$ Show that
$$\frac{1}{a-2}+\frac{1}{b-2}+\frac{1}{c-2}\geq 1$$Let $a,b,c >2 $ and $ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq \frac{6}{7}.$ Show that
$$\frac{1}{a-2}+\frac{1}{b-2}+\frac{1}{c-2}\geq 2$$
4 replies
sqing
Yesterday at 11:31 AM
DAVROS
21 minutes ago
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Cheesy's math casino and probability
pithon_with_an_i   0
an hour ago
Source: Revenge JOM 2025 Problem 4, Revenge JOMSL 2025 C3
There are $p$ people are playing a game at Cheesy's math casino, where $p$ is a prime number. Let $n$ be a positive integer. A subset of length $s$ from the set of integers from $1$ to $n$ inclusive is randomly chosen, with an equal probability ($s \leq n$ and is fixed). The winner of Cheesy's game is person $i$, if the sum of the chosen numbers are congruent to $i \pmod p$ for $0 \leq i \leq p-1$.
For each $n$, find all values of $s$ such that no person will sue Cheesy for creating unfair games (i.e. all the winning outcomes are equally likely).

(Proposed by Jaydon Chieng, Yeoh Teck En)

Remark
0 replies
pithon_with_an_i
an hour ago
0 replies
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Partitioning coprime integers to arithmetic sequences
sevket12   4
N an hour ago by bochidd
Source: 2025 Turkey EGMO TST P3
For a positive integer $n$, let $S_n$ be the set of positive integers that do not exceed $n$ and are coprime to $n$. Define $f(n)$ as the smallest positive integer that allows $S_n$ to be partitioned into $f(n)$ disjoint subsets, each forming an arithmetic progression.

Prove that there exist infinitely many pairs $(a, b)$ satisfying $a, b > 2025$, $a \mid b$, and $f(a) \nmid f(b)$.
4 replies
sevket12
Feb 8, 2025
bochidd
an hour ago
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Coaxal Circles
fattypiggy123   30
N an hour ago by Ilikeminecraft
Source: China TSTST Test 2 Day 1 Q3
Let $ABCD$ be a quadrilateral and let $l$ be a line. Let $l$ intersect the lines $AB,CD,BC,DA,AC,BD$ at points $X,X',Y,Y',Z,Z'$ respectively. Given that these six points on $l$ are in the order $X,Y,Z,X',Y',Z'$, show that the circles with diameter $XX',YY',ZZ'$ are coaxal.
30 replies
fattypiggy123
Mar 13, 2017
Ilikeminecraft
an hour ago
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Weird n-variable extremum problem
pithon_with_an_i   0
an hour ago
Source: Revenge JOM 2025 Problem 3, Revenge JOMSL 2025 A4
Let $n$ be a positive integer greater or equal to $2$ and let $a_1$, $a_2$, ..., $a_n$ be a sequence of non-negative real numbers. Find the maximum value of $3(a_1 + a_2 + \cdots + a_n) - (a_1^2 + a_2^2 + \cdots + a_n^2) - a_1a_2 \cdots a_n$ in terms of $n$.

(Proposed by Cheng You Seng)
0 replies
pithon_with_an_i
an hour ago
0 replies
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Inequality with a^2 + b^2 + c^2 + abc = 4
Nguyenhuyen_AG   1
N an hour ago by TNKT
Let $a,\,b,\,c$ positive real numbers such that $a^2+b^2+c^2+abc=4.$ Prove that
\[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+(k+5)(a+b+c) \geqslant 3(k+6),\]for all $0 \leqslant k \leqslant k_0 = \frac{3\big(\sqrt[3]{2}+\sqrt[3]{4}\big)-7}{2}.$
hide
1 reply
Nguyenhuyen_AG
Oct 1, 2020
TNKT
an hour ago
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2025 IMO TEAMS
Oksutok   1
N an hour ago by BR1F1SZ
Good Luck in Sunshine Coast, Australia
1 reply
Oksutok
2 hours ago
BR1F1SZ
an hour ago
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polonomials
Ducksohappi   2
N an hour ago by Ducksohappi
$P\in \mathbb{R}[x] $ with even-degree
Prove that there is a non-negative integer k such that
$Q_k(x)=P(x)+P(x+1)+...+P(x+k)$
has no real root
2 replies
Ducksohappi
May 8, 2025
Ducksohappi
an hour ago
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Problem 7
SlovEcience   1
N an hour ago by GreekIdiot
Consider the sequence \((u_n)\) defined by \(u_0 = 5\) and
\[ u_{n+1} = \frac{1}{2}u_n^2 - 4 \quad \text{for all } n \in \mathbb{N}. \]a) Prove that there exist infinitely many positive integers \(n\) such that \(u_n > 2020n\).

b) Compute
\[ \lim_{n \to \infty} \frac{2u_{n+1}}{u_0u_1\cdots u_n}. \]
1 reply
SlovEcience
3 hours ago
GreekIdiot
an hour ago
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Three lines meet at one point
TUAN2k8   0
2 hours ago
Source: Own
Let $ABC$ be an acute triangle incribed in a circle $\omega$.Let $M$ be the midpoint of $BC$.Let $AD,BE$ and $CF$ be altitudes from $A,B$ and $C$ of triangle $ABC$, respectively, and let them intersect at $H$.Let $K$ be the intersection point of tangents to the circle $\omega$ at points $B,C$.Prove that $MH,KD$ and $EF$ are concurrent.
0 replies
TUAN2k8
2 hours ago
0 replies
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Combinatorics Problem
P.J   8
N 2 hours ago by MITDragon
Source: Mexican Mathematical Olympiad Problems Book
Calculate the sum of 1 x 1000 + 2 x 999 + ... + 999 x 2 + 1000 x 1
8 replies
P.J
Dec 28, 2024
MITDragon
2 hours ago
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Trig Identity
gauss202   1
N 2 hours ago by Lankou
Simplify $\dfrac{1-\cos \theta + \sin \theta}{\sqrt{1 - \cos \theta + \sin \theta - \sin \theta \cos \theta}}$
1 reply
gauss202
2 hours ago
Lankou
2 hours ago
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Complex Number Geometry
gauss202   0
2 hours ago
Describe the locus of complex numbers, $z$, such that $\arg \left(\dfrac{z+i}{z-1} \right) = \dfrac{\pi}{4}$.
0 replies
gauss202
2 hours ago
0 replies
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Trunk of cone
soruz   1
N 5 hours ago by Mathzeus1024
One hemisphere is putting a truncated cone, with the base circles hemisphere. How height should have truncated cone as its lateral area to be minimal side?
1 reply
soruz
May 6, 2015
Mathzeus1024
5 hours ago
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An interesting sum
ThePirate1   6
N Dec 1, 2021 by fungarwai
Let ,
$a=-\sqrt{3}+\sqrt{5}+\sqrt{7} $
$ b= \sqrt{3}-\sqrt{5}+\sqrt{7} ,$
$c= \sqrt{3}+\sqrt{5}-\sqrt{7} $
Then what is, $\sum_\text{cyc} \frac{a^4}{(a-b)(a-c)}$ ?
6 replies
ThePirate1
Nov 29, 2021
fungarwai
Dec 1, 2021
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An interesting sum
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Reveal topic tags
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ThePirate1
122 posts
ThePirate1
#1 h  Nov 29, 2021, 3:37 PM
Y by
Let ,
$a=-\sqrt{3}+\sqrt{5}+\sqrt{7} $
$ b= \sqrt{3}-\sqrt{5}+\sqrt{7} ,$
$c= \sqrt{3}+\sqrt{5}-\sqrt{7} $
Then what is, $\sum_\text{cyc} \frac{a^4}{(a-b)(a-c)}$ ?
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fungarwai
865 posts
fungarwai
#2 h  Nov 30, 2021, 11:32 AM • 1 Y
Y by teomihai
Summation with polynomial roots (Lagrange polynomial type)

$\sigma_1=a+b+c=\sqrt{3}+\sqrt{5}+\sqrt{7}$
$ab=7-(\sqrt{3}-\sqrt{5})^2=7-3-5+2\sqrt{15}=-1+2\sqrt{15}$
$ac=5-3-7+2\sqrt{21}=-5+2\sqrt{21}$
$bc=3-5-7+2\sqrt{35}=-9+2\sqrt{35}$
$\sigma_2=-15+2\sqrt{15}+2\sqrt{21}+2\sqrt{35}$

$S_{4,3}=\sigma_1^2-\sigma_2=3+5+7+15=30$
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ThePirate1
122 posts
ThePirate1
#3 h  Nov 30, 2021, 11:59 AM
Y by
Wow, I solved this using plain factorization techniques, but I am stumped by this method! I visited your blog too, I loved the techniques!
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BackToSchool
1640 posts
BackToSchool
#4 h  Nov 30, 2021, 1:31 PM
Y by
It should be some elementary method with some factorization tricks other than this "super advanced" approach.
It's good to know it, but definitely it's beyond the high school level.
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fungarwai
865 posts
fungarwai
#5 h  Nov 30, 2021, 3:03 PM
Y by
The proof starts from $x_i^{s-m+1}\prod_{j=1\atop i\neq j}^m (x_i-x_j)=x_i^s+\sum_{k=2}^m (-1)^{k-1}(k-1)\sigma_k x_i^{s-k}$
In this case, $a^2 (a-b)(a-c)=a^4-(ab+ac+bc)a^2+2(abc)a$
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BackToSchool
1640 posts
BackToSchool
#6 h  Nov 30, 2021, 3:17 PM
Y by
fungarwai wrote:
The proof starts from $x_i^{s-m+1}\prod_{j=1\atop i\neq j}^m (x_i-x_j)=x_i^s+\sum_{k=2}^m (-1)^{k-1}(k-1)\sigma_k x_i^{s-k}$
In this case, $a^2 (a-b)(a-c)=a^4-(ab+ac+bc)a^2+2(abc)a$

Yes. It is important to understand how these formulae are obtained.
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fungarwai
865 posts
fungarwai
#7 h  Dec 1, 2021, 1:26 AM
Y by
I focused on the mathematical induction and forgot to prove the first identity at all.

I add Lemma1 on my blog now.

Lemma1
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