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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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Combi Proof Mathematical Games/Algorithm
CatalanThinker   0
4 minutes ago
Source: Olympiad_Combinatorics_by_Pranav_A_Sriram
3. [Russia 1961]
Real numbers are written in an $m \times n$ table. It is permissible to reverse the signs of all the numbers in any row or column. Prove that after a number of these operations, we can make the sum of the numbers along each line (row or column) nonnegative.
0 replies
+1 w
CatalanThinker
4 minutes ago
0 replies
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Unexpecredly Quick-Solve Inequality
Primeniyazidayi   1
N 12 minutes ago by Ritwin
Source: German MO 2025,Round 4,Grade 11/12 Day 2 P1
If $a, b, c>0$, prove that $$\frac{a^5}{b^2}+\frac{b}{c}+\frac{c^3}{a^2}>2a$$
1 reply
Primeniyazidayi
24 minutes ago
Ritwin
12 minutes ago
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Easy Taiwanese Geometry
USJL   14
N 35 minutes ago by Want-to-study-in-NTU-MATH
Source: 2024 Taiwan Mathematics Olympiad
Suppose $O$ is the circumcenter of $\Delta ABC$, and $E, F$ are points on segments $CA$ and $AB$ respectively with $E, F \neq A$. Let $P$ be a point such that $PB = PF$ and $PC = PE$.
Let $OP$ intersect $CA$ and $AB$ at points $Q$ and $R$ respectively. Let the line passing through $P$ and perpendicular to $EF$ intersect $CA$ and $AB$ at points $S$ and $T$ respectively. Prove that points $Q, R, S$, and $T$ are concyclic.

Proposed by Li4 and usjl
14 replies
USJL
Jan 31, 2024
Want-to-study-in-NTU-MATH
35 minutes ago
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Problem 7
SlovEcience   6
N 44 minutes ago by Li0nking
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
\[ \lim_{n \to \infty} \frac{2u_{n+1}}{u_0u_1\cdots u_n}. \]
6 replies
SlovEcience
May 14, 2025
Li0nking
44 minutes ago
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Strange circles in an orthocenter config
VideoCake   1
N 44 minutes ago by KrazyNumberMan
Source: 2025 German MO, Round 4, Grade 12, P3
Let \(\overline{AD}\) and \(\overline{BE}\) be altitudes in an acute triangle \(ABC\) which meet at \(H\). Suppose that \(DE\) meets the circumcircle of \(ABC\) at \(P\) and \(Q\) such that \(P\) lies on the shorter arc of \(BC\) and \(Q\) lies on the shorter arc of \(CA\). Let \(AQ\) and \(BE\) meet at \(S\). Show that the circumcircles of \(BPE\) and \(QHS\) and the line \(PH\) concur.
1 reply
VideoCake
Monday at 5:10 PM
KrazyNumberMan
44 minutes ago
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Inspired by Adhyayan Jana
sqing   0
an hour ago
Source: Own
Let $a,b,c,d>0,a^2 + d^2+ad = b^2 + c^2 $ aand $ a^2 + b^2 = c^2 + d^2+cd$ Prove that $$ \frac{ab+cd}{ad+bc} =1$$
0 replies
sqing
an hour ago
0 replies
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Impossible Infinite Sequence
Rijul saini   4
N an hour ago by guptaamitu1
Source: India IMOTC 2024 Day 1 Problem 3
Let $P(x) \in \mathbb{Q}[x]$ be a polynomial with rational coefficients and degree $d\ge 2$. Prove there is no infinite sequence $a_0, a_1, \ldots$ of rational numbers such that $P(a_i)=a_{i-1}+i$ for all $i\ge 1$.

Proposed by Pranjal Srivastava and Rohan Goyal
4 replies
Rijul saini
May 31, 2024
guptaamitu1
an hour ago
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Functional xf(x+f(y))=(y-x)f(f(x)) for all reals x,y
cretanman   59
N an hour ago by math-olympiad-clown
Source: BMO 2023 Problem 1
Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$,
\[xf(x+f(y))=(y-x)f(f(x)).\]
Proposed by Nikola Velov, Macedonia
59 replies
cretanman
May 10, 2023
math-olympiad-clown
an hour ago
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Inspired by Adhyayan Jana
sqing   2
N 2 hours ago by sqing
Source: Own
Let $a,b,c,d>0,a^2 + d^2-ad = (b + c)^2 $ aand $ a^2 +c^2 = b^2 + d^2.$ Prove that$$ \frac{ab+cd}{ad+bc} \geq \frac{ 4}{5}$$Let $a,b,c,d>0,a^2 + d^2-ad = b^2 + c^2 + bc $ aand $ a^2 +c^2 = b^2 + d^2.$ Prove that$$ \frac{ab+cd}{ad+bc} \geq \frac{\sqrt 3}{2}$$Let $a,b,c,d>0,a^2 + d^2 - ad = b^2 + c^2 + bc $ aand $ a^2 + b^2 = c^2 + d^2.$ Prove that $$ \frac{ab+cd}{ad+bc} =\frac{\sqrt 3}{2}$$
2 replies
sqing
3 hours ago
sqing
2 hours ago
J
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Concurrent lines
syk0526   28
N 2 hours ago by alexanderchew
Source: North Korea Team Selection Test 2013 #1
The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $ BC, CA, AB$ at $ A_1 , B_1 , C_1 $ respectively. The line $AI$ meets the circumcircle of $ABC$ at $A_2 $. The line $B_1 C_1 $ meets the line $BC$ at $A_3 $ and the line $A_2 A_3 $ meets the circumcircle of $ABC$ at $A_4 (\ne A_2 ) $. Define $B_4 , C_4 $ similarly. Prove that the lines $ AA_4 , BB_4 , CC_4 $ are concurrent.
28 replies
syk0526
May 17, 2014
alexanderchew
2 hours ago
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Equal angles (a very old problem)
April   56
N 2 hours ago by Ilikeminecraft
Source: ISL 2007, G3, VAIMO 2008, P5
The diagonals of a trapezoid $ ABCD$ intersect at point $ P$. Point $ Q$ lies between the parallel lines $ BC$ and $ AD$ such that $ \angle AQD = \angle CQB$, and line $ CD$ separates points $ P$ and $ Q$. Prove that $ \angle BQP = \angle DAQ$.

Author: Vyacheslav Yasinskiy, Ukraine
56 replies
April
Jul 13, 2008
Ilikeminecraft
2 hours ago
J
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Inspired by Adhyayan Jana
sqing   0
3 hours ago
Source: Own
Let $a,b,c,d>0,a^2 + d^2-ad = b^2 + c^2 $ aand $ a^2 +b^2 =c^2 + d^2.$ Prove that$$ \frac{ab+cd}{ad+bc} \leq \frac{2\sqrt 2}{3}$$Let $a,b,c,d>0,a^2 + d^2 = b^2 + c^2 + bc $ aand $ a^2 + b^2 = c^2 + d^2.$ Prove that $$ \frac{ab+cd}{ad+bc} \geq \frac{2\sqrt 6}{5}$$
0 replies
sqing
3 hours ago
0 replies
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2025 Zhejiang Women's Mathematical Olympiad ,Q4
sqing   2
N 4 hours ago by sqing
Source: China
Let $ a_1, a_2,\dots ,a_n\geq 0 $ and $ \sum _{i=1}^{n}a^3_i=n $ $(n\geq 3) .$ Prove that $$\sum_{1\le i<j<k\le n} \frac{1}{n-a_ia_ja_k}\leq \frac{n(n-2)}{6}$$
APMO 2012 #5
Inequalities Marathon
2 replies
sqing
Yesterday at 2:31 PM
sqing
4 hours ago
J
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Nice inequality
TUAN2k8   1
N 4 hours ago by sqing
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
$\sum_{1 \le i < j \le n} \frac{x_ix_j}{x_i^2+x_j^2+1} \ge \frac{-n}{6}$
1 reply
TUAN2k8
4 hours ago
sqing
4 hours ago
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Minimal number of divisors
Garfield   7
N May 15, 2025 by Omerking
Source: Serbian junior TST 2nd
Find minimal number of divisors that can number $|2016^m-36^n|$ have,where $m$ and $n$ are natural numbers.
7 replies
Garfield
May 21, 2016
Omerking
May 15, 2025
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Minimal number of divisors
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Reveal topic tags
number theory
JBMO
JBMO TST
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G H BBookmark kLocked kLocked NReply
Source: Serbian junior TST 2nd
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Garfield
243 posts
Garfield
#1 h  May 21, 2016, 7:15 PM • 3 Y
Y by Adventure10, Mango247, AlexCenteno2007
Find minimal number of divisors that can number $|2016^m-36^n|$ have,where $m$ and $n$ are natural numbers.
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tenplusten
1000 posts
tenplusten
#2 h  May 21, 2016, 7:21 PM • 2 Y
Y by Adventure10, Mango247
Hi dude where can I find other Serbian Junior TST problems?
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Garfield
243 posts
Garfield
#3 h  May 21, 2016, 7:26 PM • 2 Y
Y by Adventure10, Mango247
Here http://www.dms.rs/DMS/data/takmicenja/matematika/osnovna/2016/JSMO16.pdf but on Serbian language.
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mihajlon
374 posts
mihajlon
#4 h  May 21, 2016, 7:58 PM • 2 Y
Y by Adventure10, AlexCenteno2007
Murad.Aghazade wrote:
Hi dude where can I find other Serbian Junior TST problems?

You can find here other Serbia JTST's, but there are not many of those...

PS
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MilosMilicev
241 posts
MilosMilicev
#5 h  Oct 15, 2016, 11:10 AM • 2 Y
Y by Adventure10, Mango247
Let x=our number. For m,n>1, we have that 81Ix, 16Ix, 5Ix, it means that x has at least (4+1)(4+1)(1+1)=50 positive divisors, and for m=1, n=2,x=720(30 positive divisors).It is not hard to prove that it is the minimum number...
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IvanMitrovic
22 posts
IvanMitrovic
#6 h  Mar 8, 2020, 1:51 PM
Y by
MilosMilicev wrote:
Let x=our number. For m,n>1, we have that 81Ix, 16Ix, 5Ix, it means that x has at least (4+1)(4+1)(1+1)=50 positive divisors, and for m=1, n=2,x=720(30 positive divisors).It is not hard to prove that it is the minimum number...

You did not check all of the cases
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Namisgood
32 posts
Namisgood
#7 h  Apr 12, 2025, 1:25 PM
Y by
MilosMilicev wrote:
Let x=our number. For m,n>1, we have that 81Ix, 16Ix, 5Ix, it means that x has at least (4+1)(4+1)(1+1)=50 positive divisors, and for m=1, n=2,x=720(30 positive divisors).It is not hard to prove that it is the minimum number...

sorry I am late



how did you conclude 81|x ?
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Omerking
11 posts
Omerking
#8 h  May 15, 2025, 10:52 AM • 1 Y
Y by Nuran2010
$Claim$
Firstly, I claimed the answer is 720 where $m=1$ and $n=2$.

$Proof$
As it says in the problem statement $n\ge2$ . If, $m\ge2$, $|2016^m-36^n|\vdots16$ and $|2016^m-36^n|\vdots3$. Also, both $2016^m$ 's and $36^n$ 's last digits are $6$. So, $|2016^m-36^n|$ is divisible by 5 too. If, we check both $240$ and $480$ we get conclusion. So, the answer is $720$.
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