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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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IMO96/2 [the lines AP, BD, CE meet at a point]
Arne   47
N 29 minutes ago by Bridgeon
Source: IMO 1996 problem 2, IMO Shortlist 1996, G2
Let $ P$ be a point inside a triangle $ ABC$ such that
\[ \angle APB - \angle ACB = \angle APC - \angle ABC. \]
Let $ D$, $ E$ be the incenters of triangles $ APB$, $ APC$, respectively. Show that the lines $ AP$, $ BD$, $ CE$ meet at a point.
47 replies
Arne
Sep 30, 2003
Bridgeon
29 minutes ago
J
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A sharp one with 3 var (3)
mihaig   4
N 41 minutes ago by aaravdodhia
Source: Own
Let $a,b,c\geq0$ satisfying
$$\left(a+b+c-2\right)^2+8\leq3\left(ab+bc+ca\right).$$Prove
$$a^2+b^2+c^2+5abc\geq8.$$
4 replies
mihaig
Yesterday at 5:17 PM
aaravdodhia
41 minutes ago
J
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Cup of Combinatorics
M11100111001Y1R   1
N an hour ago by Davdav1232
Source: Iran TST 2025 Test 4 Problem 2
There are \( n \) cups labeled \( 1, 2, \dots, n \), where the \( i \)-th cup has capacity \( i \) liters. In total, there are \( n \) liters of water distributed among these cups such that each cup contains an integer amount of water. In each step, we may transfer water from one cup to another. The process continues until either the source cup becomes empty or the destination cup becomes full.

$a)$ Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most \( \frac{4n}{3} \) steps.

$b)$ Prove that in at most \( \frac{5n}{3} \) steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
1 reply
M11100111001Y1R
Yesterday at 7:24 AM
Davdav1232
an hour ago
J
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Bulgaria National Olympiad 1996
Jjesus   7
N an hour ago by reni_wee
Find all prime numbers $p,q$ for which $pq$ divides $(5^p-2^p)(5^q-2^q)$.
7 replies
Jjesus
Jun 10, 2020
reni_wee
an hour ago
J
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Can't be power of 2
shobber   31
N an hour ago by LeYohan
Source: APMO 1998
Show that for any positive integers $a$ and $b$, $(36a+b)(a+36b)$ cannot be a power of $2$.
31 replies
shobber
Mar 17, 2006
LeYohan
an hour ago
J
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Brilliant Problem
M11100111001Y1R   4
N an hour ago by IAmTheHazard
Source: Iran TST 2025 Test 3 Problem 3
Find all sequences \( (a_n) \) of natural numbers such that for every pair of natural numbers \( r \) and \( s \), the following inequality holds:
\[ \frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2 \]
4 replies
M11100111001Y1R
Yesterday at 7:28 AM
IAmTheHazard
an hour ago
J
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Own made functional equation
Primeniyazidayi   1
N 2 hours ago by Primeniyazidayi
Source: own(probably)
Find all functions $f:R \rightarrow R$ such that $xf(x^2+2f(y)-yf(x))=f(x)^3-f(y)(f(x^2)-2f(x))$ for all $x,y \in \mathbb{R}$
1 reply
Primeniyazidayi
May 26, 2025
Primeniyazidayi
2 hours ago
J
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not fun equation
DottedCaculator   13
N 2 hours ago by Adywastaken
Source: USA TST 2024/6
Find all functions $f\colon\mathbb R\to\mathbb R$ such that for all real numbers $x$ and $y$,
\[f(xf(y))+f(y)=f(x+y)+f(xy).\]
Milan Haiman
13 replies
DottedCaculator
Jan 15, 2024
Adywastaken
2 hours ago
J
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Serbian selection contest for the IMO 2025 - P6
OgnjenTesic   12
N 3 hours ago by atdaotlohbh
Source: Serbian selection contest for the IMO 2025
For an $n \times n$ table filled with natural numbers, we say it is a divisor table if:
- the numbers in the $i$-th row are exactly all the divisors of some natural number $r_i$,
- the numbers in the $j$-th column are exactly all the divisors of some natural number $c_j$,
- $r_i \ne r_j$ for every $i \ne j$.

A prime number $p$ is given. Determine the smallest natural number $n$, divisible by $p$, such that there exists an $n \times n$ divisor table, or prove that such $n$ does not exist.

Proposed by Pavle Martinović
12 replies
OgnjenTesic
May 22, 2025
atdaotlohbh
3 hours ago
J
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Geometry with fix circle
falantrng   33
N 3 hours ago by zuat.e
Source: RMM 2018 Problem 6
Fix a circle $\Gamma$, a line $\ell$ to tangent $\Gamma$, and another circle $\Omega$ disjoint from $\ell$ such that $\Gamma$ and $\Omega$ lie on opposite sides of $\ell$. The tangents to $\Gamma$ from a variable point $X$ on $\Omega$ meet $\ell$ at $Y$ and $Z$. Prove that, as $X$ varies over $\Omega$, the circumcircle of $XYZ$ is tangent to two fixed circles.
33 replies
falantrng
Feb 25, 2018
zuat.e
3 hours ago
J
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USAMO 2001 Problem 2
MithsApprentice   54
N 3 hours ago by lpieleanu
Let $ABC$ be a triangle and let $\omega$ be its incircle. Denote by $D_1$ and $E_1$ the points where $\omega$ is tangent to sides $BC$ and $AC$, respectively. Denote by $D_2$ and $E_2$ the points on sides $BC$ and $AC$, respectively, such that $CD_2=BD_1$ and $CE_2=AE_1$, and denote by $P$ the point of intersection of segments $AD_2$ and $BE_2$. Circle $\omega$ intersects segment $AD_2$ at two points, the closer of which to the vertex $A$ is denoted by $Q$. Prove that $AQ=D_2P$.
54 replies
MithsApprentice
Sep 30, 2005
lpieleanu
3 hours ago
J
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German-Style System of Equations
Primeniyazidayi   1
N 4 hours ago by Primeniyazidayi
Source: German MO 2025 11/12 Day 1 P1
Solve the system of equations in $\mathbb{R}$

\begin{align*} \frac{a}{c} &= b-\sqrt{b}+c \\ \sqrt{\frac{a}{c}} &= \sqrt{b}+1 \\ \sqrt[4]{\frac{a}{c}} &=\sqrt[3]{b}-1 \end{align*}
1 reply
Primeniyazidayi
4 hours ago
Primeniyazidayi
4 hours ago
J
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gcd nt from switzerland
AshAuktober   5
N 4 hours ago by Siddharthmaybe
Source: Swiss 2025 Second Round
Let $a, b$ be positive integers. Prove that the expression
\[\frac{\gcd(a+b,ab)}{\gcd(a,b)}\]is always a positive integer, and determine all possible values it can take.
5 replies
AshAuktober
5 hours ago
Siddharthmaybe
4 hours ago
J
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Shortlist 2017/G1
fastlikearabbit   92
N 4 hours ago by Ilikeminecraft
Source: Shortlist 2017
Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.
92 replies
fastlikearabbit
Jul 10, 2018
Ilikeminecraft
4 hours ago
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J
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Number of permutative
nmtruong1986   8
N Jun 7, 2020 by Brudder
Source: When solving a problem
Find out the number of permutations of $ n$ first positive integer such that they contain no block $ (i,i + 1)$ or $ (i + 1,i)$ for all $ i$.
8 replies
nmtruong1986
Dec 14, 2005
Brudder
Jun 7, 2020
J
Number of permutative
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Reveal topic tags
calculus
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combinatorics unsolved
combinatorics
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G H BBookmark kLocked kLocked NReply
Source: When solving a problem
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nmtruong1986
186 posts
nmtruong1986
#1 h  Dec 14, 2005, 6:22 PM • 2 Y
Y by Adventure10, Mango247
Find out the number of permutations of $ n$ first positive integer such that they contain no block $ (i,i + 1)$ or $ (i + 1,i)$ for all $ i$.
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maxal
629 posts
maxal
#2 h  Jan 29, 2006, 10:38 AM • 1 Y
Y by Adventure10
By inclusion-exclusion it is
\[ n! + \sum_{k=1}^n (-1)^k \sum_{t=1}^k {k-1\choose t-1}{n-k\choose t} 2^t (n-k)! \]
See also sequence A002464 for recurrent formula and other references.
This post has been edited 1 time. Last edited by maxal, Aug 9, 2013, 1:45 PM
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keira_khtn
485 posts
keira_khtn
#3 h  Jan 30, 2006, 3:25 AM • 2 Y
Y by Adventure10, Mango247
maxal wrote:
By inclusion-exclusion it is
\[ n! + \sum_{k=1}^n (-1)^k \sum_{t=1}^k {k-1\choose t-1}{n-k\choose t} 2^t (n-k)! \]
.
Could you explain me how you got it???
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maxal
629 posts
maxal
#4 h  Jan 30, 2006, 4:27 AM • 2 Y
Y by Adventure10, Mango247
k stands for the number of pairs (i,i+1) that are adjacent in the permutation, such pairs form chains (e.g., if i,i+1 are adjacent and i+1,i+2 are adjacent then i,i+1,i+2 is a chain in the permutation), and t stands for the number of such chains. Summands are the number of permutations with k adjacent pairs forming t chains. In particular,
${k-1\choose t-1}$ is the number of distributions of k adacent pairs among t chains such that each chain contains at least one pair;
${n-k\choose t}$ is the number of ways to select t chains of total length k+t out of elements $1,2,\dots,n$;
$2^t$ is the number of ways to orient the chains in the permutation (e.g., a chain i,i+1,i+2 can appear in the permutation as i,i+1,i+2 or as i+2,i+1,i);
$(n-k)!$ is the number of permutations of n-k-t elements not included into the chains and t chains (chains here can be viewed as single elements, hence, it's simply the number of permutations of n-k-t+t=n-k elements).
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keira_khtn
485 posts
keira_khtn
#5 h  Jan 30, 2006, 11:07 AM • 2 Y
Y by Adventure10, Mango247
As I understand now:$(^{k-1}_{t-1})$ is number of oisitive intergral roots of the Diophant equation:$x_1+x_2+...+x_t=k$.
Is it right??
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Bluesea
59 posts
Bluesea
#6 h  Jan 30, 2006, 12:02 PM • 2 Y
Y by Adventure10, Mango247
it's the way to divide k pairs into t chain.I use alsi this way to solve this pro:
http://www.mathlinks.ro/Forum/viewtopic.php?t=4110
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maxal
629 posts
maxal
#7 h  Jan 30, 2006, 12:10 PM • 2 Y
Y by Adventure10, Mango247
keira_khtn wrote:
As I understand now:$(^{k-1}_{t-1})$ is number of oisitive intergral roots of the Diophant equation:$x_1+x_2+...+x_t=k$.
Is it right??
That's correct.
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Stormersyle
2786 posts
Stormersyle
#8 h  Jun 7, 2020, 7:25 PM • 3 Y
Y by fidgetboss_4000, RadiantCheddar, IAmTheHazard
bruh $ $
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Brudder
416 posts
Brudder
#9 h  Jun 7, 2020, 8:43 PM • 2 Y
Y by fidgetboss_4000, IAmTheHazard
wish id seen this like 2 days ago
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