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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
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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Two midpoints and the circumcenter are collinear.
ricarlos   0
11 minutes ago
Let $ABC$ be a triangle with circumcenter $O$. Let $P$ be a point on the perpendicular bisector of $AB$ (see figure) and $Q$, $R$ be the intersections of the perpendicular bisectors of $AC$ and $BC$, respectively, with $PA$ and $PB$. Prove that the midpoints of $PC$ and $QR$ and the point $O$ are collinear.

0 replies
+1 w
ricarlos
11 minutes ago
0 replies
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Valuable subsets of segments in [1;n]
NO_SQUARES   1
N 25 minutes ago by NO_SQUARES
Source: Russian May TST to IMO 2023; group of candidates P6; group of non-candidates P8
The integer $n \geqslant 2$ is given. Let $A$ be set of all $n(n-1)/2$ segments of real line of type $[i, j]$, where $i$ and $j$ are integers, $1\leqslant i<j\leqslant n$. A subset $B \subset A$ is said to be valuable if the intersection of any two segments from $B$ is either empty, or is a segment of nonzero length belonging to $B$. Find the number of valuable subsets of set $A$.
1 reply
NO_SQUARES
Thursday at 8:34 PM
NO_SQUARES
25 minutes ago
J
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geometry party
pnf   0
27 minutes ago
https://artofproblemsolving.com/community/c4260013_geometry_party
0 replies
1 viewing
pnf
27 minutes ago
0 replies
J
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All heads to tails?
smartvong   0
an hour ago
Source: CEMC Euclid Contest 2025
An equilateral triangle is formed using $n$ rows of coins. There is 1 coin in the first row, 2 coins in the second row, 3 coins in the third row, and so on, up to $n$ coins in the $n$th row. Initially, all of the coins show heads (H). Carley plays a game in which, on each turn, she chooses three mutually adjacent coins and flips these three coins over. To win the game, all of the coins must be showing tails (T) after a sequence of turns. An example game with 4 rows of coins after a sequence of two turns is shown.

IMAGE

Below (a), (b) and (c), you will find instructions about how to refer to these turns in your solutions.

(a) If there are 3 rows of coins, give a sequence of 4 turns that results in a win.

(b) Suppose that there are 4 rows of coins. Determine whether or not there is a sequence of turns that results in a win.

(c) Determine all values of $n$ for which it is possible to win the game starting with $n$ rows of coins.

Note: For a triangle with 4 rows of coins, there are 9 possibilities for the set of three coins that Carley can flip on a given turn. These 9 possibilities are shown as shaded triangles below:

IMAGE

IMAGE

[You should use the names for these moves shown inside the 9 shaded triangles when answering (b). You should adapt this naming convention in a suitable way when answering parts (a) and (c).]
0 replies
smartvong
an hour ago
0 replies
J
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Inequality with three variables
crazyfehmy   14
N an hour ago by TopGbulliedU
Source: Turkey JBMO Team Selection Test 2013, P4
For all positive real numbers $a, b, c$ satisfying $a+b+c=1$, prove that

\[ \frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca \]
14 replies
crazyfehmy
May 31, 2013
TopGbulliedU
an hour ago
J
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Factorial: n!|a^n+1
Nima Ahmadi Pour   66
N an hour ago by cursed_tangent1434
Source: IMO Shortlist 2005 N4, Iran preparation exam
Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property:
\[ n!\mid a^n + 1 \]

Proposed by Carlos Caicedo, Colombia
66 replies
Nima Ahmadi Pour
Apr 24, 2006
cursed_tangent1434
an hour ago
J
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Problems for v_p(n)
xytunghoanh   15
N 2 hours ago by AshAuktober
Hello everyone. I need some easy problems for $v_p(n)$ (use in a problem for junior) to practice. Can anyone share to me?
Thanks :>
15 replies
1 viewing
xytunghoanh
5 hours ago
AshAuktober
2 hours ago
J
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BMO2 1997 Q4
rstather   1
N 2 hours ago by aidan0626
Source: BMO2 1997 Q4
Problem Statement:
The set S= {1/r : r = 1, 2, 3,...} of reciprocals of the positive integers contains arithmetic progressions of various lengths. For instance, 1/20, 1/8, 1/5 is such a progression, of length 3 (and common difference 3/40). Moreover, this is a maximal progression in S of length 3 since it cannot be extended to the left or right within S (−1/40 and 11/40 not
being members of S).
(i) Find a maximal progression in
S of length 1996.
(ii) Is there a maximal progression in
S of length 1997?


For part (i) I constructed the sequence: 1/1996!, 2/1996!, 3/1996!, ..., 1996/1996!. Which has common difference of 1/1996! and because of the nature of the factorial function, each of them are elements in S, and continuing to the left we get 0, which is not in S, and continuing to the right we get 1997/1996! which is not in S because 1997 is prime. Therefore we have found a maximal progression in S of length 1996 as desired.

For part (ii), let A = (4000!)/(2003!). Now consider the sequence: 2004/A, 2005/A, 2006/A, ... , 4000/A. Each is an element of S, by the nature of the factorial function and the common difference is 1/A. Continuing to the left gives 2003/A, 2003 is prime which is not a factor of A, so 2003/A is not an element of S. Continuing to the right gives 4001/A, 4001 is prime which is not a factor of A, so 4001/A is not an element of S. Therefore we have found a maximal progression in S of length 1997, and so the answer is Yes.


Is this proof sound?
1 reply
rstather
2 hours ago
aidan0626
2 hours ago
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Geo to make por people happy
AlephG_64   1
N 2 hours ago by ND_
Source: 2025 Finals Portuguese Mathematical Olympiad P4
Let $ABCD$ be a square with $2cm$ side length and with center $T$. A rhombus $ARTE$ is drawn where point $E$ lies on line $DC$. What is the area of $ARTE$?
1 reply
AlephG_64
4 hours ago
ND_
2 hours ago
J
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Modular Matching Pairs
steven_zhang123   3
N 3 hours ago by flower417477
Source: China TST 2025 P20
Let \( n \) be an odd integer, \( m = \frac{n+1}{2} \). Consider \( 2m \) integers \( a_1, a_2, \ldots, a_m, b_1, b_2, \ldots, b_m \) such that for any \( 1 \leq i < j \leq m \), \( a_i \not\equiv a_j \pmod{n} \) and \( b_i \not\equiv b_j \pmod{n} \). Prove that the number of \( k \in \{0, 1, \ldots, n-1\} \) for which satisfy \( a_i + b_j \equiv k \pmod{n} \) for some \( i \neq j \), $i, j \in \left \{ 1,2,\cdots,m \right \} $ is greater than \( n - \sqrt{n} - \frac{1}{2} \).
3 replies
steven_zhang123
Mar 29, 2025
flower417477
3 hours ago
J
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Find the function!
Johann Peter Dirichlet   13
N 3 hours ago by bin_sherlo
Source: Problem 3, Brazilian Olympic Revenge 2005
Find all functions $f: R \rightarrow R$ such that
\[f(x+yf(x))+f(xf(y)-y)=f(x)-f(y)+2xy\]
for all $x,y \in R$
13 replies
Johann Peter Dirichlet
Jun 1, 2005
bin_sherlo
3 hours ago
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inequality ( 4 var
SunnyEvan   3
N 3 hours ago by arqady
Let $ a,b,c,d \in R $ , such that $ a+b+c+d=4 . $ Prove that :
$$ a^4+b^4+c^4+d^4+3 \geq \frac{7}{4}(a^3+b^3+c^3+d^3) $$$$ a^4+b^4+c^4+d^4+ \frac{252}{25} \geq \frac{88}{25}(a^3+b^3+c^3+d^3) $$equality cases : ?
3 replies
SunnyEvan
Yesterday at 5:19 AM
arqady
3 hours ago
J
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Adding 2006 to the product of any two numbers gives a square
WakeUp   3
N 3 hours ago by Nari_Tom
Source: Baltic Way 2006
Are there $4$ distinct positive integers such that adding the product of any two of them to $2006$ yields a perfect square?
3 replies
WakeUp
Dec 4, 2010
Nari_Tom
3 hours ago
J
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Sequence and prime factors
USJL   6
N 4 hours ago by shanelin-sigma
Source: 2025 Taiwan TST Round 2 Independent Study 1-N
Let $a_0,a_1,\ldots$ be a sequence of positive integers with $a_0=1$, $a_1=2$ and
\[a_n = a_{n-1}^{a_{n-1}a_{n-2}}-1\]for all $n\geq 2$. Show that if $p$ is a prime less than $2^k$ for some positive integer $k$, then there exists $n\leq k+1$ such that $p\mid a_n$.
6 replies
USJL
Mar 26, 2025
shanelin-sigma
4 hours ago
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J
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sidelenghts wanted when altitudes are integers and circumradius =2
parmenides51   1
N Mar 28, 2021 by Firestar98k
Source: 2009 Tournament "Mathematical Multiathlon" / Reggata Seniors 3.2 / Математическое многоборье
The radius of the circumscribed circle of the triangle is $2$, and the lengths of all altitudes are integers. Find the sides of the triangle.
1 reply
parmenides51
Mar 28, 2021
Firestar98k
Mar 28, 2021
J
sidelenghts wanted when altitudes are integers and circumradius =2
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Reveal topic tags
geometry
altitudes
Integers
circumradius
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G H BBookmark kLocked kLocked NReply
Source: 2009 Tournament "Mathematical Multiathlon" / Reggata Seniors 3.2 / Математическое многоборье
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parmenides51
30629 posts
parmenides51
#1 h  Mar 28, 2021, 5:51 AM
Y by
The radius of the circumscribed circle of the triangle is $2$, and the lengths of all altitudes are integers. Find the sides of the triangle.
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Firestar98k
21 posts
Firestar98k
#2 h  Mar 28, 2021, 10:04 AM
Y by
you can think about Helen formula
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