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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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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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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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Problems for v_p(n)
xytunghoanh   15
N 10 minutes 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
3 viewing
xytunghoanh
3 hours ago
AshAuktober
10 minutes ago
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BMO2 1997 Q4
rstather   1
N 12 minutes 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
1 viewing
rstather
17 minutes ago
aidan0626
12 minutes ago
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Geo to make por people happy
AlephG_64   1
N 24 minutes 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
1 viewing
AlephG_64
3 hours ago
ND_
24 minutes ago
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Modular Matching Pairs
steven_zhang123   3
N an hour 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
an hour ago
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Find the function!
Johann Peter Dirichlet   13
N an hour 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
1 viewing
Johann Peter Dirichlet
Jun 1, 2005
bin_sherlo
an hour ago
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inequality ( 4 var
SunnyEvan   3
N an hour 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
an hour ago
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Adding 2006 to the product of any two numbers gives a square
WakeUp   3
N an hour 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
an hour ago
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Sequence and prime factors
USJL   6
N 2 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
2 hours ago
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ab + c + d = 3, bc + d + a = 5, cd + a + b = 2, da + b + c = 6
parmenides51   9
N 2 hours ago by DensSv
Source: 2021 JBMO TST Bosnia and Herzegovina P1
Determine all real numbers $a, b, c, d$ for which
$$ab + c + d = 3$$$$bc + d + a = 5$$$$cd + a + b = 2$$$$da + b + c = 6$$
9 replies
parmenides51
Oct 7, 2022
DensSv
2 hours ago
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Prime number and composite number
mingzhehu   3
N 2 hours ago by mingzhehu
I have one topic on how to identify Prime Number and Composite Number quickly? Maybe the number is more than 100 or 1000.......!
If there are some formula that can be used to verify the number easily, it will be highly appreciated.
Does anybody has any good idea for that?

3 replies
mingzhehu
Today at 8:06 AM
mingzhehu
2 hours ago
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Parallel Lines and Q Point
taptya17   13
N 2 hours ago by everythingpi3141592
Source: India EGMO TST 2025 Day 1 P3
Let $\Delta ABC$ be an acute angled scalene triangle with circumcircle $\omega$. Let $O$ and $H$ be the circumcenter and orthocenter of $\Delta ABC,$ respectively. Let $E,F$ and $Q$ be points on segments $AB,AC$ and $\omega$, respectively, such that
$$\angle BHE=\angle CHF=\angle AQH=90^\circ.$$Prove that $OQ$ and $AH$ intersect on the circumcircle of $\Delta AEF$.

Proposed by Antareep Nath
13 replies
taptya17
Dec 13, 2024
everythingpi3141592
2 hours ago
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Cool combinatorial problem (grid)
Anto0110   0
2 hours ago
Suppose you have an $m \cdot n$ grid with $m$ rows and $n$ columns, and each square of the grid is filled with a non-negative integer. Let $a$ be the average of all the numbers in the grid. Prove that if $m >(10a+10)^n$ the there exist two identical rows (meaning same numbers in the same order).
0 replies
Anto0110
2 hours ago
0 replies
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Cards and combi
AlephG_64   0
2 hours ago
Source: 2025 Finals Portuguese Mathematical Olympiad P6
Maria wants to solve a challenge with a deck of cards, each with a different figure. Initially, the cards are distributed randomly into two piles, not necessarily in equal parts. Maria's goal is to get all the cards into the same pile.

On each turn, Maria takes the top card from each pile and compares them. In the rule book, there's a table that indicates, for each card match, which of the two wins. Both cards are then placed on the bottom of the winning card in the order Maria chooses. The challenge ends when all the cards are in one pile.

Show that it is always possible for Maria to solve the challenge. Regardless of the initial distribution of the cards and the table in the rule book.
0 replies
AlephG_64
2 hours ago
0 replies
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Cool number condition
AlephG_64   0
2 hours ago
Source: 2025 Finals Portuguese Mathematical Olympiad P5
An integer number $n \geq 2$ is called feirense if it is possible to write on a sheet of paper some integers such that every positive divisor of $n$ less than $n$ is the difference between two numbers on the sheet, and no other positive number is.
Find all the feirense numbers.
0 replies
AlephG_64
2 hours ago
0 replies
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impossible construction of trapezoids with legs equal to bases
parmenides51   0
Aug 5, 2018
Source: Quantum - English M92
Prove that it's impossible to construct two trapezoids (not parallelograms!) such that the legs of each are congruent
to the bases of the other.

(V. Proizvolov)
0 replies
parmenides51
Aug 5, 2018
0 replies
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impossible construction of trapezoids with legs equal to bases
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parmenides51
30629 posts
parmenides51
#1 h  Aug 5, 2018, 5:52 PM • 1 Y
Y by Adventure10
Prove that it's impossible to construct two trapezoids (not parallelograms!) such that the legs of each are congruent
to the bases of the other.

(V. Proizvolov)
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