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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
1 viewing
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
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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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product of the first n terms
FireBreathers   5
N 3 minutes ago by ihategeo_1969
Does there exist an infinite sequence of positive integers $a_i$ such that every positive integer appears exactly once and the product of the first $n$ terms is a perfect $n - th$ power ?
5 replies
FireBreathers
Dec 16, 2024
ihategeo_1969
3 minutes ago
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Inequality with three conditions
oVlad   1
N 8 minutes ago by Haris1
Source: Romania EGMO TST 2019 Day 1 P3
Let $a,b,c$ be non-negative real numbers such that \[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]Prove that $a^2+b^2+c^2\leqslant 2abc+1.$
1 reply
oVlad
5 hours ago
Haris1
8 minutes ago
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Paint and Optimize: A Grid Strategy Problem
mojyla222   2
N 8 minutes ago by sami1618
Source: Iran 2025 second round p2
Ali and Shayan are playing a turn-based game on an infinite grid. Initially, all cells are white. Ali starts the game, and in the first turn, he colors one unit square black. In the following turns, each player must color a white square that shares at least one side with a black square. The game continues for exactly 2808 turns, after which each player has made 1404 moves. Let $A$ be the set of black cells at the end of the game. Ali and Shayan respectively aim to minimize and maximise the perimeter of the shape $A$ by playing optimally. (The perimeter of shape $A$ is defined as the total length of the boundary segments between a black and a white cell.)

What are the possible values of the perimeter of $A$, assuming both players play optimally?
2 replies
mojyla222
Yesterday at 4:25 AM
sami1618
8 minutes ago
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n + k are composites for all nice numbers n, when n+1, 8n+1 both squares
parmenides51   1
N 10 minutes ago by Nuran2010
Source: 2022 Saudi Arabia JBMO TST 1.1
The positive $n > 3$ called ‘nice’ if and only if $n +1$ and $8n + 1$ are both perfect squares. How many positive integers $k \le 15$ such that $4n + k$ are composites for all nice numbers $n$?
1 reply
parmenides51
Nov 3, 2022
Nuran2010
10 minutes ago
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Distinct Integers with Divisibility Condition
tastymath75025   16
N 13 minutes ago by ihategeo_1969
Source: 2017 ELMO Shortlist N3
For each integer $C>1$ decide whether there exist pairwise distinct positive integers $a_1,a_2,a_3,...$ such that for every $k\ge 1$, $a_{k+1}^k$ divides $C^ka_1a_2...a_k$.

Proposed by Daniel Liu
16 replies
tastymath75025
Jul 3, 2017
ihategeo_1969
13 minutes ago
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GCD of a sequence
oVlad   6
N 28 minutes ago by Rohit-2006
Source: Romania EGMO TST 2017 Day 1 P2
Determine all pairs $(a,b)$ of positive integers with the following property: all of the terms of the sequence $(a^n+b^n+1)_{n\geqslant 1}$ have a greatest common divisor $d>1.$
6 replies
1 viewing
oVlad
5 hours ago
Rohit-2006
28 minutes ago
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Maximum with the condition $x^2+y^2+z^2=1$
hlminh   1
N 30 minutes ago by rchokler
Let $x,y,z$ be real numbers such that $x^2+y^2+z^2=1,$ find the largest value of $$E=|x-2y|+|y-2z|+|z-2x|.$$
1 reply
hlminh
Today at 9:20 AM
rchokler
30 minutes ago
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Mock 22nd Thailand TMO P10
korncrazy   2
N 30 minutes ago by korncrazy
Source: own
Prove that there exists infinitely many triples of positive integers $(a,b,c)$ such that $a>b>c,\,\gcd(a,b,c)=1$ and $$a^2-b^2,a^2-c^2,b^2-c^2$$are all perfect square.
2 replies
korncrazy
Apr 13, 2025
korncrazy
30 minutes ago
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IMO Shortlist 2014 N6
hajimbrak   26
N 31 minutes ago by ihategeo_1969
Let $a_1 < a_2 < \cdots <a_n$ be pairwise coprime positive integers with $a_1$ being prime and $a_1 \ge n + 2$. On the segment $I = [0, a_1 a_2 \cdots a_n ]$ of the real line, mark all integers that are divisible by at least one of the numbers $a_1 , \ldots , a_n$ . These points split $I$ into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by $a_1$.

Proposed by Serbia
26 replies
hajimbrak
Jul 11, 2015
ihategeo_1969
31 minutes ago
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An easy FE
oVlad   2
N 39 minutes ago by BR1F1SZ
Source: Romania EGMO TST 2017 Day 1 P3
Determine all functions $f:\mathbb R\to\mathbb R$ such that \[f(xy-1)+f(x)f(y)=2xy-1,\]for any real numbers $x{}$ and $y{}.$
2 replies
oVlad
5 hours ago
BR1F1SZ
39 minutes ago
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Nationalist Combo
blacksheep2003   15
N 2 hours ago by cj13609517288
Source: USEMO 2019 Problem 5
Let $\mathcal{P}$ be a regular polygon, and let $\mathcal{V}$ be its set of vertices. Each point in $\mathcal{V}$ is colored red, white, or blue. A subset of $\mathcal{V}$ is patriotic if it contains an equal number of points of each color, and a side of $\mathcal{P}$ is dazzling if its endpoints are of different colors.

Suppose that $\mathcal{V}$ is patriotic and the number of dazzling edges of $\mathcal{P}$ is even. Prove that there exists a line, not passing through any point in $\mathcal{V}$, dividing $\mathcal{V}$ into two nonempty patriotic subsets.

Ankan Bhattacharya
15 replies
blacksheep2003
May 24, 2020
cj13609517288
2 hours ago
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UIL Number Sense problem
Potato512   2
N 2 hours ago by buddy2007
I keep seeing a certain type of problem in UIL Number Sense, though I can't figure out how to do it (I aim to do it in my head in about 7-8 seconds).

The problem is x^((p+1)/2) mod p, where p is prime.
For example 11^15 mod 29
I know it technically doesn't work this way, but using fermats little theorem (on √x^(p+1)) always gives either the number itself, x, or the modular inverse, p-x.
By using the theorem i mean √x^28 mod 29 = 1, and then youre left with √x^2 mod 29 or x, but then its + or -.
I was wondering if there is a way to figure out whether its + or -, a slow or fast way if its slow maybe its possible to speed it up.
2 replies
Potato512
Today at 12:17 AM
buddy2007
2 hours ago
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Concurrency with 10 lines
oVlad   1
N 2 hours ago by kokcio
Source: Romania EGMO TST 2017 Day 1 P1
Consider five points on a circle. For every three of them, we draw the perpendicular from the centroid of the triangle they determine to the line through the remaining two points. Prove that the ten lines thus formed are concurrent.
1 reply
oVlad
5 hours ago
kokcio
2 hours ago
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Advanced topics in Inequalities
va2010   21
N 2 hours ago by Novmath
So a while ago, I compiled some tricks on inequalities. You are welcome to post solutions below!
21 replies
va2010
Mar 7, 2015
Novmath
2 hours ago
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integer right triangle with area twice the perimeter
parmenides51   4
N Oct 10, 2021 by pi271828
Source: 1994 Argentina IberoAmerican Training List p30 https://artofproblemsolving.com/community/c2434876_argentina_training_lists__geom
Find all right triangles with integer sides such that their area is twice their perimeter.
4 replies
parmenides51
Oct 10, 2021
pi271828
Oct 10, 2021
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integer right triangle with area twice the perimeter
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Reveal topic tags
geometry
Integers
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Source: 1994 Argentina IberoAmerican Training List p30 https://artofproblemsolving.com/community/c2434876_argentina_training_lists__geom
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parmenides51
30630 posts
parmenides51
#1 h  Oct 10, 2021, 5:02 PM
Y by
Find all right triangles with integer sides such that their area is twice their perimeter.
This post has been edited 1 time. Last edited by parmenides51, Oct 10, 2021, 5:21 PM
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OlympusHero
17020 posts
OlympusHero
#2 h  Oct 10, 2021, 10:05 PM
Y by
Solution
This post has been edited 2 times. Last edited by OlympusHero, Oct 10, 2021, 10:56 PM
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James_HN
51 posts
James_HN
#3 h  Oct 10, 2021, 10:40 PM
Y by
Let the sides of the triangle be $a,b,\sqrt{a^2 + b^2}$. Then we have
$\frac{ab}{2} = 2(a + b + \sqrt{a^2 + b^2})$

$(\frac{ab}{4} - a - b) = \sqrt{a^2 + b^2}$

$(\frac{ab}{4} - a - b)^2 = a^2 + b^2$

$\frac{a^2b^2}{16} + a^2 + b^2 + 2ab - \frac{a^2b}{2}- \frac{ab^2}{2} = a^2 + b^2$

$\frac{a^2b^2}{16} + 2ab - \frac{a^2b}{2}- \frac{ab^2}{2} = 0$

$a^2b^2 + 32ab - 8a^2b- 8ab^2 = 0$

$ab- 8a- 8b + 64 = 32$

$(a-8)(b-8)= 32$

hence solving for a and b we get the following triples,
$$(9,40,41),(10,24,26),(12,16,20)$$
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OlympusHero
17020 posts
OlympusHero
#4 h  Oct 10, 2021, 10:46 PM
Y by
Oh my god I'm such an idiot
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pi271828
3363 posts
pi271828
#5 h  Oct 10, 2021, 10:54 PM
Y by
OlympusHero wrote:
Oh my god I'm such an idiot

smh you reversed the condition

sketch
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