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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.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

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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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4 variables with quadrilateral sides
mihaig   3
N a minute ago by mihaig
Source: VL
Let $a,b,c,d\geq0$ satisfying
$$\frac1{a+1}+\frac1{b+1}+\frac1{c+1}+\frac1{d+1}=2.$$Prove
$$4\left(abc+abd+acd+bcd\right)\geq3\left(a+b+c+d\right)+4.$$
3 replies
mihaig
Today at 5:11 AM
mihaig
a minute ago
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Calculate the distance of chess king!!
egxa   5
N 25 minutes ago by Tesla12
Source: All Russian 2025 9.4
A chess king was placed on a square of an \(8 \times 8\) board and made $64$ moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called $\emph{pleasant}$ if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)
5 replies
+1 w
egxa
Apr 18, 2025
Tesla12
25 minutes ago
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How many cases did you check?
avisioner   17
N 30 minutes ago by sansgankrsngupta
Source: 2023 ISL N2
Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square.

Proposed by Tahjib Hossain Khan, Bangladesh
17 replies
avisioner
Jul 17, 2024
sansgankrsngupta
30 minutes ago
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IMO Shortlist 2014 N5
hajimbrak   59
N 31 minutes ago by Ilikeminecraft
Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$.

Proposed by Belgium
59 replies
hajimbrak
Jul 11, 2015
Ilikeminecraft
31 minutes ago
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Number theory
falantrng   38
N 32 minutes ago by Ilikeminecraft
Source: RMM 2018 D2 P4
Let $a,b,c,d$ be positive integers such that $ad \neq bc$ and $gcd(a,b,c,d)=1$. Let $S$ be the set of values attained by $\gcd(an+b,cn+d)$ as $n$ runs through the positive integers. Show that $S$ is the set of all positive divisors of some positive integer.
38 replies
1 viewing
falantrng
Feb 25, 2018
Ilikeminecraft
32 minutes ago
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USAMO 2001 Problem 5
MithsApprentice   23
N 34 minutes ago by Ilikeminecraft
Let $S$ be a set of integers (not necessarily positive) such that

(a) there exist $a,b \in S$ with $\gcd(a,b)=\gcd(a-2,b-2)=1$;
(b) if $x$ and $y$ are elements of $S$ (possibly equal), then $x^2-y$ also belongs to $S$.

Prove that $S$ is the set of all integers.
23 replies
1 viewing
MithsApprentice
Sep 30, 2005
Ilikeminecraft
34 minutes ago
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IMO 2016 Shortlist, N6
dangerousliri   67
N 35 minutes ago by Ilikeminecraft
Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$.

Proposed by Dorlir Ahmeti, Albania
67 replies
dangerousliri
Jul 19, 2017
Ilikeminecraft
35 minutes ago
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IMO ShortList 1998, number theory problem 1
orl   54
N 35 minutes ago by Ilikeminecraft
Source: IMO ShortList 1998, number theory problem 1
Determine all pairs $(x,y)$ of positive integers such that $x^{2}y+x+y$ is divisible by $xy^{2}+y+7$.
54 replies
orl
Oct 22, 2004
Ilikeminecraft
35 minutes ago
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IMO Shortlist 2011, Number Theory 3
orl   47
N 36 minutes ago by Ilikeminecraft
Source: IMO Shortlist 2011, Number Theory 3
Let $n \geq 1$ be an odd integer. Determine all functions $f$ from the set of integers to itself, such that for all integers $x$ and $y$ the difference $f(x)-f(y)$ divides $x^n-y^n.$

Proposed by Mihai Baluna, Romania
47 replies
orl
Jul 11, 2012
Ilikeminecraft
36 minutes ago
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IMO ShortList 2002, number theory problem 6
orl   30
N 36 minutes ago by Ilikeminecraft
Source: IMO ShortList 2002, number theory problem 6
Find all pairs of positive integers $m,n\geq3$ for which there exist infinitely many positive integers $a$ such that \[ \frac{a^m+a-1}{a^n+a^2-1} \] is itself an integer.

Laurentiu Panaitopol, Romania
30 replies
orl
Sep 28, 2004
Ilikeminecraft
36 minutes ago
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Euclid NT
Taco12   12
N 37 minutes ago by Ilikeminecraft
Source: 2023 Fall TJ Proof TST, Problem 4
Find all pairs of positive integers $(a,b)$ such that \[ a^2b-1 \mid ab^3-1. \]
Calvin Wang
12 replies
Taco12
Oct 6, 2023
Ilikeminecraft
37 minutes ago
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A=b
k2c901_1   87
N 38 minutes ago by Ilikeminecraft
Source: Taiwan 1st TST 2006, 1st day, problem 3
Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$.

Proposed by Mohsen Jamali, Iran
87 replies
k2c901_1
Mar 29, 2006
Ilikeminecraft
38 minutes ago
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Floor of square root
v_Enhance   43
N 39 minutes ago by Ilikeminecraft
Source: APMO 2013, Problem 2
Determine all positive integers $n$ for which $\dfrac{n^2+1}{[\sqrt{n}]^2+2}$ is an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.
43 replies
v_Enhance
May 3, 2013
Ilikeminecraft
39 minutes ago
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Pair of multiples
Jalil_Huseynov   62
N 40 minutes ago by Ilikeminecraft
Source: APMO 2022 P1
Find all pairs $(a,b)$ of positive integers such that $a^3$ is multiple of $b^2$ and $b-1$ is multiple of $a-1$.
62 replies
Jalil_Huseynov
May 17, 2022
Ilikeminecraft
40 minutes ago
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BC = BD + DA if AB=AC, <A = 100^o, BD angle bisector
parmenides51   3
N Aug 25, 2021 by zerononnatural
Source: Indonesia INAMO Shortlist 2008 G10
Given a triangle $ABC$ with $AB = AC$, angle $\angle A = 100^o$ and $BD$ bisector of angle $\angle B$. Prove that $$BC = BD + DA.$$
3 replies
parmenides51
Aug 25, 2021
zerononnatural
Aug 25, 2021
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BC = BD + DA if AB=AC, <A = 100^o, BD angle bisector
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Reveal topic tags
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G H BBookmark kLocked kLocked NReply
Source: Indonesia INAMO Shortlist 2008 G10
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parmenides51
30631 posts
parmenides51
#1 h  Aug 25, 2021, 9:09 PM
Y by
Given a triangle $ABC$ with $AB = AC$, angle $\angle A = 100^o$ and $BD$ bisector of angle $\angle B$. Prove that $$BC = BD + DA.$$
This post has been edited 2 times. Last edited by parmenides51, Aug 25, 2021, 9:10 PM
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OlympusHero
17020 posts
OlympusHero
#2 h  Aug 25, 2021, 10:07 PM
Y by
There's probably a super slick solution but we'll just take the obvious route.

Solution
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i3435
1350 posts
i3435
#3 h  Aug 25, 2021, 10:24 PM
Y by
Sol
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zerononnatural
5 posts
zerononnatural
#4 h  Aug 25, 2021, 11:19 PM
Y by
Let $T$ be on $BC$ such that $BD = BT$. With a simple angle chasing we found that $TD = TC$, so we just need to prove that $TC = TD = DA$, but this is simple since $BADT$ is cyclic, then $\angle TAD = \angle ATD = 20º$. And we are done.
This post has been edited 1 time. Last edited by zerononnatural, Aug 25, 2021, 11:19 PM
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