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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.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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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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2n^2+4n-1 and 3n+4 have common powers
bin_sherlo   4
N a minute ago by CM1910
Source: Türkiye 2025 JBMO TST P5
Find all positive integers $n$ such that a positive integer power of $2n^2+4n-1$ equals to a positive integer power of $3n+4$.
4 replies
bin_sherlo
Yesterday at 7:13 PM
CM1910
a minute ago
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An interesting geometry
k.vasilev   19
N 8 minutes ago by Ilikeminecraft
Source: All-Russian Olympiad 2019 grade 10 problem 4
Let $ABC$ be an acute-angled triangle with $AC<BC.$ A circle passes through $A$ and $B$ and crosses the segments $AC$ and $BC$ again at $A_1$ and $B_1$ respectively. The circumcircles of $A_1B_1C$ and $ABC$ meet each other at points $P$ and $C.$ The segments $AB_1$ and $A_1B$ intersect at $S.$ Let $Q$ and $R$ be the reflections of $S$ in the lines $CA$ and $CB$ respectively. Prove that the points $P,$ $Q,$ $R,$ and $C$ are concyclic.
19 replies
k.vasilev
Apr 23, 2019
Ilikeminecraft
8 minutes ago
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Squeezing Between Perfect Squares and Modular Arithmetic(JBMO TST Turkey 2025)
HeshTarg   0
11 minutes ago
Source: Turkey JBMO TST Problem 4
If $p$ and $q$ are primes and $pq(p+1)(q+1)+1$ is a perfect square, prove that $pq+1$ is a perfect square.
0 replies
HeshTarg
11 minutes ago
0 replies
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sum (a+b)/(a^2+ab+b^2) <=2 if 1/a+1/b+1/c =3 for a,b,c>0
parmenides51   15
N 11 minutes ago by AylyGayypow009
Source: 2020 Greek JBMO TST p2
Let $a,b,c$ be positive real numbers such that $\frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}=3$. Prove that
$$\frac{a+b}{a^2+ab+b^2}+ \frac{b+c}{b^2+bc+c^2}+ \frac{c+a}{c^2+ca+a^2}\le 2$$When is the equality valid?
15 replies
parmenides51
Nov 14, 2020
AylyGayypow009
11 minutes ago
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Bosnia and Herzegovina JBMO TST 2016 Problem 3
gobathegreat   3
N an hour ago by Sh309had
Source: Bosnia and Herzegovina Junior Balkan Mathematical Olympiad TST 2016
Let $O$ be a center of circle which passes through vertices of quadrilateral $ABCD$, which has perpendicular diagonals. Prove that sum of distances of point $O$ to sides of quadrilateral $ABCD$ is equal to half of perimeter of $ABCD$.
3 replies
gobathegreat
Sep 16, 2018
Sh309had
an hour ago
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Power Of Factorials
Kassuno   180
N an hour ago by maromex
Source: IMO 2019 Problem 4
Find all pairs $(k,n)$ of positive integers such that \[ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). \]Proposed by Gabriel Chicas Reyes, El Salvador
180 replies
1 viewing
Kassuno
Jul 17, 2019
maromex
an hour ago
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100 card with 43 having odd integers on them
falantrng   7
N an hour ago by Just1
Source: Azerbaijan JBMO TST 2018, D2 P4
In the beginning, there are $100$ cards on the table, and each card has a positive integer written on it. An odd number is written on exactly $43$ cards. Every minute, the following operation is performed: for all possible sets of $3$ cards on the table, the product of the numbers on these three cards is calculated, all the obtained results are summed, and this sum is written on a new card and placed on the table. A day later, it turns out that there is a card on the table, the number written on this card is divisible by $2^{2018}.$ Prove that one hour after the start of the process, there was a card on the table that the number written on that card is divisible by $2^{2018}.$
7 replies
falantrng
Aug 1, 2023
Just1
an hour ago
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GCD and LCM operations
BR1F1SZ   1
N an hour ago by WallyWalrus
Source: 2025 Francophone MO Juniors P4
Charlotte writes the integers $1,2,3,\ldots,2025$ on the board. Charlotte has two operations available: the GCD operation and the LCM operation.
[list]
[*]The GCD operation consists of choosing two integers $a$ and $b$ written on the board, erasing them, and writing the integer $\operatorname{gcd}(a, b)$.
[*]The LCM operation consists of choosing two integers $a$ and $b$ written on the board, erasing them, and writing the integer $\operatorname{lcm}(a, b)$.
[/list]
An integer $N$ is called a winning number if there exists a sequence of operations such that, at the end, the only integer left on the board is $N$. Find all winning integers among $\{1,2,3,\ldots,2025\}$ and, for each of them, determine the minimum number of GCD operations Charlotte must use.

Note: The number $\operatorname{gcd}(a, b)$ denotes the greatest common divisor of $a$ and $b$, while the number $\operatorname{lcm}(a, b)$ denotes the least common multiple of $a$ and $b$.
1 reply
BR1F1SZ
Saturday at 11:24 PM
WallyWalrus
an hour ago
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x^2+4 = y^5
Valentin Vornicu   14
N an hour ago by Rayvhs
Source: Balkan MO 1998, Problem 4
Prove that the following equation has no solution in integer numbers: \[ x^2 + 4 = y^5. \] Bulgaria
14 replies
Valentin Vornicu
Apr 24, 2006
Rayvhs
an hour ago
J
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2023 Japan Mathematical Olympiad Preliminary
parkjungmin   0
2 hours ago
Please help me if I can solve the Chinese question
0 replies
parkjungmin
2 hours ago
0 replies
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Polynomial divisible by x^2+1
Miquel-point   1
N 3 hours ago by luutrongphuc
Source: Romanian IMO TST 1981, P1 Day 1
Consider the polynomial $P(X)=X^{p-1}+X^{p-2}+\ldots+X+1$, where $p>2$ is a prime number. Show that if $n$ is an even number, then the polynomial \[-1+\prod_{k=0}^{n-1} P\left(X^{p^k}\right)\]is divisible by $X^2+1$.

Mircea Becheanu
1 reply
Miquel-point
Apr 6, 2025
luutrongphuc
3 hours ago
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the same prime factors
andria   5
N 3 hours ago by bin_sherlo
Source: Iranian third round number theory P4
$a,b,c,d,k,l$ are positive integers such that for every natural number $n$ the set of prime factors of $n^k+a^n+c,n^l+b^n+d$ are same. prove that $k=l,a=b,c=d$.
5 replies
andria
Sep 6, 2015
bin_sherlo
3 hours ago
J
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A sharp estimation of the product
mihaig   0
3 hours ago
Source: VL
Let $n\geq4$ and let $a_1,a_2,\ldots, a_n\geq0$ be reals such that $\sum_{i=1}^{n}{\frac{1}{2a_i+n-2}}=1.$
Prove
$$a_1+\cdots+a_n+2^{n-1}\geq n+2^{n-1}\cdot\prod_{i=1}^{n}{a_i}.$$
0 replies
mihaig
3 hours ago
0 replies
J
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xf(x) + f ^2(y) +2 xf(y) perfect square for all positive integers x,y
parmenides51   9
N 3 hours ago by Gaunter_O_Dim_of_math
Source: Balkan BMO Shortlist 2017 N2
Find all functions $f :Z_{>0} \to Z_{>0}$ such that the number $xf(x) + f ^2(y) + 2xf(y)$ is a perfect square for all positive integers $x,y$.
9 replies
parmenides51
Aug 1, 2019
Gaunter_O_Dim_of_math
3 hours ago
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J
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116 problems
momed66   18
N Aug 31, 2020 by Gaussian_cyber
Source: By Mohammad Jafari
Problem 85 was one of IRAN TST 2011 problems.
18 replies
momed66
May 14, 2011
Gaussian_cyber
Aug 31, 2020
J
116 problems
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Reveal topic tags
function
inequalities
induction
Problem Sets
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G H BBookmark kLocked kLocked NReply
Source: By Mohammad Jafari
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momed66
14 posts
momed66
#1 h  May 14, 2011, 4:40 PM • 23 Y
Y by NewAlbionAcademy, airplanes1, Neversmile123, bryanbr2, Yousefi, sanabhai, HMoghaddas, dantx5, Jerry37284, Wizard_32, fjm30, Functional_equation, Math-wiz, Adventure10, Mango247, and 8 other users
Problem 85 was one of IRAN TST 2011 problems.
Attachments:
116 problems.pdf (215kb)
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Dr Sonnhard Graubner
16100 posts
Dr Sonnhard Graubner
#2 h  May 14, 2011, 4:42 PM • 2 Y
Y by Adventure10, Mango247
hello, nice problems, thank you.
Sonnhard.
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goodar2006
1347 posts
goodar2006
#3 h  May 14, 2011, 5:51 PM • 2 Y
Y by Adventure10, Mango247
Thanks Mr.Jafari for the problems.
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Amir Hossein
5452 posts
Amir Hossein
#4 h  May 14, 2011, 6:03 PM • 2 Y
Y by Adventure10, Mango247
Great collection, thanks Mr. Jafari! :)
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Ligouras
826 posts
Ligouras
#5 h  May 14, 2011, 8:42 PM • 2 Y
Y by Adventure10, Mango247
Nice collection, thanks Mr. Jafari!!! :)
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abch42
127 posts
abch42
#6 h  May 15, 2011, 1:57 PM • 2 Y
Y by Adventure10, Mango247
Very nice! Thanks Mr.Jafari :)
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xiyuguzu
25 posts
xiyuguzu
#7 h  May 20, 2011, 1:02 AM • 2 Y
Y by Adventure10, Mango247
where to find the solutions?
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Merlinaeus
162 posts
Merlinaeus
#8 h  May 21, 2011, 5:08 PM • 1 Y
Y by Adventure10
Mods - What's your preferred way for us to discuss these problems?

Dang it, I can't do #1 !

Merlin
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applepi2000
2226 posts
applepi2000
#9 h  Nov 8, 2011, 10:26 PM • 6 Y
Y by Yoni2b, socrates, Amir Hossein, Adventure10, Mango247, and 1 other user
Here are a few of my solutions :lol:
Problem 2
Problem 11
Problem 9
Problem 37
Problem 39
Problem 69
Problem 70
Problem 71
Problem 79
Problem 105
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socrates
2105 posts
socrates
#10 h  Nov 9, 2011, 3:23 PM • 3 Y
Y by Amir Hossein, Adventure10, and 1 other user
Problem 43
Problem 30
Problem 31
Problem 26
Problem 81, Stronger Version
Problem 60
Problem 85
Problem 42
Problem 6
Problem 5
Problem 2
Problem 20
Problem 4
Problem 34
Problem 33

Other problems:
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=36&t=378717
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=36&t=378367
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=36&t=378164
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=36&t=378718
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socrates
2105 posts
socrates
#11 h  Nov 9, 2011, 8:12 PM • 5 Y
Y by Amir Hossein, applepi2000, kienthuckiquai, Adventure10, Mango247
Problem 7
Problem 8
Problem 15
Problem 16
Problem 17
Problem 19
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 27
Problem 29
Problem 12
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applepi2000
2226 posts
applepi2000
#12 h  Nov 10, 2011, 12:19 AM • 4 Y
Y by socrates, Amir Hossein, Adventure10, Mango247
Problem 49
Problem 50
Problem 52
Problem 59
Problem 62
Problem 64
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socrates
2105 posts
socrates
#13 h  Nov 10, 2011, 2:19 AM • 5 Y
Y by applepi2000, Amir Hossein, a123, Adventure10, Mango247
Problem 18
Problem 28
Problem 36
Problem 38
Problem 44
Problem 45
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Amir Hossein
5452 posts
Amir Hossein
#14 h  Nov 10, 2011, 9:14 AM • 1 Y
Y by Adventure10
Please post your solutions in this topic.
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Functional_equation
530 posts
Functional_equation
#15 h  Aug 31, 2020, 1:39 PM • 1 Y
Y by Mango247
Great handout! :)
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Functional_equation
530 posts
Functional_equation
#17 h  Aug 31, 2020, 1:47 PM • 4 Y
Y by Bumblebee60, amar_04, Wizard0001, Mango247
In fact, I didn't do it to congratulate him. I just wanted other people (newcomers to Aops) to see it :)
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franzliszt
23531 posts
franzliszt
#18 h  Aug 31, 2020, 1:54 PM • 1 Y
Y by Mango247
Ya good bump man! :)
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SardorNarpay
106 posts
SardorNarpay
#19 h  Aug 31, 2020, 3:16 PM
Y by
Thank you very much
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Gaussian_cyber
162 posts
Gaussian_cyber
#20 h  Aug 31, 2020, 3:21 PM • 2 Y
Y by Mango247, Mango247
You can see This. It's is the newer version Ig.
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