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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

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Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
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0 replies
jlacosta
Mar 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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Find min
hunghd8   2
N a few seconds ago by ehuseyinyigit
Let $a,b,c$ be nonnegative real numbers such that $ a+b+c\geq 2+abc $. Find min
$$P=a^2+b^2+c^2.$$
2 replies
hunghd8
3 hours ago
ehuseyinyigit
a few seconds ago
J
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Interesting inequality
sqing   5
N 20 minutes ago by sqing
Source: Own
Let $ a,b >0. $ Prove that
$$ \frac{1}{\frac{a}{a+b}+\frac{a}{2b}} +\frac{1}{\frac{b}{a+b}+\frac{1}{2}} +\frac{a}{2b} \geq \frac{5}{2} $$
5 replies
1 viewing
sqing
Feb 26, 2025
sqing
20 minutes ago
J
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sum of divisors nt
Soupboy0   0
27 minutes ago
Source: own
Let $\epsilon(n)$ denote the sum of the sum of the factors of all positive $\mathbb Z \le n$, for example, $\epsilon(5) $ is the sum of the factors of $5$ added to the sum of the factors of $4$ and so on until the sum of the factors of $1$, which would be $(1+5)+(1+2+4)+(1+3)+(1+2)+(1) = 21$. Let $M(n)$ denote $\sum_{i=1}^{n} n \pmod{i}$. Show that $\epsilon(n) + M(n) = n^2$ or find a counterexample
0 replies
Soupboy0
27 minutes ago
0 replies
J
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euler-totient function
Laan   2
N 28 minutes ago by Laan
Proof that there are infinitely many positive integers $n$ such that
$\varphi(n)<\varphi(n+1)<\varphi(n+2)$
2 replies
Laan
Today at 7:13 AM
Laan
28 minutes ago
J
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Prime for square numbers
giangtruong13   0
32 minutes ago
Source: City&rsquo;s Specialized Math Examination
Given that $a,b$ are natural numbers satisfy that: $\frac{a^3}{a+b}$ and $\frac{b^3}{a+b}$ are prime numbers. Prove that $$a^2+3ab+3a+b+1$$is a perfect squared number
0 replies
giangtruong13
32 minutes ago
0 replies
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2 var inquality
sqing   5
N an hour ago by sqing
Source: Own
Let $ a,b $ be nonnegative real numbers such that $ a^2+ab+b^2+a+b=1. $ Prove that
$$ (ab+1)(a+b)\leq \frac{ 20}{27} $$$$ (ab+1)(a+b-1)\leq - \frac{ 10}{27} $$Let $ a,b $ be nonnegative real numbers such that $ a^2+b^2+a+b=1. $ Prove that
$$ (ab+1)(a+b)\leq \frac{ 5\sqrt 3-7}{2} $$$$ (ab+1)(a+b-1)\leq 3\sqrt 3- \frac{ 11}{2} $$
5 replies
1 viewing
sqing
Yesterday at 3:00 PM
sqing
an hour ago
J
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Is it fake? how can someone score 12 in AIME but can&#039;t qualify RMO
Bruce_wayne123   3
N an hour ago by Bruce_wayne123
Source: https://www.reddit.com/r/JEENEETards/comments/1jgduci/op_qualified_for_usamo/#lightbox
He also claims to have scored 93 percentile in JEEM maths another thing which makes it more doubtful and also he didn't got any letter from MAA
3 replies
Bruce_wayne123
an hour ago
Bruce_wayne123
an hour ago
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AZE JBMO TST
IstekOlympiadTeam   9
N an hour ago by DensSv
Source: AZE JBMO TST
Prove that there are not intgers $a$ and $b$ with conditions,
i) $16a-9b$ is a prime number.
ii) $ab$ is a perfect square.
iii) $a+b$ is also perfect square.
9 replies
IstekOlympiadTeam
May 2, 2015
DensSv
an hour ago
J
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Iterates of function give distinct residues
Tintarn   6
N an hour ago by AshAuktober
Source: Abelkonkurransen Finale 2024, Problem 1b
Find all functions $f:\mathbb{Z} \to \mathbb{Z}$ such that the numbers
\[n, f(n),f(f(n)),\dots,f^{m-1}(n)\]are distinct modulo $m$ for all integers $n,m$ with $m>1$.
(Here $f^k$ is defined by $f^0(n)=n$ and $f^{k+1}(n)=f(f^{k}(n))$ for $k \ge 0$.)
6 replies
Tintarn
Mar 8, 2024
AshAuktober
an hour ago
J
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Interesting NT function
amogususususus   5
N an hour ago by AshAuktober
Source: Indonesia TSTST - Number Theory
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that for every prime number $p$ and natural number $x$,
$$\{ x,f(x),\cdots f^{p-1}(x) \} $$is a complete residue system modulo $p$. With $f^{k+1}(x)=f(f^k(x))$ for every natural number $k$ and $f^1(x)=f(x)$.

Proposed by IndoMathXdZ
5 replies
amogususususus
Feb 29, 2024
AshAuktober
an hour ago
J
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4 free variables ineq
RainbowNeos   0
an hour ago
Given $0\leq a,b,c,d\leq 1$, show that
\[abc\sqrt{1-d}+bcd\sqrt{1-a}+cda\sqrt{1-b}+dab\sqrt{1-c}\leq 1.\]
0 replies
RainbowNeos
an hour ago
0 replies
J
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An interesting inequality
JK1603JK   0
an hour ago
Source: unknown
Let a,b,c>=0 and a^2+b^2+c^2+abc=4 then prove \frac{1}{a+b+2}+\frac{1}{b+c+2}+\frac{1}{c+a+2} \le \frac{6-(a+b+c)}{4}
When does equality occur?
0 replies
JK1603JK
an hour ago
0 replies
J
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Romania Junior TST 2021 Day 3 P2
oVlad   4
N 2 hours ago by DensSv
Let $O$ be the circumcenter of triangle $ABC$ and let $AD$ be the height from $A$ ($D\in BC$). Let $M,N,P$ and $Q$ be the midpoints of $AB,AC,BD$ and $CD$ respectively. Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be the circumcircles of triangles $AMN$ and $POQ$. Prove that $\mathcal{C}_1\cap \mathcal{C}_2\cap AD\neq \emptyset$.
4 replies
oVlad
Jun 7, 2021
DensSv
2 hours ago
J
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A lot of z
Anulick   4
N 2 hours ago by quasar_lord
Source: CMI 2024
(a) FInd the number of complex roots of $Z^6 = Z + \bar{Z}$
(b) Find the number of complex solutions of $Z^n = Z + \bar{Z}$ for $n \in \mathbb{Z}^+$
4 replies
Anulick
May 19, 2024
quasar_lord
2 hours ago
J
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J
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Functional Equation
IstekOlympiadTeam   5
N Feb 10, 2025 by Nuran2010
Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that \[f(f(n))=n+2015\]where $n\in \mathbb{N}.$
5 replies
IstekOlympiadTeam
Feb 19, 2016
Nuran2010
Feb 10, 2025
J
Functional Equation
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Reveal topic tags
function
algebra
functional equation
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IstekOlympiadTeam
542 posts
IstekOlympiadTeam
#1 h  Feb 19, 2016, 2:57 PM • 5 Y
Y by Davi-8191, Amir Hossein, megarnie, Adventure10, Mango247
Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that \[f(f(n))=n+2015\]where $n\in \mathbb{N}.$
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Ankoganit
3070 posts
Ankoganit
#2 h  Feb 19, 2016, 3:01 PM • 3 Y
Y by muradmurad, megarnie, Adventure10
From this post, we conclude there are no solutions.
Also, this is similar to IMO 1987 P4.
This post has been edited 1 time. Last edited by Ankoganit, Feb 19, 2016, 3:04 PM
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Takeya.O
769 posts
Takeya.O
#3 h  Jul 14, 2016, 2:13 PM • 3 Y
Y by megarnie, Adventure10, Mango247
I'm going to write the proof refer to IMO 1987 P4.

Suppose that there is such f.Let A=N－f（N）, B=f（A）.Apparently f is injective.We claim that B=f（N）－f（f（N））.Because if k∈B, k∈f（N） and from injectivity of f, k∈f（N）－f（f（N））.On the other hand, if k∈f（N）－f（f（N））, k∈f（A）=B.

Since A and B are disjoint, |A∪B|=|A|＋|B|.And from injectivity of f, |A|=|B|, so |A∪B| is even.On the other hand, A∪B=N－f（f（N））=｛1,...,2015｝ which is absurd.

Therefore there is no such solution f.
This post has been edited 1 time. Last edited by Takeya.O, Jul 14, 2016, 2:38 PM
Reason: adding explanation
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strong_boy
261 posts
strong_boy
#4 h  Jul 29, 2022, 5:39 PM • 5 Y
Y by hectorleo123, Mango247, Mango247, Mango247, wizixez
It can be easily seen that the function $f$ is injective

Our problem function produces natural values from $2016$ onwards

Suppose the function $f$ alone does not generate $N$ values. So the function $f(f(n))$ does not produce $2N$ value.

So, if a function wants to be true in the problem, the following equation must have a solution


$2N=2015$
And this issue has no natural answer!

So the function does not work in the problem
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MuradSafarli
60 posts
MuradSafarli
#7 h  Feb 8, 2025, 5:54 PM
Y by
Also, this question is the first question in Tutu's book on functions. Book's link-https://www.isinj.com/mt-usamo/Functional%20Equations%20-%20Andresscru,%20Boreico%20(2007).pdf
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Nuran2010
49 posts
Nuran2010
#9 h  Feb 10, 2025, 8:22 PM
Y by
MuradSafarli wrote:
Also, this question is the first question in Tutu's book on functions. Book's link-https://www.isinj.com/mt-usamo/Functional%20Equations%20-%20Andresscru,%20Boreico%20(2007).pdf

Tutu's book on functions
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