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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
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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amazing balkan combi
egxa   7
N a few seconds ago by Assassino9931
Source: BMO 2025 P4
There are $n$ cities in a country, where $n \geq 100$ is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities $A$ and $B$ we define:

$(i)$ A $\emph{path}$ between $A$ and $B$ as a sequence of distinct cities $A = C_0, C_1, \dots, C_k, C_{k+1} = B$, $k \geq 0$, such that there are direct flights between $C_i$ and $C_{i+1}$ for every $0 \leq i \leq k$;
$(ii)$ A $\emph{long path}$ between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has more cities;
$(iii)$ A $\emph{short path}$ between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has fewer cities.
Assume that for any pair of cities $A$ and $B$ in the country, there exist a long path and a short path between them that have no cities in common (except $A$ and $B$). Let $F$ be the total number of pairs of cities in the country that are connected by direct flights. In terms of $n$, find all possible values $F$

Proposed by David-Andrei Anghel, Romania.
7 replies
egxa
Apr 27, 2025
Assassino9931
a few seconds ago
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Question on Balkan SL
Fmimch   2
N 3 minutes ago by Assassino9931
Does anyone know where to find the Balkan MO Shortlist 2024? If you have the file, could you send in this thread? Thank you!
2 replies
Fmimch
Today at 12:13 AM
Assassino9931
3 minutes ago
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Or statement function
ItzsleepyXD   1
N 5 minutes ago by Haris1
Source: Own , Mock Thailand Mathematic Olympiad P2
Find all $f: \mathbb{R} \to \mathbb{Z^+}$ such that $$f(x+f(y))=f(x)+f(y)+1\quad\text{ or }\quad f(x)+f(y)-1$$for all real number $x$ and $y$
1 reply
ItzsleepyXD
an hour ago
Haris1
5 minutes ago
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Add a digit to obtain a new perfect square
Lukaluce   2
N 15 minutes ago by TopGbulliedU
Source: 2024 Junior Macedonian Mathematical Olympiad P4
Let $a_1, a_2, ..., a_n$ be a sequence of perfect squares such that $a_{i + 1}$ can be obtained by concatenating a digit to the right of $a_i$. Determine all such sequences that are of maximum length.

Proposed by Ilija Jovčeski
2 replies
Lukaluce
Apr 14, 2025
TopGbulliedU
15 minutes ago
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D1025 : Can you do that?
Dattier   2
N 19 minutes ago by CerealCipher
Source: les dattes à Dattier
Let $x_{n+1}=x_n^3$ and $x_0=3$.

Can you calculate $\sum\limits_{i=1}^{2^{2025}} x_i \mod 10^{30}$?
2 replies
Dattier
Yesterday at 8:24 PM
CerealCipher
19 minutes ago
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Simple inequality
sqing   7
N 25 minutes ago by sqing
Source: Daniel Sitaru
Let $a,b,c>0$ . Prove that$$\frac{a^3}{b^3}+\frac{b^3}{c^3}+\frac{c^3}{a^3}+9>\frac{3}{2}\left(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}+ \frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)$$
7 replies
2 viewing
sqing
Feb 10, 2017
sqing
25 minutes ago
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Vector Vortex
steven_zhang123   1
N 25 minutes ago by Mathzeus1024
Source: NS Issue 1 P3 (2014.4)
Let $v_{1}, v_{2}, \cdots, v_{n}$ be $n$ unit vectors on a plane, where $n$ is an odd number. Prove that there exist $\varepsilon _i\in \left \{ -1,1 \right \} $ for $i=1,2,\cdots,n$ such that $\left | \sum_{i=1}^{n} \varepsilon_i v_i \right | \le 1.$
1 reply
1 viewing
steven_zhang123
Feb 15, 2025
Mathzeus1024
25 minutes ago
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China Northern MO 2009 p4 CNMO
parkjungmin   0
25 minutes ago
Source: China Northern MO 2009 p4 CNMO P4
The problem is too difficult.
0 replies
parkjungmin
25 minutes ago
0 replies
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The Appetizer of Iran NT2023
alinazarboland   6
N 38 minutes ago by A22-
Source: Iran MO 3rd round 2023 NT exam , P1
Find all integers $n > 4$ st for every two subsets $A,B$ of $\{0,1,....,n-1\}$ , there exists a polynomial $f$ with integer coefficients st either $f(A) = B$ or $f(B) = A$ where the equations are considered mod n.
We say two subsets are equal mod n if they produce the same set of reminders mod n. and the set $f(X)$ is the set of reminders of $f(x)$ where $x \in X$ mod n.
6 replies
alinazarboland
Aug 17, 2023
A22-
38 minutes ago
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Do not try to case bash lol
ItzsleepyXD   1
N 39 minutes ago by Haris1
Source: Own , Mock Thailand Mathematic Olympiad P3
Let $n,d\geqslant 6$ be a positive integer such that $d\mid 6^{n!}+1$ .
Prove that $d>2n+6$ .
1 reply
ItzsleepyXD
an hour ago
Haris1
39 minutes ago
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Rutthee on some APMO style
ItzsleepyXD   0
40 minutes ago
Source: Own , Mock Thailand Mathematic Olympiad P10 (Not Rutthee problem , Idk what to name a sequence)
Let $a_0,a_1,\dots$ be Rutthee sequence if $a_0$ be a positive integer and
$$a_{i}\in\left\{3a_{i-1}+2,\frac{2a_{i-1}+1}{a_{i-1}+2},\frac{a_{i-1}}{2a_{i-1}+3}\right\}$$for all $i \in \mathbb{Z^+}$ and there are some positive integer $n$ sastisfied $a_n\in\{2025,2568\}$
Is it possible that there are Rutthee sequence such that there exist positive integer $m\neq n$ such that $a_m=2025$ and $a_n=2568$ also find all possible value of $a_0$ in Rutthee sequence
0 replies
ItzsleepyXD
40 minutes ago
0 replies
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3 var inequality
sqing   0
an hour ago
Source: Own
Let $ a,b,c>0 ,\frac{a}{b} +\frac{b}{c} +\frac{c}{a} \leq 2\left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right). $ Prove that
$$a+b+c+2\geq abc$$Let $ a,b,c>0 , a^3+b^3+c^3\leq 2(ab+bc+ca). $ Prove that
$$a+b+c+2\geq abc$$
0 replies
sqing
an hour ago
0 replies
J
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Do not try to bash on beautiful geometry
ItzsleepyXD   0
an hour ago
Source: Own , Mock Thailand Mathematic Olympiad P9
Let $ABC$be triangle with point $D,E$ and $F$ on $BC,AB,CA$
such that $BE=CF$ and $E,F$ are on the same side of $BC$
Let $M$ be midpoint of segment $BC$ and $N$ be midpoint of segment $EF$
Let $G$ be intersection of $BF$ with $CE$ and $\dfrac{BD}{DC}=\dfrac{AC}{AB}$
Prove that $MN\parallel DG$
0 replies
ItzsleepyXD
an hour ago
0 replies
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already well-known, but yet strangely difficult
Valentin Vornicu   37
N an hour ago by cursed_tangent1434
Source: Romanian ROM TST 2004, problem 6
Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]
37 replies
Valentin Vornicu
May 1, 2004
cursed_tangent1434
an hour ago
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J
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A very nice inequality
KhuongTrang   4
N Apr 6, 2025 by arqady
Source: own
Problem. Let $a,b,c\in \mathbb{R}:\ a+b+c=3.$ Prove that $$\color{black}{\sqrt{5a^{2}-ab+5b^{2}}+\sqrt{5b^{2}-bc+5c^{2}}+\sqrt{5c^{2}-ca+5a^{2}}\le 2(a^2+b^2+c^2)+ab+bc+ca.}$$When does equality hold?
4 replies
KhuongTrang
Apr 6, 2025
arqady
Apr 6, 2025
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A very nice inequality
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Reveal topic tags
inequalities
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Source: own
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KhuongTrang
729 posts
KhuongTrang
#1 h  Apr 6, 2025, 1:35 PM
Y by
Problem. Let $a,b,c\in \mathbb{R}:\ a+b+c=3.$ Prove that $$\color{black}{\sqrt{5a^{2}-ab+5b^{2}}+\sqrt{5b^{2}-bc+5c^{2}}+\sqrt{5c^{2}-ca+5a^{2}}\le 2(a^2+b^2+c^2)+ab+bc+ca.}$$When does equality hold?
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Mathdreams
1469 posts
Mathdreams
#3 h  Apr 6, 2025, 2:01 PM
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Solution
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MS_asdfgzxcvb
70 posts
MS_asdfgzxcvb
#4 h  Apr 6, 2025, 2:39 PM
Y by
Mathdreams wrote:
Notice that $2ab \le a^2 + b^2$ by the Trivial Inequality. Hence, we have that
\[ \sqrt{5a^2 - ab + 5b^2} + \sqrt{5b^2 - bc + 5c^2} \le \sqrt{9ab} + \sqrt{9bc} + \sqrt{9ac}.~~~(1) \]
?
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Mathdreams
1469 posts
Mathdreams
#5 h  Apr 6, 2025, 2:49 PM
Y by
MS_asdfgzxcvb wrote:
Mathdreams wrote:
Notice that $2ab \le a^2 + b^2$ by the Trivial Inequality. Hence, we have that
\[ \sqrt{5a^2 - ab + 5b^2} + \sqrt{5b^2 - bc + 5c^2} \le \sqrt{9ab} + \sqrt{9bc} + \sqrt{9ac}.~~~(1) \]
?

:oops_sign:
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arqady
30219 posts
arqady
#6 h  Apr 6, 2025, 4:33 PM • 1 Y
Y by MS_asdfgzxcvb
KhuongTrang wrote:
Problem. Let $a,b,c\in \mathbb{R}:\ a+b+c=3.$ Prove that $$\color{black}{\sqrt{5a^{2}-ab+5b^{2}}+\sqrt{5b^{2}-bc+5c^{2}}+\sqrt{5c^{2}-ca+5a^{2}}\le 2(a^2+b^2+c^2)+ab+bc+ca.}$$When does equality hold?
By C-S $$\sum_{cyc}\sqrt{5a^2-ab+5b^2}\leq\sqrt{\sum_{cyc}\frac{5a^2-ab+5b^2}{a^2+b^2+kc^2+mab}\sum_{cyc}(a^2+b^2+kc^2+mab)},$$where $k=\frac{60}{61}$ and $m=1.$
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