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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.

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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
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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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Another Number Theory!
matinyousefi   7
N 7 minutes ago by MR.1
Source: Iran MO 2024 second round P6
Find all natural numbers $x,y>1$and primes $p$ that satisfy $$\frac{x^2-1}{y^2-1}=(p+1)^2. $$
7 replies
matinyousefi
Apr 19, 2024
MR.1
7 minutes ago
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IMO Shortlist 2014 A2
hajimbrak   40
N 38 minutes ago by ezpotd
Define the function $f:(0,1)\to (0,1)$ by \[\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.\] Let $a$ and $b$ be two real numbers such that $0 < a < b < 1$. We define the sequences $a_n$ and $b_n$ by $a_0 = a, b_0 = b$, and $a_n = f( a_{n -1})$, $b_n = f (b_{n -1} )$ for $n > 0$. Show that there exists a positive integer $n$ such that \[(a_n - a_{n-1})(b_n-b_{n-1})<0.\]

Proposed by Denmark
40 replies
hajimbrak
Jul 11, 2015
ezpotd
38 minutes ago
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sequence positive
malinger   38
N an hour ago by ezpotd
Source: ISL 2006, A2, VAIMO 2007, P4, Poland 2007
The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$.

Proposed by Mariusz Skalba, Poland
38 replies
malinger
Apr 22, 2007
ezpotd
an hour ago
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3 numbers have their fractional parts lying in the interval
orl   13
N an hour ago by ezpotd
Source: IMO Shortlist 2000, A2
Let $ a, b, c$ be positive integers satisfying the conditions $ b > 2a$ and $ c > 2b.$ Show that there exists a real number $ \lambda$ with the property that all the three numbers $ \lambda a, \lambda b, \lambda c$ have their fractional parts lying in the interval $ \left(\frac {1}{3}, \frac {2}{3} \right].$
13 replies
orl
Aug 10, 2008
ezpotd
an hour ago
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"Eulerian" closed walk with of length less than v+e
Miquel-point   0
Yesterday at 4:56 PM
Source: IMAR 2019 P4
Show that a connected graph $G=(V, E)$ has a closed walk of length at most $|V|+|E|-1$ passing through each edge of $G$ at least once.

Proposed by Radu Bumbăcea
0 replies
Miquel-point
Yesterday at 4:56 PM
0 replies
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Cycle in a graph with a minimal number of chords
GeorgeRP   5
N Yesterday at 3:05 AM by Photaesthesia
Source: Bulgaria IMO TST 2025 P3
In King Arthur's court every knight is friends with at least $d>2$ other knights where friendship is mutual. Prove that King Arthur can place some of his knights around a round table in such a way that every knight is friends with the $2$ people adjacent to him and between them there are at least $\frac{d^2}{10}$ friendships of knights that are not adjacent to each other.
5 replies
GeorgeRP
May 14, 2025
Photaesthesia
Yesterday at 3:05 AM
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Shortest cycle if sum d^2 = n^2 - n
Miquel-point   0
May 14, 2025
Source: KoMaL B. 4218
In a graph, no vertex is connected to all of the others. For any pair of vertices not connected there is a vertex adjacent to both. The sum of the squares of the degrees of vertices is $n^2-n$ where $n$ is the number of vertices. What is the length of the shortest possible cycle in the graph?

Proposed by B. Montágh, Memphis
0 replies
Miquel-point
May 14, 2025
0 replies
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Counting monochromatic squares in K_n
Miquel-point   0
May 14, 2025
Source: KoMaL B. 5035
The edges of a complete graph on $n \ge 8$ vertices are coloured in two colours. Prove that the number of cycles formed by four edges of the same colour is more than $\frac{(n-5)^4}{64}$.

Based on a problem proposed by M. Pálfy
0 replies
Miquel-point
May 14, 2025
0 replies
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Graph theory
VicKmath7   5
N May 14, 2025 by CBMaster
Source: St Petersburg 2007 MO
Find the maximal number of edges a connected graph $G$ with $n$ vertices may have, so that after deleting an arbitrary cycle, $G$ is not connected anymore.
5 replies
VicKmath7
Aug 30, 2021
CBMaster
May 14, 2025
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forced vertices in graphs
Davdav1232   3
N May 14, 2025 by CBMaster
Source: Israel TST 7 2025 p2
Let \( G \) be a graph colored using \( k \) colors. We say that a vertex is forced if it has neighbors in all the other \( k - 1 \) colors.

Prove that for any \( 2024 \)-regular graph \( G \) that contains no triangles or quadrilaterals, there exists a coloring using \( 2025 \) colors such that at least \( 1013 \) of the colors have a forced vertex of that color.

Note: The graph coloring must be valid, this means no \( 2 \) vertices of the same color may be adjacent.
3 replies
Davdav1232
May 8, 2025
CBMaster
May 14, 2025
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Disconnected Tree Subsets
AwesomeYRY   25
N May 13, 2025 by john0512
Source: TSTST 2021/5
Let $T$ be a tree on $n$ vertices with exactly $k$ leaves. Suppose that there exists a subset of at least $\frac{n+k-1}{2}$ vertices of $T$, no two of which are adjacent. Show that the longest path in $T$ contains an even number of edges. *

Vincent Huang
25 replies
AwesomeYRY
Dec 13, 2021
john0512
May 13, 2025
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Social Club with 2k+1 Members
v_Enhance   24
N May 11, 2025 by mathwiz_1207
Source: USA December TST for IMO 2013, Problem 1
A social club has $2k+1$ members, each of whom is fluent in the same $k$ languages. Any pair of members always talk to each other in only one language. Suppose that there were no three members such that they use only one language among them. Let $A$ be the number of three-member subsets such that the three distinct pairs among them use different languages. Find the maximum possible value of $A$.
24 replies
v_Enhance
Jul 30, 2013
mathwiz_1207
May 11, 2025
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Short combi omg
Davdav1232   6
N May 11, 2025 by DeathIsAwe
Source: Israel TST 2025 test 4 p3
Let \( n \) be a positive integer. A graph on \( 2n - 1 \) vertices is given such that the size of the largest clique in the graph is \( n \). Prove that there exists a vertex that is present in every clique of size \( n\)
6 replies
Davdav1232
Feb 3, 2025
DeathIsAwe
May 11, 2025
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Croatian mathematical olympiad, day 1, problem 2
Matematika   6
N May 9, 2025 by Cqy00000000
There were finitely many persons at a party among whom some were friends. Among any $4$ of them there were either $3$ who were all friends among each other or $3$ who weren't friend with each other. Prove that you can separate all the people at the party in two groups in such a way that in the first group everyone is friends with each other and that all the people in the second group are not friends to anyone else in second group. (Friendship is a mutual relation).
6 replies
Matematika
Apr 10, 2011
Cqy00000000
May 9, 2025
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Inequalities
idomybest   3
N Apr 23, 2025 by damyan
Source: The Interesting Around Technical Analysis Three Variable Inequalities
The problem is in the attachment below.
3 replies
idomybest
Oct 15, 2021
damyan
Apr 23, 2025
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Inequalities
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Reveal topic tags
Inequality
inequalities
inequalities unsolved
algebra unsolved
algebra
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Source: The Interesting Around Technical Analysis Three Variable Inequalities
The post below has been deleted. Click to close.
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idomybest
96 posts
idomybest
#1 h  Oct 15, 2021, 9:16 PM
Y by
The problem is in the attachment below.
Attachments:
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arqady
30253 posts
arqady
#2 h  Oct 18, 2021, 9:32 AM • 1 Y
Y by Hoto_Mukai
It's $$\sum_{cyc}ab(a-b)^4+6abc\sum_{cyc}a(a-b)(a-c)\geq0.$$
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VMF-er
512 posts
VMF-er
#3 h  Oct 18, 2021, 11:01 AM
Y by
idomybest wrote:
The problem is in the attachment below.

Stronger
$$LHS-RHS\geq \frac{abc(a^{3}+b^{3}+c^{3}-3abc)}{(b^{2}+bc+c^{2})(c^{2}+ca+a^{2})(a^{2}+ab+b^{2})}.$$
Z K Y
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damyan
18 posts
damyan
#4 h  Apr 23, 2025, 4:01 PM
Y by
arqady wrote:
It's $$\sum_{cyc}ab(a-b)^4+6abc\sum_{cyc}a(a-b)(a-c)\geq0.$$

This is just the left part which is $\geq 0$ and the right is schur
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