Stay ahead of learning milestones! Enroll in a class over the summer!

G
Topic
First Poster
Last Poster
k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
Yesterday at 11:16 PM
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.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]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]
Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
Tuesday, May 13 - Aug 26
Thursday, May 29 - Sep 11
Sunday, Jun 15 - Oct 12
Monday, Jun 30 - Oct 20
Wednesday, Jul 16 - Oct 29

Prealgebra 2 Self-Paced

Prealgebra 2
Wednesday, May 7 - Aug 20
Monday, Jun 2 - Sep 22
Sunday, Jun 29 - Oct 26
Friday, Jul 25 - Nov 21

Introduction to Algebra A Self-Paced

Introduction to Algebra A
Sunday, May 11 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, May 14 - Aug 27
Friday, May 30 - Sep 26
Monday, Jun 2 - Sep 22
Sunday, Jun 15 - Oct 12
Thursday, Jun 26 - Oct 9
Tuesday, Jul 15 - Oct 28

Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Thursday, May 15 - Jul 31
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Wednesday, Jul 9 - Sep 24
Sunday, Jul 27 - Oct 19

Introduction to Number Theory
Friday, May 9 - Aug 1
Wednesday, May 21 - Aug 6
Monday, Jun 9 - Aug 25
Sunday, Jun 15 - Sep 14
Tuesday, Jul 15 - Sep 30

Introduction to Algebra B Self-Paced

Introduction to Algebra B
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
Sunday, May 11 - Nov 9
Tuesday, May 20 - Oct 28
Monday, Jun 16 - Dec 8
Friday, Jun 20 - Jan 9
Sunday, Jun 29 - Jan 11
Monday, Jul 14 - Jan 19

Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)

Intermediate: Grades 8-12

Intermediate Algebra
Sunday, Jun 1 - Nov 23
Tuesday, Jun 10 - Nov 18
Wednesday, Jun 25 - Dec 10
Sunday, Jul 13 - Jan 18
Thursday, Jul 24 - Jan 22

Intermediate Counting & Probability
Wednesday, May 21 - Sep 17
Sunday, Jun 22 - Nov 2

Intermediate Number Theory
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
Friday, May 16 - Oct 24
Sunday, Jun 1 - Nov 9
Monday, Jun 30 - Dec 8

Advanced: Grades 9-12

Olympiad Geometry
Tuesday, Jun 10 - Aug 26

Calculus
Tuesday, May 27 - Nov 11
Wednesday, Jun 25 - Dec 17

Group Theory
Thursday, Jun 12 - Sep 11

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Friday, May 23 - Aug 15
Monday, Jun 2 - Aug 18
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

MATHCOUNTS/AMC 8 Advanced
Sunday, May 11 - Aug 10
Tuesday, May 27 - Aug 12
Wednesday, Jun 11 - Aug 27
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Problem Series
Friday, May 9 - Aug 1
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Tuesday, Jun 17 - Sep 2
Sunday, Jun 22 - Sep 21 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Jun 23 - Sep 15
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Final Fives
Sunday, May 11 - Jun 8
Tuesday, May 27 - Jun 17
Monday, Jun 30 - Jul 21

AMC 12 Problem Series
Tuesday, May 27 - Aug 12
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Wednesday, Aug 6 - Oct 22

AMC 12 Final Fives
Sunday, May 18 - Jun 15

AIME Problem Series A
Thursday, May 22 - Jul 31

AIME Problem Series B
Sunday, Jun 22 - Sep 21

F=ma Problem Series
Wednesday, Jun 11 - Aug 27

WOOT Programs
Visit the pages linked for full schedule details for each of these programs!


MathWOOT Level 1
MathWOOT Level 2
ChemWOOT
CodeWOOT
PhysicsWOOT

Programming

Introduction to Programming with Python
Thursday, May 22 - Aug 7
Sunday, Jun 15 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Tuesday, Jun 17 - Sep 2
Monday, Jun 30 - Sep 22

Intermediate Programming with Python
Sunday, Jun 1 - Aug 24
Monday, Jun 30 - Sep 22

USACO Bronze Problem Series
Tuesday, May 13 - Jul 29
Sunday, Jun 22 - Sep 1

Physics

Introduction to Physics
Wednesday, May 21 - Aug 6
Sunday, Jun 15 - Sep 14
Monday, Jun 23 - Sep 15

Physics 1: Mechanics
Thursday, May 22 - Oct 30
Monday, Jun 23 - Dec 15

Relativity
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
0 replies
jlacosta
Yesterday at 11:16 PM
0 replies
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
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
Inspired by Kunihiko_Chikaya
sqing   0
3 minutes ago
Source: Own
Let $a,\ b,\ c$ be real numbers such that $ a+b+c=1  $ and $ a^2+b^2+c^2=23 . $ Prove that$$ -47\leq a^3+b^3-3c\leq 78$$
0 replies
1 viewing
sqing
3 minutes ago
0 replies
Distributing coins in a circle
quacksaysduck   2
N 11 minutes ago by genius_007
Source: JOM 2025 Mock 1 P4
There are $n$ people arranged in a circle, and $n^{n^n}$ coins are distributed among them, where each person has at least $n^n$ coins. Each person is then assigned a random index number in $\{1,2,...n\}$ such that no two people have the same number. Then every minute, if $i$ is the number of minutes passed, the person with index number congruent to $i$ mod $n$ will give a coin to the person on his left or right. After some time, everyone has the same number of coins.

For what $n$ is this always possible, regardless of the original distribution of coins and index numbers?

(Proposed by Ho Janson)
2 replies
quacksaysduck
Jan 26, 2025
genius_007
11 minutes ago
Familiar cyclic quad config
Rijul saini   11
N 15 minutes ago by ihategeo_1969
Source: India IMOTC Practice Test 1 Problem 2
Let $ABCD$ be a convex cyclic quadrilateral with circumcircle $\omega$. Let $BA$ produced beyond $A$ meet $CD$ produced beyond $D$, at $L$. Let $\ell$ be a line through $L$ meeting $AD$ and $BC$ at $M$ and $N$ respectively, so that $M,D$ (respectively $N,C$) are on opposite sides of $A$ (resp. $B$). Suppose $K$ and $J$ are points on the arc $AB$ of $\omega$ not containing $C,D$ so that $MK, NJ$ are tangent to $\omega$. Prove that $K,J,L$ are collinear.

Proposed by Rijul Saini
11 replies
Rijul saini
May 31, 2024
ihategeo_1969
15 minutes ago
Inspired by Nice inequality
sqing   1
N 17 minutes ago by sqing
Source: Own
Let $  a,b,c >0  $. Show that
$$\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+1\right)^2 \geq \frac{16}{k+3}\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}+k\right)$$Where $ k\geq 1.$
$$\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+1\right)^2 \geq  \frac{b}{a}+\frac{c}{b}+\frac{a}{c}+13$$$$\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+1\right)^2 \geq \frac{16}{5}\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}+2\right)$$
1 reply
sqing
5 hours ago
sqing
17 minutes ago
Hard inequality
ys33   5
N 25 minutes ago by sqing
Let $a, b, c, d>0$. Prove that
$\sqrt[3]{ab}+ \sqrt[3]{cd} < \sqrt[3]{(a+b+c)(b+c+d)}$.
5 replies
1 viewing
ys33
4 hours ago
sqing
25 minutes ago
China Northern MO 2009 p4 CNMO
parkjungmin   1
N 31 minutes ago by exoticc
Source: China Northern MO 2009 p4 CNMO
China Northern MO 2009 p4 CNMO

The problem is too difficult.
Is there anyone who can help me?
1 reply
parkjungmin
Apr 30, 2025
exoticc
31 minutes ago
Hojoo Lee problem 73
Leon   25
N 35 minutes ago by sqing
Source: Belarus 1998
Let $a$, $b$, $c$ be real positive numbers. Show that \[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq \frac{a+b}{b+c}+\frac{b+c}{a+b}+1\]
25 replies
Leon
Aug 21, 2006
sqing
35 minutes ago
Almost Squarefree Integers
oVlad   3
N 42 minutes ago by Primeniyazidayi
Source: Romania Junior TST 2025 Day 1 P1
A positive integer $n\geqslant 3$ is almost squarefree if there exists a prime number $p\equiv 1\bmod 3$ such that $p^2\mid n$ and $n/p$ is squarefree. Prove that for any almost squarefree positive integer $n$ the ratio $2\sigma(n)/d(n)$ is an integer.
3 replies
oVlad
Apr 12, 2025
Primeniyazidayi
42 minutes ago
Math camp combi
ErTeeEs06   3
N 42 minutes ago by genius_007
Source: BxMO 2025 P2
Let $N\geq 2$ be a natural number. At a mathematical olympiad training camp the same $N$ courses are organised every day. Each student takes exactly one of the $N$ courses each day. At the end of the camp, every student has takes each course exactly once, and any two students took the same course on at least one day, but took different courses on at least one other day. What is, in terms of $N$, the largest possible number of students at the camp?
3 replies
ErTeeEs06
Apr 26, 2025
genius_007
42 minutes ago
Benelux fe
ErTeeEs06   10
N an hour ago by genius_007
Source: BxMO 2025 P1
Does there exist a function $f:\mathbb{R}\to \mathbb{R}$ such that $$f(x^2+f(y))=f(x)^2-y$$for all $x, y\in \mathbb{R}$?
10 replies
ErTeeEs06
Apr 26, 2025
genius_007
an hour ago
IMO Shortlist Problems
ABCD1728   0
an hour ago
Source: IMO official website
Where can I get the official solution for ISL before 2005? The official website only has solutions after 2006. Thanks :)
0 replies
ABCD1728
an hour ago
0 replies
Geometric inequality in quadrilateral
BBNoDollar   0
an hour ago
Source: Romanian Mathematical Gazette 2025
Let ABCD be a convex quadrilateral with angles BAD and BCD obtuse, and let the points E, F ∈ BD, such that AE ⊥ BD and CF ⊥ BD.
Prove that 1/(AE*CF) ≥ 1/(AB*BC) + 1/(AD*CD) .
0 replies
BBNoDollar
an hour ago
0 replies
A coincidence about triangles with common incenter
flower417477   2
N 2 hours ago by flower417477
$\triangle ABC,\triangle ADE$ have the same incenter $I$.Prove that $BCDE$ is concyclic iff $BC,DE,AI$ is concurrent
2 replies
flower417477
Wednesday at 2:08 PM
flower417477
2 hours ago
Function equation
LeDuonggg   5
N 2 hours ago by luutrongphuc
Find all functions $f: \mathbb{R^+} \rightarrow \mathbb{R^+}$ , such that for all $x,y>0$:
\[ f(x+f(y))=\dfrac{f(x)}{1+f(xy)}\]
5 replies
LeDuonggg
Yesterday at 2:59 PM
luutrongphuc
2 hours ago
Connected graph with k edges
orl   26
N Apr 16, 2025 by Maximilian113
Source: IMO 1991, Day 2, Problem 4, IMO ShortList 1991, Problem 10 (USA 5)
Suppose $ \,G\,$ is a connected graph with $ \,k\,$ edges. Prove that it is possible to label the edges $ 1,2,\ldots ,k\,$ in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1.

Note: Graph-Definition. A graph consists of a set of points, called vertices, together with a set of edges joining certain pairs of distinct vertices. Each pair of vertices $ \,u,v\,$ belongs to at most one edge. The graph $ G$ is connected if for each pair of distinct vertices $ \,x,y\,$ there is some sequence of vertices $ \,x = v_{0},v_{1},v_{2},\cdots ,v_{m} = y\,$ such that each pair $ \,v_{i},v_{i + 1}\;(0\leq i < m)\,$ is joined by an edge of $ \,G$.
26 replies
orl
Nov 11, 2005
Maximilian113
Apr 16, 2025
Connected graph with k edges
G H J
Source: IMO 1991, Day 2, Problem 4, IMO ShortList 1991, Problem 10 (USA 5)
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
orl
3647 posts
#1 • 3 Y
Y by Adventure10, megarnie, GeoKing
Suppose $ \,G\,$ is a connected graph with $ \,k\,$ edges. Prove that it is possible to label the edges $ 1,2,\ldots ,k\,$ in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1.

Note: Graph-Definition. A graph consists of a set of points, called vertices, together with a set of edges joining certain pairs of distinct vertices. Each pair of vertices $ \,u,v\,$ belongs to at most one edge. The graph $ G$ is connected if for each pair of distinct vertices $ \,x,y\,$ there is some sequence of vertices $ \,x = v_{0},v_{1},v_{2},\cdots ,v_{m} = y\,$ such that each pair $ \,v_{i},v_{i + 1}\;(0\leq i < m)\,$ is joined by an edge of $ \,G$.
This post has been edited 1 time. Last edited by orl, Aug 15, 2008, 1:51 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Peter
3615 posts
#2 • 4 Y
Y by Wizard_32, Adventure10, Mango247, GeoKing
Click to reveal hidden text

Start at any vertex A and number 1,2,3,... along any path. every vertex you pass will have edges k,k+1 so has gcd=1, until you reach a dead end. (which is either a vertex with degree 1 (no prob) or a vertex on which all edges are labeled yet (no prob). Now if there are unlabeled edges left, start at any vertex of such an edge, and repeat the procedure. Like that you can fill the graph such that all vertices have gcd(edges)=1.

remark
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
perfect_radio
2607 posts
#3 • 2 Y
Y by Adventure10, Mango247
Can I make a small remark? It's kind of late now, maybe I'm missing something obvious.

Small (Redundant) Remark
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Peter
3615 posts
#4 • 2 Y
Y by Adventure10, Mango247
I don't see your point... where would I use non-randomness? Note that we are labeling edges, not vertices.

Either a vertex has only one edge (in which case there's nothing to prove), or either there are at least 2, and by the algorithm we will ever run through it and make that vertex thus have at least 2 coprime edges.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
tim1234133
523 posts
#5 • 1 Y
Y by Adventure10
I don't think this is the point that perfect_radio is making, but anyway... I don't see where your proof uses connectivity. When we have gone through the first set of labelling, you say that if there are unlabelled edges (Sorry about the typo that was here...) 'start at any vertex of such an edge'. How do we know that this vertex (the one we start on the second time we make a path) has gcd(edges)=1?
Can we get round this by saying 'start at a vertex that has some already labelled vertices', and hence use the fact that the graph is connected? Or am I just missing something altogether?
This post has been edited 1 time. Last edited by tim1234133, Aug 25, 2006, 7:22 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
JBL
16123 posts
#6 • 2 Y
Y by Adventure10, Mango247
tim1234133 wrote:
I don't think this is the point that perfect_radio is making, but anyway... I don't see where your proof uses connectivity. When we have gone through the first set of labelling, you say that if there are unlabelled vertices 'start at any vertex of such an edge'. How do we know that this vertex (the one we start on the second time we make a path) has gcd(edges)=1?
Can we get round this by saying 'start at a vertex that has some already labelled vertices', and hence use the fact that the graph is connected? Or am I just missing something altogether?

"Start at a vertex that has already labelled edges" is what he meant. Such a vertex exists by connectivity.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Peter
3615 posts
#7 • 2 Y
Y by Adventure10, Mango247
[here was a wrong idea, thx tim1234133.]

For the rest of your remark, tim1234133, I will say it again: we are labeling edges, NOT vertices. If it was only a typo please reformulate your question in the right way as there are many interpretations possible.
This post has been edited 1 time. Last edited by Peter, Aug 25, 2006, 7:35 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
tim1234133
523 posts
#8 • 2 Y
Y by Adventure10, Mango247
How do we label the (unconnected) graph formed of two unconnected triangles (If we call the vertices A-F we have A connected to B and C with B and C also connected to each other, and D connected to E and F with E and F connected to each other) in the required manner?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Peter
3615 posts
#9 • 1 Y
Y by Adventure10
Err... never mind, we really need connectedness indeed. :oops:
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
perfect_radio
2607 posts
#10 • 2 Y
Y by Adventure10, Mango247
Yes, that was also my point. It's too bad I couldn't explain it in a more understandable manner :(
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
GoldenFrog1618
667 posts
#11 • 1 Y
Y by Adventure10
I will induct on the number of edges and keep the number of verticies fixed.

For a base case, we need to prove that all trees can be labeled. Notice that there is a path between all the verticies of degree >=2. Thus, we should label these edges in order 1,2,3,4 ..., n. (Include the two edges connected to a vertex of degree 1).
Call vertex with no edges labeled "unlabeled" since any permutation of the labels will work.
Note that there are only 2 labeled degree one verticies.

Now we progressively add edges to the graph until we get the desired number of edges.

Case 1: A vertex of degree >=2 connected to a vertex of degree >=2.
There is nothing needed to be done.
Case 2: An unlabeled vertex of degree 1 connected to a vertex of degree >=2.
Label the new edge and the unlabeled edge two consectutive numbers.
Case 3: An unlabeled vertex of degree 1 connected to an unlabeled vertex of degree 1.
Label the two unlabeled edges and the new edge three consecutive numbers.
Case 4: A labeled vertex of degree 1 connected to a vertex of degree >=2.
Label this new edge either n+1 if the label is n or keep it unlabeled if the label is 1.
Case 5: A labled vertex of degree 1 connected to a labeled vertex of degree 1.
Label the connection n+1.
Case 6: A labeled vertex of degree 1 connected to an unlabeled vertex of degree 1.
If the labled edge is n, label the two edges n+1 and n+2 (increase other edges if necessary).
If the labled edge is 1, label the two edges consecutive numbers.

Notice that there are at most two instances of connecting a labeled vertex to another vertex, so we are done.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
shekast-istadegi
81 posts
#12 • 2 Y
Y by Adventure10, Mango247
This is easy probelm_hint_>we are doing the graph Euler Garlic by Connect odd edges of the graph and we use induaction.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Generic_Username
1088 posts
#13 • 2 Y
Y by Adventure10, Mango247
wrong
This post has been edited 1 time. Last edited by Generic_Username, Jan 4, 2017, 6:11 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
dgrozev
2463 posts
#14 • 3 Y
Y by Generic_Username, Adventure10, Mango247
It's not so trivial. Suppose, we have 4 vertices: $a,b,c,d$ s.t. $\{b,c,d\}$ are connected with each other and $a$ is connected only with $b$. Following the above, we label $ab$ as $1$; $bc, bd$ as $2$ and $3$ respectively and $cd$ as 4. Look at the vertex $c$, its edges are labeled as $2,4$ !?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
yayups
1614 posts
#15 • 1 Y
Y by Adventure10
Not 100\% sure if this works, but here it is.

It suffices to have two adjacent labels at every vertex. Consider the path of maximum length. Label its edges consecutively, and then color all the edges red. Now, consider the maximum length path in non-red edges and repeat. Call path $k$ the maximum length path that we chose on the $k$th iteration. We claim that this works.

Note that all vertices that are not on the ends of any paths are automatically taken care of. The only potential issue is that there could be some vertex $v$ that is always only on the end of paths. If $v$ has degree $1$. then this isn't actually a problem. So suppose $v$ has degree at least $2$. It only appears on the end of paths, so there exist $i,j$ such that $v$ is on an end of path $i$ and path $j$ (WLOG $i<j$). But then, gluing together paths $i$ and $j$ results in a path of longer length at stage $i$ (all of path $j$ is still not red since it was not red at stage $j$). One possible issue is that if paths $i$ and $j$ also share their other respective ends, but this can easily solved by deleting one edge in the cycle that's formed when we glued them (still leads to a longer path on stage $i$). Therefore, we must have $\mathrm{deg}v=1$, so every vertex now has two adjacent labels adjacent to it.

EDIT: This is wrong, consider a cycle. I will try to fix soon...
This post has been edited 1 time. Last edited by yayups, Sep 1, 2018, 4:28 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
MathStudent2002
934 posts
#16 • 4 Y
Y by cosmicgenius, AlastorMoody, Adventure10, Mango247
Epic problem. Solution with ewan.

Say all degrees are even. Then, we have an Eulerian circuit, which we just label $1, 2, \ldots, k$ in succession. Then, each vertex is labeled either with 2 consecutive integers or with $1$ and $k$ (or both) so we are done.

Else, consider the graph $G'$ where we add edges to $G$ between pairs of odd-degree vertices. Then, $G'$ has an Eulerian circuit. Removing the added edges from the circuit turns it into a set of walks with union $E(G)$ such that each walk starts and ends at odd-degree vertices and each odd-degree is the endpoint of exactly one walk.

Now, if the walks have lengths $k_1, \ldots, k_m$, then label the first one with $[1, k_1]$ in order, the second one with $[k_1+1, k_1+k_2]$, etc., and the last one with $[k_1+\cdots+k_{m-1}+1, k_1+\cdots+k_m]$. We see that each non-leaf vertex must appear in the interior (i.e. non-endpoint) of some walk and thus have two edges labeled with consecutive numbers, and hence have $\gcd$ of edge labels equal to one. $\blacksquare$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
dgrozev
2463 posts
#17 • 1 Y
Y by Adventure10
MathStudent2002 wrote:
...Now, if the walks have lengths $k_1, \ldots, k_m$, then label the first one with $[1, k_1]$ in order, the second one with $[k_1+1, k_1+k_2]$, etc., and the last one with $[k_1+\cdots+k_{m-1}+1, k_1+\cdots+k_m]$. We see that each non-leaf vertex must appear in the interior (i.e. non-endpoint) of some walk and thus have two edges labeled with consecutive numbers, and hence have $\gcd$ of edge labels equal to one. $\blacksquare$

What if the second walk, for example, begins and ends to one and the same vertex? So you assign $k_1+1$ and $k_1+k_2$ to those first and last edges and of course the may not be coprime.
I know it can be patched somehow, but then that idea of Euler cycles will lost its originality and it would be dissolved into something like in post #2.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Pathological
578 posts
#18 • 1 Y
Y by Adventure10
I think what he/she is doing in the third paragraph is the following.

Let $v_1, v_2, \cdots, v_{2t}$ be the vertices of odd degree of $G$ (there's an even number of them by the Handshake Lemma, say). If $t = 0$, then simply take an Eulerian cycle and label the edges $1, 2, \cdots, k$ in order. Otherwise, we will add the edges $v_1v_2, v_3v_4, \cdots, v_{2t-1} v_{2t}.$ Let this modified graph be $G'.$ Note that this may result in the graph not being simple anymore, but that's fine since the problem still holds. Now, observe that $G'$ has all degree evens, and hence contains an Eulerian cycle. Then, it's clear that the removal of $v_1v_2, v_3v_4, \cdots, v_{2t-1} v_{2t}$ from the cycle would partition the cycle into walks from $v_2$ to $v_3$, from $v_4$ to $v_5$, $\cdots$, from $v_{2t}$ to $v_1.$ This makes it such that none of the walks can be cycles since the $v_i$'s are distinct, and so we finish as above.

Edit: Also, as far as I can tell, the idea in post #2 does not work, since it does not account for cycles (what if the walk ends at the same vertex it began with, and we label the edges $2, 3, 4$?). The solution in post #16, on the other hand, obviates the possibility of cycles using the odd degree trick.
This post has been edited 1 time. Last edited by Pathological, Jun 20, 2019, 11:12 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
SnowPanda
186 posts
#19 • 2 Y
Y by hakN, jelena_ivanchic
Solution
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
vsamc
3789 posts
#20 • 1 Y
Y by Captainscrubz
Solution
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
HamstPan38825
8857 posts
#21 • 1 Y
Y by chenghaohu
The idea here is that fixing two edges around a vertex essentially makes it work, so we can work greedily.

Call a vertex valid if we have labeled two edges through it with relatively prime indices. It suffices to make all vertices valid.

Consider the longest walk in $G$, and label the edges along this walk in the order $1, 2, \dots, k$ consecutively. By maximality of the walk, every vertex in the walk (including the endpoints) is now valid; now, consider the subgraph $G'$ formed by removing all edges and any leaves in the walk, and continue the process until all edges are exhausted.
This post has been edited 2 times. Last edited by HamstPan38825, Aug 2, 2023, 3:05 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
thdnder
197 posts
#22 • 2 Y
Y by Upwgs_2008, taki09
An inductive solution.
Let $G = (V,E)$. We induct on $|V|$, assuming $|V| \geq 3$. Assume the problem statement is true on all connected graph with the number of vertices less than $|V|$. Since $G$ is connected, let $T$ be the spanning tree of $G$. Let $v$ be a leaf of $T$. Then $G-v$ is still connected, so by the induction hypothesis, we can label the edges of $G-v$ satisfying the problem condition. Let $\deg v = s$. Then if $s \geq 2$, $\gcd(k, k-1, \dots, k-s+1) = 1$ so labeling the remaining edges in arbitrary order will be satisfying the condition. Thus we're done.
This post has been edited 1 time. Last edited by thdnder, Dec 29, 2023, 11:48 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
kamatadu
480 posts
#23 • 3 Y
Y by GeoKing, channing421, taki09
We induct with the base case being clearly true.

If $G$ has a cycle and the cycle is connected to some other component, then there is a vertex with $\ge 3$ degree.

Now remove an edge from this vertex thus breaking the cycle but the graph still remaining connected. Then by our induction hypothesis, we can get a number of the edges using $1,\ldots, k-1$.

Now, after removing the edge, we note taht the gcd of the vertex $=1$. So adding back the removed edge, the $\gcd$ would still be $=1$.

Now if the graph has only a cycle (i.e. $G$ has $k$ edges and $k$ vertices as shown below).
https://i.imgur.com/bslnMTq.png

Then we can just number the edges in an increasing order by $1$.

Otherwise, if $G$ is a tree, then also we can pick a path between the leaves of the tree and label the edges in a increasing order and also the subtrees in a similar increasing by $1$ order. It is easy to see that this works.
https://i.imgur.com/QCzuiz8.png
:yoda:
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
lelouchvigeo
181 posts
#24 • 1 Y
Y by taki09
Sketch of the solution
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
renrenthehamster
40 posts
#25 • 1 Y
Y by taki09
If you are familiar with depth first search (DFS) algorithm, simply perform depth-first search starting at any chosen root $v$ and label the edges in the order of discovery. Other than the root $v$, all other vertices with at least degree 2 must have a pair of consecutive integer labelled edges (when you first enter the vertex by an edge, you still have at least an undiscovered edge and DFS forces you to immediately discover one of your undiscovered edge).

The exception to this argument is the root, but the root has an edge with label 1. By the way this is why you need the graph to be connected - so that you only have 1 root to deal with.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
IMD2
41 posts
#26 • 1 Y
Y by taki09
thdnder wrote:
An inductive solution.
Let $G = (V,E)$. We induct on $|V|$, assuming $|V| \geq 3$. Assume the problem statement is true on all connected graph with the number of vertices less than $|V|$. Since $G$ is connected, let $T$ be the spanning tree of $G$. Let $v$ be a leaf of $T$. Then $G-v$ is still connected, so by the induction hypothesis, we can label the edges of $G-v$ satisfying the problem condition. Let $\deg v = s$. Then if $s \geq 2$, $\gcd(k, k-1, \dots, k-s+1) = 1$ so labeling the remaining edges in arbitrary order will be satisfying the condition. Thus we're done.

What if the vertex $s$ is connected to goes from degree 1 to degree 2 when you include $s$?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Maximilian113
574 posts
#27
Y by
Call a vertex with degree at least $2$ "satisfied" if the labels of its adjacent edges have greatest common divisor $1.$

Consider the following algorithm:
For some $k$-cycle, assign the edges the $k$ largest available labels, in increasing order through one direction of the cycle. This ensures that we can remove the second-largest edge and both of its vertices will already have been satisfied. (as consecutive numbers are coprime)
Keep doing this until there are no cycles and the graph is a tree. Then each vertex with degree $1$ in this tree but with degree more than $1$ in the original graph is satisfied.
At this point if there are $r$ edges in this tree, we have the first $r$ numbers left as labels. For each vertice adjacent to a leaf, if it is adjacent to $\ell$ leaves assign their connections the $\ell$ largest available labels, and we can proceed inductively as this vertice will have been satisfied, and we are left with a smaller tree of the same scenario. QED
Z K Y
N Quick Reply
G
H
=
a