arcs with all integer lengths from 1 to 89 by 10 points on circle

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.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. 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 events:
[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
[*]April 8th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS State Discussion
April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/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
Sunday, Apr 13 - Aug 10
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
Sunday, Apr 13 - Aug 10
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
Monday, Apr 7 - Jul 28
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
Wednesday, Apr 16 - Jul 2
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
Thursday, Apr 17 - Jul 3
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
Wednesday, Apr 16 - Jul 30
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
Wednesday, Apr 23 - Oct 1
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

Intermediate: Grades 8-12

Intermediate Algebra
Monday, Apr 21 - Oct 13
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
Friday, Apr 11 - Jun 27
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
Wednesday, Apr 9 - Sep 3
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
Wednesday, Apr 16 - Jul 2
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
Friday, Apr 11 - Jun 27
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

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
Sat & Sun, Apr 26 - Apr 27 (4:00 - 7:00 pm ET/1:00 - 4:00pm PT)
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)

0 replies

jlacosta

Apr 2, 2025

0 replies

Inequality with a,b,c

GeoMorocco 1

N 8 minutes ago by Natrium

Source: Morocco Training

Let $a,b,c$ be positive real numbers such that : $ab+bc+ca=3$ . Prove that : $\frac{\sqrt{1+a^2}}{1+ab}+\frac{\sqrt{1+b^2}}{1+bc}+\frac{\sqrt{1+c^2}}{1+ca}\ge \sqrt{\frac{3(a+b+c)}{2}}$

1 reply

GeoMorocco

Yesterday at 10:05 PM

Natrium

8 minutes ago

NEPAL TST 2025 DAY 2

Tony_stark0094 2

N 9 minutes ago by ThatApollo777

Consider an acute triangle $\Delta ABC$ . Let $D$ and $E$ be the feet of the altitudes from $A$ to $BC$ and from $B$ to $AC$ respectively.

Define $D_1$ and $D_2$ as the reflections of $D$ across lines $AB$ and $AC$ , respectively. Let $\Gamma$ be the circumcircle of $\Delta AD_1D_2$ . Denote by $P$ the second intersection of line $D_1B$ with $\Gamma$ , and by $Q$ the intersection of ray $EB$ with $\Gamma$ .

If $O$ is the circumcenter of $\Delta ABC$ , prove that $O$ , $D$ , and $Q$ are collinear if and only if quadrilateral $BCQP$ can be inscribed within a circle.

2 replies

Tony_stark0094

Today at 8:40 AM

ThatApollo777

9 minutes ago

IMO Shortlist 2013, Number Theory #1

lyukhson 149

N 14 minutes ago by SSS_123

Source: IMO Shortlist 2013, Number Theory #1

Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that
$m^2 + f(n) \mid mf(m) +n$
for all positive integers $m$ and $n$ .

149 replies

lyukhson

Jul 10, 2014

SSS_123

14 minutes ago

Number Theory Chain!

JetFire008 32

N 21 minutes ago by TopGbulliedU

I will post a question and someone has to answer it. Then they have to post a question and someone else will answer it and so on. We can only post questions related to Number Theory and each problem should be more difficult than the previous. Let's start!

Question 1

Starting with the simplest
What is $2+3$ ?

32 replies

JetFire008

Apr 7, 2025

TopGbulliedU

21 minutes ago

No more topics!

arcs with all integer lengths from 1 to 89 by 10 points on circle

parmenides51 1

N Feb 28, 2025 by yamamaya

Source: Champions Tournament (Ukraine) - Турнір чемпіонів - 2018 Juniors p4

Given a circle of length $90$ . Is it possible to mark $10$ points on it so that among the arcs with ends at these points there are arcs with all integer lengths from $1$ to $89$ ?

1 reply

parmenides51

May 21, 2022

yamamaya

Feb 28, 2025

arcs with all integer lengths from 1 to 89 by 10 points on circle

G H J

Reveal topic tags

combinatorial geometry

combinatorics

geometry

Champions Tournament

G H BBookmark kLocked kLocked NReply

Source: Champions Tournament (Ukraine) - Турнір чемпіонів - 2018 Juniors p4

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

parmenides51

30630 posts

parmenides51

#1 h May 21, 2022, 7:53 AM

Y by

Given a circle of length $90$ . Is it possible to mark $10$ points on it so that among the arcs with ends at these points there are arcs with all integer lengths from $1$ to $89$ ?

This post has been edited 2 times. Last edited by parmenides51, May 21, 2022, 7:56 AM

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

yamamaya

76 posts

yamamaya

#2 h Feb 28, 2025, 8:31 AM

Y by

1. There are a total of $10$ x $9=90$ arcs with these $10$ points as endpoints. If there is an arc of length $a$ , $1\le a \le 89$ , there must be an equal number of arcs of length $90-a$ . So if the length of the arc covers $1$ to $89$ , then the arc with a length of $45$ appears twice, and the arcs of other lengths appear exactly once. So there are $46$ arcs with odd lengths.
2. Take any point $A$ and put all points with even distances to $A$ , as well as $A$ itself, into the set $X$ . Put the other points into set $Y$ . Note that $|X|=x$ , $|Y|=y$ , and then $x+y=10$ .
3. Note that arcs with any point in $X$ and any point in $Y$ as endpoints (there are $2$ such arcs) have an odd length. So $2xy=46$ .
4. There is no such integer $x$ , $y$ . Therefore, there is no such circular division.

Consider if the circle of length $n(n-1)+1$ could be marked by $n$ points, and then the arcs length could cover from $1$ to $n(n-1)$ exactly once? We named the problem as $P(n)$ , and consider for which kind of $n$ , $P(n)$ is solvable.
This is a completely different problem and I have not solved it yet.

Z K Y

N Quick Reply