,![\[
-\frac{1}{3} \leq \frac{\sin(n\theta)}{n \sin \theta} \leq \frac{\sqrt{6}}{9}
\quad \text{for } \frac{\pi}{n} \leq \theta \leq \pi - \frac{\pi}{n},
\]](http://latex.artofproblemsolving.com/d/e/3/de361adc16c54cc154fda34c66436fd15978e55b.png)
![\[
-\frac{1}{3} \leq \frac{\sin(n\theta)}{n \sin \theta} \leq \frac{1}{5}
\quad \text{for } \frac{\pi}{n} \leq \theta \leq \frac{\pi}{2}.
\]](http://latex.artofproblemsolving.com/9/f/d/9fd0aeb1cf4958d14a076656fd034c33cdc16ead.png)
Y by Demetri
In algebra chapter 19 videos Graphing exponents part 2 is before graphing exponentials part 1 as you can see here https://artofproblemsolving.com/videos/algebra1/chapter19/589 please fix this asap site support.
G
Topic
First Poster
Last Poster
k a June Highlights and 2025 AoPS Online Class Information
jlacosta 0
Congratulations to all the mathletes who competed at National MATHCOUNTS! If you missed the exciting Countdown Round, you can watch the video at this link. Are you interested in training for MATHCOUNTS or AMC 10 contests? How would you like to train for these math competitions in half the time? We have accelerated sections which meet twice per week instead of once starting on July 8th (7:30pm ET). These sections fill quickly so enroll today!
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC 10 Problem Series[/list]
For those interested in Olympiad level training in math, computer science, physics, and chemistry, be sure to enroll in our WOOT courses before August 19th to take advantage of early bird pricing!
Summer camps are starting this 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 a transformative 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][*]June 5th, Thursday, 7:30pm ET: Open Discussion with Ben Kornell and Andrew Sutherland, Art of Problem Solving's incoming CEO Ben Kornell and CPO Andrew Sutherland host an Ask Me Anything-style chat. Come ask your questions and get to know our incoming CEO & CPO!
[*]June 9th, Monday, 7:30pm ET, Game Jam: Operation Shuffle!, Come join us to play our second round of Operation Shuffle! If you enjoy number sense, logic, and a healthy dose of luck, this is the game for you. No specific math background is required; all are welcome.[/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.
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Prealgebra 1 Self-Paced
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[list][*]MATHCOUNTS/AMC 8 Basics
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[*]AMC 10 Problem Series[/list]
For those interested in Olympiad level training in math, computer science, physics, and chemistry, be sure to enroll in our WOOT courses before August 19th to take advantage of early bird pricing!
Summer camps are starting this 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 a transformative 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][*]June 5th, Thursday, 7:30pm ET: Open Discussion with Ben Kornell and Andrew Sutherland, Art of Problem Solving's incoming CEO Ben Kornell and CPO Andrew Sutherland host an Ask Me Anything-style chat. Come ask your questions and get to know our incoming CEO & CPO!
[*]June 9th, Monday, 7:30pm ET, Game Jam: Operation Shuffle!, Come join us to play our second round of Operation Shuffle! If you enjoy number sense, logic, and a healthy dose of luck, this is the game for you. No specific math background is required; all are welcome.[/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
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0 replies
Answer isn’t displayed MATHCOUNTS
SandCanyon 2
N
by cz1917
The answer isn’t displayed when I type it in. I will type in the answer and the text cursor will move to the right clearly showing that I typed it in but no text will be shown. I can press submit and it will still register the answer properly how I entered it, but I can’t actually see what I entered in until it shows me the answer I put in on the solution screen.
I have a screen recording but I don’t know how to submit it
I have a screen recording but I don’t know how to submit it
2 replies
maximal order
loup blanc 1
N
by loup blanc
Let 
be the set of the permutation matrices of dimension 
. If 
, then 
.
Find
.
be the set of the permutation matrices of dimension 
. If 
, then 
.Find
.
1 reply
Putnam 2000 B2
ahaanomegas 22
N
by mudkip42
Prove that the expression ![\[ \dfrac {\text {gcd}(m, n)}{n} \dbinom {n}{m} \]](//latex.artofproblemsolving.com/c/0/a/c0a6877cd49e904728b24e7743fe8021efd6d312.png)
is an integer for all pairs of integers 
.
![\[ \dfrac {\text {gcd}(m, n)}{n} \dbinom {n}{m} \]](http://latex.artofproblemsolving.com/c/0/a/c0a6877cd49e904728b24e7743fe8021efd6d312.png)
is an integer for all pairs of integers 
.
22 replies
f(x) = x^4 - 2x^3 + 10x + 15
fxandi 3
N
by fxandi
By using the intermediate value theorem, show that 
reaches a global minimum
reaches a global minimum
3 replies
Show that if \( d_3 < \frac{d_1}{3} \), then there exist two other positive diag
Martin.s 1
N
by Martin.s
Let
![\[
D = \begin{bmatrix}
d_1 & 0 & 0 \\
0 & d_2 & 0 \\
0 & 0 & d_3
\end{bmatrix}, \quad
T = \begin{bmatrix}
2 & -1 & 0 \\
-1 & 2 & -1 \\
0 & -1 & 2
\end{bmatrix}
\]](//latex.artofproblemsolving.com/d/6/f/d6fb5d5c862514bf28eaa74e1a330b42d70315fd.png)
where 
are positive and 
.
Show that if
, then there exist two other positive diagonal matrices 
and 
such that 
are distinct but 
have the same eigenvalues.
Show also that if
and is a positive diagonal matrix distinct from 
, then 
and 
must have different eigenvalues.
![\[
D = \begin{bmatrix}
d_1 & 0 & 0 \\
0 & d_2 & 0 \\
0 & 0 & d_3
\end{bmatrix}, \quad
T = \begin{bmatrix}
2 & -1 & 0 \\
-1 & 2 & -1 \\
0 & -1 & 2
\end{bmatrix}
\]](http://latex.artofproblemsolving.com/d/6/f/d6fb5d5c862514bf28eaa74e1a330b42d70315fd.png)
where 
are positive and 
.Show that if
, then there exist two other positive diagonal matrices 
and 
such that 
are distinct but 
have the same eigenvalues.Show also that if
and is a positive diagonal matrix distinct from 
, then 
and 
must have different eigenvalues.
1 reply
AMM 12503 (Juggling Sequences and Derangements)
kgator 0
Source: American Mathematical Monthly Volume 132 (2025), Issue 1: https://doi.org/10.1080/00029890.2024.2412964
12503. Proposed by Richard Ehrenborg, University of Kentucky, Lexington, KY. Fix a positive integer 
. For an arbitrary integer 
, let ![$[t]_d$](//latex.artofproblemsolving.com/9/7/f/97f4a203c1bef2928ba924f7e7543923f3359e10.png)
be the least nonnegative residue of modulo . A -tuple 
of nonnegative integers is a juggling sequence if the -tuple 
defined by ![$\pi_t = [t + a_t]_d$](//latex.artofproblemsolving.com/0/2/e/02e0796a4b8db22cef2c02097a3e964eaee6732d.png)
is a permutation of 
. Let 
be the number of juggling sequences of length with entries in 
.
(a) Prove
for any positive integer 
.
(b) Prove
for any positive integer .
. For an arbitrary integer 
, let ![$[t]_d$](http://latex.artofproblemsolving.com/9/7/f/97f4a203c1bef2928ba924f7e7543923f3359e10.png)
be the least nonnegative residue of modulo . A -tuple 
of nonnegative integers is a juggling sequence if the -tuple 
defined by ![$\pi_t = [t + a_t]_d$](http://latex.artofproblemsolving.com/0/2/e/02e0796a4b8db22cef2c02097a3e964eaee6732d.png)
is a permutation of 
. Let 
be the number of juggling sequences of length with entries in 
.(a) Prove
for any positive integer 
.(b) Prove
for any positive integer .
0 replies
Divisibility of k+1 integers by n
TheIntegral 1
N
by heed
Source: UM Mathematical Olympiad 2025 (B1)
Show that among any 
integers, we can find at least 
of them such that their sum is divisible by for any nonnegative integers 
and .
integers, we can find at least 
of them such that their sum is divisible by for any nonnegative integers 
and .
1 reply
Putnam 1980 B3
sqrtX 3
N
by Ilkin_H
Source: Putnam 1980
For which real numbers 
does the sequence 
defined by the initial condition 
and the recursion 
have 
for all 
does the sequence 
defined by the initial condition 
and the recursion 
have 
for all 
3 replies
BA = A^2 + B, then AB=BA?
Cubastic_Boy 8
N
by Ilkin_H
Let 
2 matrices such that 
. Prove that
2 matrices such that 
. Prove that
8 replies
Asymptote caching problem
char0221 8
N
by Ravensrule8
Summary of the problem: Asymptote caches the wrong diagram.
Page URL: Any community URL
Steps to reproduce:
1. Type the following inside asy tags:
[code]
draw(unitcircle);
[/code]
2. Preview/post the result
3. Watch as your unit circle becomes a weird triangle diagram
Expected behavior: It should display a unit circle. This is caused by caching, because comments can revert the effects.
Frequency: 100%
Operating system(s): macOS Sequoia 15.5
Browser(s), including version: Safari 18.5
Additional information:
IMAGE
Page URL: Any community URL
Steps to reproduce:
1. Type the following inside asy tags:
[code]
draw(unitcircle);
[/code]
2. Preview/post the result
3. Watch as your unit circle becomes a weird triangle diagram
Expected behavior: It should display a unit circle. This is caused by caching, because comments can revert the effects.
Frequency: 100%
Operating system(s): macOS Sequoia 15.5
Browser(s), including version: Safari 18.5
Additional information:
IMAGE
8 replies
k Is there a forum?
CurlyFalcon55 1
N
by jlacosta
I'm not sure where to post this, but does anyone know if there is a forum where you can ask for help on problems? I finished my class and the Message Board is locked, but I still have access to the problems.
I don't think you can post in Alcumus, can you? Can you post in the high school forum or middle school forum for questions like this? Thanks!
(Sorry if this is the wrong forum.)
I don't think you can post in Alcumus, can you? Can you post in the high school forum or middle school forum for questions like this? Thanks!
(Sorry if this is the wrong forum.)
1 reply
k Why Did Gamebot Spam?
PikaPika999 2
N
by PikaPika999
So basically a few minutes ago, gamebot posted a new topic ~8 times, why did it do this? I got screenshot to prove:
Also the screenshot isn't full, it wouldn't fit in my computer screen.
Also the screenshot isn't full, it wouldn't fit in my computer screen.
2 replies
wrong video order
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