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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

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Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

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[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
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0 replies
jlacosta
Mar 2, 2025
0 replies
J
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Concavity of a function
pii-oner   0
5 hours ago
Hi everyone,

I am studying the concavity of the function

\[ f(x) = \sqrt{1 - x^a}, \quad a \geq 0 \]
on the interval \( x \in [0,1] \).

I computed the second derivative and found that for \( a \geq 1 \), the function appears to be concave. However, I am uncertain about the behavior at the endpoints.

Does anyone have insights on confirming concavity rigorously for \( a \geq 1 \) and understanding the behavior at the endpoints? Any help would be greatly appreciated!

Thanks!
0 replies
pii-oner
5 hours ago
0 replies
J
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Putnam 2017 B3
goveganddomath   34
N Today at 4:16 AM by OronSH
Source: Putnam
Suppose that $$f(x) = \sum_{i=0}^\infty c_ix^i$$is a power series for which each coefficient $c_i$ is $0$ or $1$. Show that if $f(2/3) = 3/2$, then $f(1/2)$ must be irrational.
34 replies
goveganddomath
Dec 3, 2017
OronSH
Today at 4:16 AM
J
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3-dimensional matrix system
loup blanc   1
N Today at 3:33 AM by alexheinis
Let $A=\begin{pmatrix}1&1&0\\0&1&1\\0&0&1\end{pmatrix}$.
i) Find the matrices $B\in M_3(\mathbb{R})$ s.t. $A^TA=B^TB,AA^T=BB^T$.
EDIT. ii) Show that each solution of i) is in $M_3(K)$, where $K=\mathbb{Q}[\cos(\dfrac{2}{3}\arctan(3\sqrt{3}))]$.
iii) Solve i) when $B\in M_3(\mathbb{C})$.
EDIT. In iii) we again consider the transpose of $B$ and not its conjugate transpose.
1 reply
loup blanc
Yesterday at 1:04 PM
alexheinis
Today at 3:33 AM
J
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Square of a rational matrix of dimension 2
loup blanc   9
N Today at 1:28 AM by ysharifi
The following exercise was posted -two months ago- on the Website StackExchange; cf.
https://math.stackexchange.com/questions/5006488/image-of-the-squaring-function-on-mathcalm-2-mathbbq
There was no solution on Stack.

-Statement of the exercise-
We consider the matrix function $f:X\in M_2(\mathbb{Q})\mapsto X^2\in M_2(\mathbb{Q})$.
Find the image of $f$.
In other words, give a method to decide whether a given matrix has or does not have at least a square root
in $M_2(\mathbb{Q})$; if the answer is yes, then give a method to calculate at least one of its roots.
9 replies
loup blanc
Feb 17, 2025
ysharifi
Today at 1:28 AM
J
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Differentiation Marathon!
LawofCosine   181
N Today at 12:24 AM by Levieee
Hello, everybody!

This is a differentiation marathon. It is just like an ordinary marathon, where you can post problems and provide solutions to the problem posted by the previous user. You can only post differentiation problems (not including integration and differential equations) and please don't make it too hard!

Have fun!

(Sorry about the bad english)
181 replies
LawofCosine
Feb 1, 2025
Levieee
Today at 12:24 AM
J
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Integrals problems and inequality
tkd23112006   2
N Yesterday at 6:02 PM by PolyaPal
Let f be a continuous function on [0,1] such that f(x) ≥ 0 for all x ∈[0,1] and
$\int_x^1 f(t) dt \geq \frac{1-x^2}{2}$ , ∀x∈[0,1].
Prove that:
$\int_0^1 (f(x))^{2021} dx \geq \int_0^1 x^{2020} f(x) dx$
2 replies
tkd23112006
Feb 16, 2025
PolyaPal
Yesterday at 6:02 PM
J
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real analysis
ay19bme   1
N Yesterday at 5:30 PM by alexheinis
.........
1 reply
ay19bme
Yesterday at 3:05 PM
alexheinis
Yesterday at 5:30 PM
J
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Proving an inequality involving cosine functions
pii-oner   3
N Yesterday at 3:33 PM by pii-oner
Hello AoPS Community,

I am curious about how to demonstrate the following inequality:

[code] \sqrt{1 - |\cos(x \pm y)|^a} \leq \sqrt{1 - |\cos(x)|^a} + \sqrt{1 - |\cos(y)|^a}, \quad \text{for } a \geq 1. [/code]

I’ve plotted the functions, and the inequality seems to hold. However, I am looking for a rigorous way to prove it.

I’d greatly appreciate insights into how to break this down analytically and references to similar problems or techniques that might be helpful.

Thank you so much for your guidance and support!
3 replies
pii-oner
Jan 22, 2025
pii-oner
Yesterday at 3:33 PM
J
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Linear algebra
Dynic   2
N Yesterday at 3:32 PM by loup blanc
Let A and B be two square matrices with the same size. Prove that if AB is an invertible matrix, then A and B are also invertible matrices
2 replies
Dynic
Yesterday at 2:48 PM
loup blanc
Yesterday at 3:32 PM
J
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Very hard group theory problem
mathscrazy   3
N Yesterday at 9:50 AM by quasar_lord
Source: STEMS 2025 Category C6
Let $G$ be a finite abelian group. There is a magic box $T$. At any point, an element of $G$ may be added to the box and all elements belonging to the subgroup (of $G$) generated by the elements currently inside $T$ are moved from outside $T$ to inside (unless they are already inside). Initially $ T$ contains only the group identity, $1_G$. Alice and Bob take turns moving an element from outside $T$ to inside it. Alice moves first. Whoever cannot make a move loses. Find all $G$ for which Bob has a winning strategy.
3 replies
mathscrazy
Dec 29, 2024
quasar_lord
Yesterday at 9:50 AM
J
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limit of u(pi/45)
EthanWYX2009   0
Yesterday at 7:08 AM
Source: 2025 Pi Day Challenge T5
Let \(\omega\) be a positive real number. Divide the positive real axis into intervals \([0, \omega)\), \([\omega, 2\omega)\), \([2\omega, 3\omega)\), \([3\omega, 4\omega)\), \(\ldots\), and color them alternately black and white. Consider the function \(u(x)\) satisfying the following differential equations:
\[ u''(x) + 9^2u(x) = 0, \quad \text{for } x \text{ in black intervals}, \]\[ u''(x) + 63^2u(x) = 0, \quad \text{for } x \text{ in white intervals}, \]with the initial conditions:
\[ u(0) = 1, \quad u'(0) = 1, \]and the continuity conditions:
\[ u(x) \text{ and } u'(x) \text{ are continuous functions}. \]Show that
\[ \lim_{\omega \to 0} u\left(\frac{\pi}{45}\right) = 0. \]
0 replies
EthanWYX2009
Yesterday at 7:08 AM
0 replies
J
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Integration Bee Kaizo
Calcul8er   42
N Yesterday at 12:57 AM by awzhang10
Hey integration fans. I decided to collate some of my favourite and most evil integrals I've written into one big integration bee problem set. I've been entering integration bees since 2017 and I've been really getting hands on with the writing side of things over the last couple of years. I hope you'll enjoy!
42 replies
Calcul8er
Mar 2, 2025
awzhang10
Yesterday at 12:57 AM
J
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maximum value
Tip_pay   3
N Friday at 9:37 PM by alexheinis
Find the value $x\in [0,4]$ at which the function $f(x)=\int_{0}^{\sqrt{x}}\ln \frac{e}{1+t^2}dt$ takes its maximum value
3 replies
Tip_pay
Friday at 8:48 PM
alexheinis
Friday at 9:37 PM
J
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Spheres and a point source of light
mofidy   3
N Friday at 9:22 PM by kiyoras_2001
How many spheres are needed to shield a point source of light?
Unfortunately, I didn't find a suitable solution for it on the page below:
https://artofproblemsolving.com/community/c4h1469498p8521602
Here too, two different solutions are given:
https://math.stackexchange.com/questions/2791186
3 replies
mofidy
Mar 11, 2025
kiyoras_2001
Friday at 9:22 PM
J
Linear algebra
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Reveal topic tags
linear algebra
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Dynic
1 post
Dynic
#1 h  Yesterday at 2:48 PM
Y by
Let A and B be two square matrices with the same size. Prove that if AB is an invertible matrix, then A and B are also invertible matrices
Z K Y
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rchokler
2930 posts
rchokler
#2 h  Yesterday at 3:10 PM
Y by
A square matrix $M$ is invertible if and only if $\det M\neq 0$.

If $AB$ is invertible, we get $\det AB\neq 0\implies\det A\cdot\det B\neq 0$. Since anything times zero is zero, we get $\det A\neq 0$ and $\det B\neq 0$. So $A,B$ are both invertible.
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loup blanc
3557 posts
loup blanc
#3 h  Yesterday at 3:32 PM
Y by
@ Dynic , Your question is high school level (at best) and not college math level.
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