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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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0 replies
jlacosta
Mar 2, 2025
0 replies
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AIME score for college apps
Happyllamaalways   56
N 5 hours ago by Countmath1
What good colleges do I have a chance of getting into with an 11 on AIME? (Any chances for Princeton)

Also idk if this has weight but I had the highest AIME score in my school.
56 replies
Happyllamaalways
Mar 13, 2025
Countmath1
5 hours ago
J
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MIT Beaverworks Summer Institute
PowerOfPi_09   0
5 hours ago
Hi! I was wondering if anyone here has completed this program, and if so, which track did you choose? Do rising juniors have a chance, or is it mainly rising seniors that they accept? Also, how long did it take you to complete the prerequisites?
Thanks!
0 replies
PowerOfPi_09
5 hours ago
0 replies
J
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Convolution of order f(n)
trumpeter   70
N 6 hours ago by HamstPan38825
Source: 2019 USAMO Problem 1
Let $\mathbb{N}$ be the set of positive integers. A function $f:\mathbb{N}\to\mathbb{N}$ satisfies the equation \[\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(n)\ldots))=\frac{n^2}{f(f(n))}\]for all positive integers $n$. Given this information, determine all possible values of $f(1000)$.

Proposed by Evan Chen
70 replies
trumpeter
Apr 17, 2019
HamstPan38825
6 hours ago
J
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Linear algebra
Dynic   2
N Today 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
Today at 2:48 PM
loup blanc
Today at 3:32 PM
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3-dimensional matrix system
loup blanc   0
Today at 1:04 PM
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. $A^TA=B^TB,AA^T=BB^T$.
ii) Show that each solution of i) is in $M_3(K)$, where $K=\mathbb{Q}[\sin^2(\dfrac{2}{3}\arctan(3\sqrt{3}))]$.
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.
0 replies
loup blanc
Today at 1:04 PM
0 replies
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k HOT TAKE: MIT SHOULD NOT RELEASE THEIR DECISIONS ON PI DAY
alcumusftwgrind   8
N Today at 10:13 AM by maxamc
rant lol

Imagine a poor senior waiting for their MIT decisions just to have their hopes CRUSHED on 3/14 and they can't even celebrate pi day...

and even worse, this year's pi day is special because this year is a very special number...

8 replies
alcumusftwgrind
Today at 2:11 AM
maxamc
Today at 10:13 AM
J
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Very hard group theory problem
mathscrazy   3
N Today 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
Today at 9:50 AM
J
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limit of u(pi/45)
EthanWYX2009   0
Today 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
Today at 7:08 AM
0 replies
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Integration Bee Kaizo
Calcul8er   42
N Today 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
Today at 12:57 AM
J
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maximum value
Tip_pay   3
N Yesterday 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
Yesterday at 8:48 PM
alexheinis
Yesterday at 9:37 PM
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Spheres and a point source of light
mofidy   3
N Yesterday 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
Yesterday at 9:22 PM
J
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Real Analysis
rljmano   4
N Yesterday at 4:38 PM by alexheinis
In [0 1], {f(t)}*{f’(t)-1} =0 and f is continuously differentiable. How do we conclude that either f is identically zero or f’(t) is identically 1 in [0 1]?
4 replies
rljmano
Yesterday at 6:32 AM
alexheinis
Yesterday at 4:38 PM
J
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find the isomorphism
nguyenalex   14
N Yesterday at 1:49 PM by Royrik123456
I have the following exercise:

Let $E$ be an algebraic extension of $K$, and let $F$ be an algebraic closure of $K$ containing $E$. Prove that if $\sigma : E \to F$ is an embedding such that $\sigma(c) = c$ for all $c \in K$, then $\sigma$ extends to an automorphism of $F$.

My attempt:

Theorem (*): Suppose that $E$ is an algebraic extension of the field $K$, $F$ is an algebraically closed field, and $\sigma: K \to F$ is an embedding. Then, there exists an embedding $\tau: E \to F$ that extends $\sigma$. Moreover, if $E$ is an algebraic closure of $K$ and $F$ is an algebraic extension of $\sigma(K)$, then $\tau$ is an isomorphism.

Back to our main problem:

Since $K \subset E$ and $F$ is an algebraic extension of $K$, it follows that $F$ is an algebraic extension of $E$. Assume that there exists an embedding $\sigma : E \to F$ such that $\sigma(c) = c$ for all $c \in K$. By Theorem (*), there exists an embedding $\tau : F \to F$ that extends $\sigma$. Since $F$ is algebraically closed, $\tau(F)$ is also an algebraically closed field.

Furthermore, because $\sigma(c) = c$ for all $c \in K$ and $\tau$ is an extension of $\sigma$, we have
$$K = \sigma(K) \subset K \subset \sigma(E) \subset \tau(F) \subset F.$$
This implies that $F$ is an algebraic extension of $\tau(F)$. We conclude that $F = \tau(F)$, meaning that $\tau$ is an automorphism. (Finished!!)

Let choose $F = A$ be the field of algebraic numbers, $K=\mathbb{Q}$. Consider the embedding $\sigma: \mathbb{Q}(\sqrt{2}) \to \mathbb{Q}(\sqrt{2}) \subset A$ defined by
$$ a + b\sqrt{2} \mapsto a - b\sqrt{2}. $$Then, according to the exercise above, $\sigma$ extends to an isomorphism
$$ \bar{\sigma}: A \to A. $$How should we interpret $\bar{\sigma}$?
14 replies
nguyenalex
Mar 12, 2025
Royrik123456
Yesterday at 1:49 PM
J
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find the convex sets satisfied
nguyenalex   2
N Yesterday at 9:54 AM by ILOVEMYFAMILY
In $\mathbb{R}^2$, let $B = \{(x, y) \mid x \geq 0\}$. Find all convex sets $C$ such that

\[\mathcal{E}(B \cup C) = B \cup C.\]
2 replies
nguyenalex
Yesterday at 8:57 AM
ILOVEMYFAMILY
Yesterday at 9:54 AM
J
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J
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Some Multiples Stuff
TheMathateer   13
N Jun 1, 2021 by megarnie
Source: 2020 AMC 10B #7
How many positive even multiples of $3$ less than $2020$ are perfect squares?

$\textbf{(A) }7 \qquad \textbf{(B) }8 \qquad \textbf{(C) }9 \qquad \textbf{(D) }10 \qquad\textbf{(E) }12$
13 replies
TheMathateer
Feb 6, 2020
megarnie
Jun 1, 2021
J
Some Multiples Stuff
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Reveal topic tags
AMC
AMC 10
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2020 AMC
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G H BBookmark kLocked kLocked NReply
Source: 2020 AMC 10B #7
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TheMathateer
561 posts
TheMathateer
#1 h  Feb 6, 2020, 5:24 PM • 1 Y
Y by Adventure10
How many positive even multiples of $3$ less than $2020$ are perfect squares?

$\textbf{(A) }7 \qquad \textbf{(B) }8 \qquad \textbf{(C) }9 \qquad \textbf{(D) }10 \qquad\textbf{(E) }12$
This post has been edited 2 times. Last edited by nsato, Mar 31, 2020, 6:25 PM
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justlearningmathog
94 posts
justlearningmathog
#2 h  Feb 6, 2020, 5:24 PM • 2 Y
Y by vsar0406, Adventure10
I thought it was like 7 and stuff not those big numbers
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aop2014
2416 posts
aop2014
#3 h  Feb 6, 2020, 5:24 PM • 1 Y
Y by Adventure10
I don't think those options are right
But I got (A) 7
This post has been edited 1 time. Last edited by aop2014, Feb 6, 2020, 5:25 PM
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Susanssluk
436 posts
Susanssluk
#4 h  Feb 6, 2020, 5:25 PM • 1 Y
Y by Adventure10
Yeah can confirm it's 7.
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greenapple
263 posts
greenapple
#5 h  Feb 6, 2020, 5:25 PM • 1 Y
Y by Adventure10
Yea, I think its 7 oof sniped :(
This post has been edited 1 time. Last edited by greenapple, Feb 6, 2020, 5:25 PM
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worldwidenoah1
302 posts
worldwidenoah1
#6 h  Feb 6, 2020, 5:25 PM • 1 Y
Y by Adventure10
These choices are from the prior question
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jacoporizzo
615 posts
jacoporizzo
#7 h  Feb 6, 2020, 5:25 PM • 1 Y
Y by Adventure10
The question was how many even multiples of 3 that are square numbers under 2020, can confirm as well
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mathkid69
617 posts
mathkid69
#8 h  Feb 6, 2020, 5:25 PM • 1 Y
Y by Adventure10
yep i got a
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kc5170
694 posts
kc5170
#9 h  Feb 6, 2020, 5:32 PM • 1 Y
Y by Adventure10
Basically multiples of 36 times perfect squares, A 7
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aie8920
1270 posts
aie8920
#10 h  Feb 6, 2020, 6:18 PM • 1 Y
Y by Adventure10
Yes, all $n^2$ where $n$ is divisible by $6$ will work, so 6-42 is $\boxed{7}$ multiples.
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Imayormaynotknowcalculus
974 posts
Imayormaynotknowcalculus
#11 h  Feb 7, 2020, 10:37 PM • 1 Y
Y by Adventure10
It's $\lfloor{\left({\sqrt{\frac{2020}{({2}\times{3})^2}}}\right)}\rfloor=7$
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IMadeYouReadThis
1541 posts
IMadeYouReadThis
#12 h  Jun 26, 2020, 4:16 AM • 2 Y
Y by Imayormaynotknowcalculus, Derpy_Creeper
Sorry for the bump, but I wanted to share my solution since there wasn't one already.
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Possible
6502 posts
Possible
#13 h  Jun 26, 2020, 4:48 AM
Y by
Sol
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megarnie
5531 posts
megarnie
#14 h  Jun 1, 2021, 7:41 PM
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