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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
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.

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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
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Integral inequality
oVlad   2
N 12 minutes ago by MS_asdfgzxcvb
Source: W
Consider the function $f:[0,1]\to\mathbb{R}$ which satisfies $f(0)=f(1)=0$ and which has a continous derivative. Prove that \[\pi^2\int_0^1f(x)^2 \ dx\leqslant\int_0^1f'(x)^2 \ dx\]
2 replies
oVlad
Dec 1, 2023
MS_asdfgzxcvb
12 minutes ago
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Putnam 1958 November B1
sqrtX   9
N 3 hours ago by KAME06
Source: Putnam 1958 November
Given
$$b_n = \sum_{k=0}^{n} \binom{n}{k}^{-1}, \;\; n\geq 1,$$prove that
$$b_n = \frac{n+1}{2n} b_{n-1} +1, \;\; n \geq 2.$$Hence, as a corollary, show
$$ \lim_{n \to \infty} b_n =2.$$
9 replies
sqrtX
Jul 19, 2022
KAME06
3 hours ago
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Distribution of prime numbers
Rainbow1971   6
N 4 hours ago by Rainbow1971
Could anybody possibly prove that the limit of $$(\frac{p_n}{p_n + p_{n-1}})$$is $\tfrac{1}{2}$, maybe even with rather elementary means? As usual, $p_n$ denotes the $n$-th prime number. The problem of that limit came up in my partial solution of this problem: https://artofproblemsolving.com/community/c7h3495516.

Thank you for your efforts.
6 replies
Rainbow1971
Apr 9, 2025
Rainbow1971
4 hours ago
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Picking a College
missionsqhc   1
N 4 hours ago by zkyao
I applied to college as a math major, and my options are Georgetown, UVA, Stony Brook, and Binghamton. I was waitlisted from CMU, Columbia, Northwestern, Berkeley, Williams, UNC, and UMich.

I’ve done competition math throughout middle school and high school and obviously am currently slotted to study math. But I am also very much interested in politics, government, history, etc. I could easily see myself double majoring or even completely switching to something like political science or history. I don’t have a clear-cut vision for a future career. I used to really want to become a mathematician, but now I think it’s more likely that I’ll do something more “practical,” like finance or law. I also have aspirations of working in government, even possibly running for elected office.

If someone has gone to one of the school’s I’ve been accepted by or has experience in one of the careers I’ve mentioned (or possesses some other characteristics that gives insight into my situation), I would greatly appreciate your thoughts. On one hand, I really like Georgetown because of its strong programs in government, international relations, and other social sciences; its DC location; and its stated goal (which I hope is genuine) or educating students for life and not just work. But the hard sciences, and particularly math, are relatively smaller programs and less of the school’s emphasis. I worry that I may end up sticking mainly with math and would have been better off picking something like UVA or even Stony or Bing.

A related question I have regards how the undergraduate math departments compare at different schools. I wouldn't be surprised if the very top-tier places, like MIT, Caltech, CMU, Harvard, Stanford, and Princeton, were significantly stronger than Georgetown. But how does Georgetown compare to places that are good for math but not necessarily hyper-elite, like a Cornell or a UMich?

Also, Georgetown has a 3 + 2 program with Columbia Engineering, in which you study for three years at Georgetown to get a BA/BS in any major in any school (but preferably in math/science) and then study for two years at Columbia to get a BS in their engineering school. This seems like a way to get the best of both worlds between humanities and STEM (and to gain connections in both DC and NYC). If anyone has done this, please do share your experience.
1 reply
missionsqhc
Today at 1:42 AM
zkyao
4 hours ago
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Matrix in terms of exp
RenheMiResembleRice   1
N Mar 29, 2025 by Mathzeus1024
$\begin{pmatrix}X\left(t\right)\\ Y\left(t\right)\end{pmatrix}=\begin{pmatrix}\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{pmatrix}\begin{pmatrix}x\left(t\right)\\ y\left(t\right)\end{pmatrix}$

$X\left(t\right)=a_1e^t+a_2e^{-t}+a_3$
Find $a_1$, $a_2$, and $a_3$.
1 reply
RenheMiResembleRice
Mar 29, 2025
Mathzeus1024
Mar 29, 2025
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Matrix in terms of exp
G H J
Reveal topic tags
linear algebra
matrix
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RenheMiResembleRice
259 posts
RenheMiResembleRice
#1 h  Mar 29, 2025, 3:05 AM
Y by
$\begin{pmatrix}X\left(t\right)\\ Y\left(t\right)\end{pmatrix}=\begin{pmatrix}\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{pmatrix}\begin{pmatrix}x\left(t\right)\\ y\left(t\right)\end{pmatrix}$

$X\left(t\right)=a_1e^t+a_2e^{-t}+a_3$
Find $a_1$, $a_2$, and $a_3$.
This post has been edited 3 times. Last edited by RenheMiResembleRice, Mar 29, 2025, 4:05 AM
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Mathzeus1024
809 posts
Mathzeus1024
#2 h  Mar 29, 2025, 4:20 PM
Y by
RenheMiResembleRice wrote:
$\begin{pmatrix}X\left(t\right)\\ Y\left(t\right)\end{pmatrix}=\begin{pmatrix}\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{pmatrix}\begin{pmatrix}x\left(t\right)\\ y\left(t\right)\end{pmatrix}$

$X\left(t\right)=a_1e^t+a_2e^{-t}+a_3$
Find $a_1$, $a_2$, and $a_3$.

Question: Are $X(t),Y(t)$ derivatives with respect to $t$?
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