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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next 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 an enriching 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][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
UMich Math
missionsqhc   1
N an hour ago by Mathzeus1024
I was recently accepted into the University of Michigan as a math major. If anyone studies math at UMich or knows anything about the program, could you share your experience? How would you rate the program? I know UMich is well-regarded for math (among many other things) but from my understanding, it is not quite at the level of an MIT or CalTech. What math programs is it comparable to? How does the rigor of the curricula compare to other top math programs? What are the other students like—is there a thriving contest math community? How accessible are research opportunities and graduate-level classes? Are most students looking to get into pure math and become research mathematicians or are most people focused on applied fields?

Also, aside from the math program, how is UMich overall? What were the advantages and disadvantages from being at such a large school? I was admitted to the Residential College (RC) within the College of Literature, Science, and the Arts. This is supposed to emulate a liberal arts college (while still allowing me access to the resources of a major research university). Could anyone speak on the RC?

How academically-inclined are UMich students? I’ve heard the school is big on sports and school spirit. I am just concerned that there may be a lot of subpar in-state students. How is the climate of Ann Arbor and how is the city in general?

Finally, how is UMich generally regarded? I’m also considering Georgetown. Am I right in viewing the latter as more well-regarded for humanities and the former better-known for STEM?
1 reply
missionsqhc
Yesterday at 4:31 PM
Mathzeus1024
an hour ago
Integral and Derivative Equation
ahaanomegas   6
N 2 hours ago by Sagnik123Biswas
Source: Putnam 1990 B1
Find all real-valued continuously differentiable functions $f$ on the real line such that for all $x$, \[ \left( f(x) \right)^2 = \displaystyle\int_0^x \left[ \left( f(t) \right)^2 + \left( f'(t) \right)^2 \right] \, \mathrm{d}t + 1990. \]
6 replies
ahaanomegas
Jul 12, 2013
Sagnik123Biswas
2 hours ago
UC Berkeley Integration Bee 2025 Bracket Rounds
Silver08   64
N 2 hours ago by vanstraelen
Regular Round

Quarterfinals

Semifinals

3rd Place Match

Finals
64 replies
Silver08
May 9, 2025
vanstraelen
2 hours ago
Integral
Martin.s   0
2 hours ago
$$\int_0^{\pi/6}\arcsin\Bigl(\sqrt{\cos(3\psi)\cos\psi}\Bigr)\,d\psi.$$
0 replies
1 viewing
Martin.s
2 hours ago
0 replies
Prove the statement
Butterfly   3
N 4 hours ago by Photaesthesia
Given an infinite sequence $\{x_n\} \subseteq  [0,1]$, there exists some constant $C$, for any $r>0$, among the sequence $x_n$ and $x_m$ could be chosen to satisfy $|n-m|\ge r $ and $|x_n-x_m|<\frac{C}{|n-m|}$.
3 replies
Butterfly
May 7, 2025
Photaesthesia
4 hours ago
Tough integral
Martin.s   1
N 6 hours ago by Martin.s
$$\int_0^{\pi/2}\ln(\tan(\theta/2))
\;\frac{4\cos\theta\cos(2\theta)}{4\sin^4\theta+1}\,d\theta.$$
1 reply
Martin.s
May 12, 2025
Martin.s
6 hours ago
Calculus
youochange   1
N Yesterday at 1:21 PM by Mathzeus1024
Find the area enclosed by the curves $e^x,e^{-x},x^2+y^2=1$

1 reply
youochange
May 10, 2025
Mathzeus1024
Yesterday at 1:21 PM
Mathematical expectation 1
Tricky123   3
N Yesterday at 1:13 PM by Tricky123
X is continuous random variable having spectrum
$(-\infty,\infty) $ and the distribution function is $F(x)$ then
$E(X)=\int_{0}^{\infty}(1-F(x)-F(-x))dx$ and find the expression of $V(x)$

Ans:- $V(x)=\int_{0}^{\infty}(2x(1-F(x)+F(-x))dx-m^{2}$

How to solve help me
3 replies
Tricky123
May 11, 2025
Tricky123
Yesterday at 1:13 PM
Derivative of unknown continuous function
smartvong   2
N Yesterday at 12:43 PM by solyaris
Source: UM Mathematical Olympiad 2024
Let $f: \mathbb{R} \to \mathbb{R}$ be a function whose derivative is continuous on $[0,1]$. Show that
$$\lim_{n \to \infty} \sum^n_{k = 1}\left[f\left(\frac{k}{n}\right) - f\left(\frac{2k - 1}{2n}\right)\right] = \frac{f(1) - f(0)}{2}.$$
2 replies
smartvong
Yesterday at 1:05 AM
solyaris
Yesterday at 12:43 PM
Divisibility of cyclic sum
smartvong   1
N Yesterday at 12:06 PM by alexheinis
Source: UM Mathematical Olympiad 2024
Let $n$ be a positive integer greater than $1$. Show that
$$4 \mid (x_1x_2 + x_2x_3 + \cdots + x_{n-1}x_n + x_nx_1 - n)$$where each of $x_1, x_2, \dots, x_n$ is either $1$ or $-1$.
1 reply
smartvong
Yesterday at 9:49 AM
alexheinis
Yesterday at 12:06 PM
Polynomial with integer coefficients
smartvong   1
N Yesterday at 10:04 AM by alexheinis
Source: UM Mathematical Olympiad 2024
Prove that there is no polynomial $f(x)$ with integer coefficients, such that $f(p) = \dfrac{p + q}{2}$ and $f(q) = \dfrac{p - q}{2}$ for some distinct primes $p$ and $q$.
1 reply
smartvong
Yesterday at 9:46 AM
alexheinis
Yesterday at 10:04 AM
Existence of scalars
smartvong   0
Yesterday at 9:44 AM
Source: UM Mathematical Olympiad 2024
Let $U$ be a finite subset of $\mathbb{R}$ such that $U = -U$. Let $f,g : \mathbb{R} \to \mathbb{R}$ be functions satisfying
$$g(x) - g(y ) = (x - y)f(x + y)$$for all $x,y \in \mathbb{R} \backslash U$.
Show that there exist scalars $\alpha, \beta, \gamma \in \mathbb{R}$ such that
$$f(x) = \alpha x + \beta$$for all $x \in \mathbb{R}$,
$$g(x) = \alpha x^2 + \beta x + \gamma$$for all $x \in \mathbb{R} \backslash U$.
0 replies
smartvong
Yesterday at 9:44 AM
0 replies
Invertible matrices in F_2
smartvong   1
N Yesterday at 9:02 AM by alexheinis
Source: UM Mathematical Olympiad 2024
Let $n \ge 2$ be an integer and let $\mathcal{S}_n$ be the set of all $n \times n$ invertible matrices in which their entries are $0$ or $1$. Let $m_A$ be the number of $1$'s in the matrix $A$. Determine the minimum and maximum values of $m_A$ in terms of $n$, as $A$ varies over $S_n$.
1 reply
smartvong
Yesterday at 12:41 AM
alexheinis
Yesterday at 9:02 AM
ISI UGB 2025 P3
SomeonecoolLovesMaths   13
N Yesterday at 8:29 AM by iced_tea
Source: ISI UGB 2025 P3
Suppose $f : [0,1] \longrightarrow \mathbb{R}$ is differentiable with $f(0) = 0$. If $|f'(x) | \leq f(x)$ for all $x \in [0,1]$, then show that $f(x) = 0$ for all $x$.
13 replies
SomeonecoolLovesMaths
May 11, 2025
iced_tea
Yesterday at 8:29 AM
Find the formula
JetFire008   4
N Apr 21, 2025 by HacheB2031
Find a formula in compact form for the general term of the sequence defined recursively by $x_1=1, x_n=x_{n-1}+n-1$ if $n$ is even.
4 replies
JetFire008
Apr 20, 2025
HacheB2031
Apr 21, 2025
Find the formula
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JetFire008
126 posts
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Find a formula in compact form for the general term of the sequence defined recursively by $x_1=1, x_n=x_{n-1}+n-1$ if $n$ is even.
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Roger.Moore
5 posts
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Just add the corresponding sides of the equality in the recursion and you get x(n)=1+2+3+...+(n-1)=n(n-1)/2
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Rohit-2006
243 posts
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Roger.Moore wrote:
Just add the corresponding sides of the equality in the recursion and you get x(n)=1+2+3+...+(n-1)=n(n-1)/2

I don't think your argument is valid....you can only handle it iff $n$ is even....but your result isn't true
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Hello_Kitty
1897 posts
#4
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the sequence is not well defined, how would you compute $x_3$ ?
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HacheB2031
396 posts
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Hello_Kitty wrote:
the sequence is not well defined, how would you compute $x_3$ ?

isn't it just \[x_3=x_2+2=x_1+1+2=1+1+2=4\]also ez sol

edit: you know that moment when you didn't read the problem statement clearly? yeah today i learned im an idiot
This post has been edited 1 time. Last edited by HacheB2031, Apr 21, 2025, 12:38 AM
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