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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
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Partitioning coprime integers to arithmetic sequences
sevket12   4
N a few seconds ago by bochidd
Source: 2025 Turkey EGMO TST P3
For a positive integer $n$, let $S_n$ be the set of positive integers that do not exceed $n$ and are coprime to $n$. Define $f(n)$ as the smallest positive integer that allows $S_n$ to be partitioned into $f(n)$ disjoint subsets, each forming an arithmetic progression.

Prove that there exist infinitely many pairs $(a, b)$ satisfying $a, b > 2025$, $a \mid b$, and $f(a) \nmid f(b)$.
4 replies
1 viewing
sevket12
Feb 8, 2025
bochidd
a few seconds ago
J
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Coaxal Circles
fattypiggy123   30
N 2 minutes ago by Ilikeminecraft
Source: China TSTST Test 2 Day 1 Q3
Let $ABCD$ be a quadrilateral and let $l$ be a line. Let $l$ intersect the lines $AB,CD,BC,DA,AC,BD$ at points $X,X',Y,Y',Z,Z'$ respectively. Given that these six points on $l$ are in the order $X,Y,Z,X',Y',Z'$, show that the circles with diameter $XX',YY',ZZ'$ are coaxal.
30 replies
fattypiggy123
Mar 13, 2017
Ilikeminecraft
2 minutes ago
J
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Weird n-variable extremum problem
pithon_with_an_i   0
7 minutes ago
Source: Revenge JOM 2025 Problem 3, Revenge JOMSL 2025 A4
Let $n$ be a positive integer greater or equal to $2$ and let $a_1$, $a_2$, ..., $a_n$ be a sequence of non-negative real numbers. Find the maximum value of $3(a_1 + a_2 + \cdots + a_n) - (a_1^2 + a_2^2 + \cdots + a_n^2) - a_1a_2 \cdots a_n$ in terms of $n$.

(Proposed by Cheng You Seng)
0 replies
pithon_with_an_i
7 minutes ago
0 replies
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Quirky tangency and line concurrence with circumcircles
pithon_with_an_i   0
11 minutes ago
Source: Revenge JOM 2025 Problem 2, Revenge JOMSL 2025 G4
Let $ABC$ be a triangle. $M$ is the midpoint of segment $BC$, and points $E$, $F$ are selected on sides $AB$, $AC$ respectively such that $E$, $F$, $M$ are collinear. The circumcircles $(ABC)$ and $(AEF)$ intersect at a point $P \neq A$. The circumcircle $(APM)$ intersects line $BC$ again at a point $D \neq M$.
Show that the lines $AD$, $EF$ and the tangent to $(AEF)$ at point $P$ concur.

(Proposed by Soo Eu Khai)
0 replies
1 viewing
pithon_with_an_i
11 minutes ago
0 replies
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UMich Math
missionsqhc   1
N 4 hours 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
4 hours ago
J
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Integral and Derivative Equation
ahaanomegas   6
N 5 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
5 hours ago
J
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UC Berkeley Integration Bee 2025 Bracket Rounds
Silver08   64
N 5 hours ago by vanstraelen
Regular Round

Quarterfinals

Semifinals

3rd Place Match

Finals
64 replies
Silver08
May 9, 2025
vanstraelen
5 hours ago
J
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Integral
Martin.s   0
6 hours ago
$$\int_0^{\pi/6}\arcsin\Bigl(\sqrt{\cos(3\psi)\cos\psi}\Bigr)\,d\psi.$$
0 replies
Martin.s
6 hours ago
0 replies
J
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Prove the statement
Butterfly   3
N Today at 6:41 AM 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
Today at 6:41 AM
J
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Tough integral
Martin.s   1
N Today at 4:49 AM 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
Today at 4:49 AM
J
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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
J
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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
J
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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
J
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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
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J
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|AP|+ |BC| =\sqrt3 |AB| if <ABP = 20^o, <PBC =<PCB = 10^o, <PCA = 40^o
parmenides51   3
N Dec 17, 2021 by BarisKoyuncu
Source: Romanian Mathematical Magazine JP 353 https://artofproblemsolving.com/community/c2691230_
In $\vartriangle ABC$, $P \in Int (\vartriangle ABC)$, $\angle ABP = 20^o$, $\angle PBC =\angle PCB = 10^o$, $\angle PCA = 40^o$. Prove that $|AP|+ |BC| =\sqrt3 |AB|$.

by Mehmet Sahin - Turkey
3 replies
parmenides51
Dec 14, 2021
BarisKoyuncu
Dec 17, 2021
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|AP|+ |BC| =\sqrt3 |AB| if <ABP = 20^o, <PBC =<PCB = 10^o, <PCA = 40^o
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Source: Romanian Mathematical Magazine JP 353 https://artofproblemsolving.com/community/c2691230_
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parmenides51
30652 posts
parmenides51
#1 h  Dec 14, 2021, 9:45 PM
Y by
In $\vartriangle ABC$, $P \in Int (\vartriangle ABC)$, $\angle ABP = 20^o$, $\angle PBC =\angle PCB = 10^o$, $\angle PCA = 40^o$. Prove that $|AP|+ |BC| =\sqrt3 |AB|$.

by Mehmet Sahin - Turkey
This post has been edited 1 time. Last edited by parmenides51, Dec 14, 2021, 9:45 PM
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sunken rock
4394 posts
sunken rock
#2 h  Dec 17, 2021, 7:10 AM
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Nobody yet? Come on!
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BarisKoyuncu
577 posts
BarisKoyuncu
#3 h  Dec 17, 2021, 7:19 AM
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Trigonometric solution
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BarisKoyuncu
577 posts
BarisKoyuncu
#4 h  Dec 17, 2021, 7:54 AM
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Non-trigonometric solution
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