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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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[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]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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2025 Mathcounts Countdown Practice
HKIS-Math   1
N 2 hours ago by HKIS-Math
Date & Time:
Sunday April 20th, 2025, 6:30pm EDT (5:30pm CDT, 3:30pm PDT)
The total duration is expected to be 3.5~4.5 hours.

Host: Dr. Jiangang Yao
Dr. Yao was fascinated with mathematics as a child and started his involvement with mathematical olympiad since then. He won the gold medal with full marks in the 35th International Mathematical Olympiad and got math PhD degree from UC Berkeley. He has been the coach for mathematical olympiad at various levels for 30 years, and has written a few popular mathematical olympiad books in Chinese.

Official Participants:
Students who have been invited to the 2025 MathCounts National Competition. Every student will receive a unique three-digit number after registration so that participation can be anonymous, though participants are welcome to show real names as well.

Guests:
Math fans who want to solve interesting math olympiad problems.

Schedule:
6:30pm ~7:30pm: 12 problems with difficulty levels similar to Mathcounts National Sprint and Target will be presented and discussed, and official participants will be given points based on speed (10 pts for the first correct answer, 9pts for 2nd correct answer, etc, 1pt for 10th correct answer.)

7:45~9:00 pm: Top 12 official participants will be identified from the first round to attend the 1-1 matchups. (#12 v.s. #5 with winner A, #11 v.s. #6 with winner B, #10 v.s. #7 with winner C, #9 v.s. #8 with winner D, #4 v.s. A, #3 v.s. B, #2 v.s. C, #1 v.s. D). In each matchup, 5 questions will be presented and the participant who first successfully gets 3 questions correct is the winner. In this round, 5x8 =40 problems will be played.

9:15~10:00pm: Two semi-finals, bronze determination, and final. Each 1-1 matchup will have 7 questions, and the participant who first successfully answers 4 questions is the winner. In this round, 4x7=28 problems will be played. The Top 4 contestants will receive awards.

All guests can submit the answers to all the questions as well. Those who submitted correct answers fast will be appraised.

2022 40 mathletes, 2023 64 mathlets, and 2024 99 mathlets for Mathcounts National attended this practice. We are looking forward to have more students participate this year.

Here is the link for registration:
https://forms.gle/xoRNMLrRnn7KjFiUA
1 reply
HKIS-Math
Apr 17, 2025
HKIS-Math
2 hours ago
J
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Differential equations , Matrix theory
c00lb0y   1
N 3 hours ago by loup blanc
Source: RUDN MATH OLYMP 2024 problem 4
Any idea?? Diff equational system combined with Matrix theory.
Consider the equation dX/dt=X^2, where X(t) is an n×n matrix satisfying the condition detX=0. It is known that there are no solutions of this equation defined on a bounded interval, but there exist non-continuable solutions defined on unbounded intervals of the form (t ,+∞) and (−∞,t). Find n.
1 reply
c00lb0y
Apr 17, 2025
loup blanc
3 hours ago
J
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OMOUS-2025 (Team Competition) P7
enter16180   1
N 3 hours ago by RobertRogo
Source: Open Mathematical Olympiad for University Students (OMOUS-2025)
Let $R$ be a ring not assumed to have an identity, with the following properties:
(i) There is an element of $R$ that is not nilpotent.
(ii) If $x_{1}, \ldots, x_{2024}$ are nonzero elements of $R$, then $\sum_{j=1}^{2024} x_{j}^{2025}=0$.

Show that $R$ is a division ring, that is, the nonzero elements of R form a group under multiplication.
1 reply
enter16180
Yesterday at 12:04 PM
RobertRogo
3 hours ago
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Question
Riptide1901   1
N 4 hours ago by Mathzeus1024
Isn't their a left hand side of the graph that they're leaving out? I'm confused why they only consider $x>\sqrt{15}/2$ and not $x<-\sqrt{15}/2.$
1 reply
Riptide1901
Jan 29, 2025
Mathzeus1024
4 hours ago
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0!??????
wizwilzo   52
N Today at 6:15 AM by Craftybutterfly
why is 0! "1" ??!
52 replies
wizwilzo
Jul 6, 2016
Craftybutterfly
Today at 6:15 AM
J
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Putnam 2019 A5
hoeij   17
N Today at 5:34 AM by Ilikeminecraft
Let $p$ be an odd prime number, and let $\mathbb{F}_p$ denote the field of integers modulo $p$. Let $\mathbb{F}_p[x]$ be the ring of polynomials over $\mathbb{F}_p$, and let $q(x) \in \mathbb{F}_p[x]$ be given by $q(x) = \sum_{k=1}^{p-1} a_k x^k$ where $a_k = k^{(p-1)/2}$ mod $p$. Find the greatest nonnegative integer $n$ such that $(x-1)^n$ divides $q(x)$ in $\mathbb{F}_p[x]$.
17 replies
hoeij
Dec 10, 2019
Ilikeminecraft
Today at 5:34 AM
J
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Projection of angle into planes
geekmath-31   0
Today at 5:14 AM
Question:
consider the angle formed by 2 half lines in the three dimensional space. Prove that the average of the projection of the angle into all of the planes is equal to the angle

The answer is in the attachments.

Please could anyone prove the answer to me in detail.
0 replies
geekmath-31
Today at 5:14 AM
0 replies
J
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Projection of angle into planes
geekmath-31   0
Today at 4:31 AM
Question:
consider the angle formed by 2 half lines in the three dimensional space. Prove that the average of the projection of the angle into all of the planes is equal to the angle

The answer is in the attachments.

Please could anyone prove the answer to me in detail.
0 replies
geekmath-31
Today at 4:31 AM
0 replies
J
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2500th post
Solocraftsolo   18
N Today at 4:00 AM by valisaxieamc
i keep forgetting to do these...


2500 is cool.

i am not very sentimental so im not going to post a math story or anything.

here are some problems though

p1p2p3

p4
18 replies
Solocraftsolo
Apr 16, 2025
valisaxieamc
Today at 4:00 AM
J
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Camp Conway acceptance
fossasor   13
N Today at 4:00 AM by fossasor
Hello! I've just been accepted into Camp Conway, but I'm not sure how popular this camp actually is, given that it's new. Has anyone else applied/has been accepted/is going? (I'm trying to figure out to what degree this acceptance was just lack of qualified applicants, so I can better predict my chances of getting into my preferred math camp.)
13 replies
fossasor
Feb 20, 2025
fossasor
Today at 4:00 AM
J
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Soviet Union University Mathematical Contest
geekmath-31   0
Today at 3:40 AM
Given a n*n matrix A, prove that there exists a matrix B such that ABA = A

Solution: I have submitted the attachment

The answer is too symbol dense for me to understand the answer.
What I have undertood:

There is use of direct product in the orthogonal decomposition. The decomposition is made with kernel and some T (which the author didn't mention) but as per orthogonal decomposition it must be its orthogonal complement.

Can anyone explain the answer in much much more detail with less use of symbols ( you can also use symbols but clearly define it).

Also what is phi | T ?
0 replies
geekmath-31
Today at 3:40 AM
0 replies
J
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Dimension of a Linear Space
EthanWYX2009   0
Today at 2:50 AM
Source: 2024 May taca-10
Let \( V \) be a $10$-dimensional inner product space of column vectors, where for \( v = (v_1, v_2, \dots, v_{10})^T \) and \( w = (w_1, w_2, \dots, w_{10})^T \), the inner product of \( v \) and \( w \) is defined as \[\langle v, w \rangle = \sum_{i=1}^{10} v_i w_i.\]For \( u \in V \), define a linear transformation \( P_u \) on \( V \) as follows:
\[ P_u : V \to V, \quad x \mapsto x - \frac{2\langle x, u \rangle u}{\langle u, u \rangle} \]Given \( v, w \in V \) satisfying
\[ 0 < \langle v, w \rangle < \sqrt{\langle v, v \rangle \langle w, w \rangle} \]let \( Q = P_v \circ P_w \). Then the dimension of the linear space formed by all linear transformations \( P : V \to V \) satisfying \( P \circ Q = Q \circ P \) is $\underline{\quad\quad}.$
0 replies
EthanWYX2009
Today at 2:50 AM
0 replies
J
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Matrices and Determinants
Saucepan_man02   5
N Today at 1:23 AM by Saucepan_man02
Hello

Can anyone kindly share some problems/handouts on matrices & determinants (problems like Putnam 2004 A3, which are simple to state and doesnt involve heavy theory)?

Thank you..
5 replies
Saucepan_man02
Apr 4, 2025
Saucepan_man02
Today at 1:23 AM
J
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Putnam 1960 B1
sqrtX   4
N Yesterday at 11:26 PM by KAME06
Source: Putnam 1960
Find all integer solutions $(m,n)$ to $m^{n}=n^{m}.$
4 replies
sqrtX
Jun 18, 2022
KAME06
Yesterday at 11:26 PM
J
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J
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An algebra math problem
AVY2024   6
N Yesterday at 6:03 PM by Roger.Moore
Solve for a,b
ax-2b=5bx-3a
6 replies
AVY2024
Apr 8, 2025
Roger.Moore
Yesterday at 6:03 PM
J
An algebra math problem
G H J
Reveal topic tags
algebra
nmtc
equation
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G H BBookmark kLocked kLocked NReply
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AVY2024
22 posts
AVY2024
#1 h  Apr 8, 2025, 10:55 AM
Y by
Solve for a,b
ax-2b=5bx-3a
Z K Y
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Inaaya
269 posts
Inaaya
#2 h  Apr 8, 2025, 1:40 PM • 1 Y
Y by Exponent11
There are 3 variables and one equation, this thing can theoretically have infinite solution sets
Z K Y
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sp0rtman00000
2 posts
sp0rtman00000
#3 h  Apr 11, 2025, 11:03 AM
Y by
On the R or on the N?
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Pengu14
505 posts
Pengu14
#5 h  Apr 11, 2025, 3:24 PM
Y by
Inaaya wrote:
There are 3 variables and one equation, this thing can theoretically have infinite solution sets

I believe they’re looking for a and b such that the equation holds for ALL x.
Z K Y
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Pengu14
505 posts
Pengu14
#6 h  Apr 11, 2025, 3:25 PM
Y by
AVY2024 wrote:
Solve for a,b
ax-2b=5bx-3a

We have a=5b and 3a=2b. The only solution is a=b=0.
Z K Y
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Runner1600
12 posts
Runner1600
#7 h  Apr 11, 2025, 3:56 PM
Y by
Pengu14 wrote:
AVY2024 wrote:
Solve for a,b
ax-2b=5bx-3a

We have a=5b and 3a=2b. The only solution is a=b=0.

That is what I got
Z K Y
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Roger.Moore
4 posts
Roger.Moore
#8 h  Yesterday at 6:03 PM
Y by
The x there are parameter?
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