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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.

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[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.
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[*]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
Polynomials algebra
Foxellar   0
14 minutes ago
\textbf{9.} The real root of the polynomial \( p(x) = 8x^3 - 3x^2 - 3x - 1 \) can be written in the form
\[
\frac{\sqrt[3]{a} + \sqrt[3]{b} + 1}{c},
\]where \( a, b, \) and \( c \) are positive integers. Find the value of \( a + b + c \).
0 replies
Foxellar
14 minutes ago
0 replies
Number theory for people who love theory
Assassino9931   3
N 21 minutes ago by NamelyOrange
Source: Bulgaria RMM TST 2019
Prove that there is no positive integer $n$ such that $2^n + 1$ divides $5^n-1$.
3 replies
1 viewing
Assassino9931
Jul 31, 2024
NamelyOrange
21 minutes ago
Interesting inequalities
sqing   0
27 minutes ago
Source: Own
Let $ a,b> 0 ,   a+b+a^2+b^2=2.$ Prove that
$$ab+ \frac{k}{a+b+ab} \geq \frac{3-k+(k-1)\sqrt{5}}{2}$$Where $ k\geq 2. $
$$ab+ \frac{2}{a+b+ab} \geq \frac{1+\sqrt{5}}{2}$$$$ab+ \frac{3}{a+b+ab} \geq  \sqrt{5} $$
0 replies
sqing
27 minutes ago
0 replies
Stability of Additive Cauchy Equation
doanquangdang   1
N 2 hours ago by jasperE3
Show that if $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfies
$$
|f(x+y)-f(x)-f(y)-x y| \leq \varepsilon\left(|x|^p+|y|^p\right)
$$for some $\varepsilon>0,$ $p \in[0,1)$ and for all $x, y \in \mathbb{R}$, then there exists a unique solution $a: \mathbb{R} \rightarrow \mathbb{R}$ of the functional equation $a(x+y)=$ $a(x)+a(y)$ for all $x, y \in \mathbb{R}$ such that
$$
\left|f(x)-a(x)-\frac{1}{2} x^2\right| \leq \frac{2}{2-2^p} \varepsilon|x|^p
$$for all $x \in \mathbb{R}$.
1 reply
doanquangdang
Aug 16, 2024
jasperE3
2 hours ago
External Direct Sum
We2592   1
N Yesterday at 3:11 PM by Acridian9
Q) 1. Let $V$ be external direct sum of vector spaces $U$ and $W$ over a field $\mathbb{F}$.let $\hat{U}={\{(u,0):u\in U\}}$ and $\hat{W}={\{(0,w):w\in W\}}$
show that
i) $\hat{U}$ and $\hat{W}$ is subspaces.
ii)$V=\hat{U}\oplus\hat{W}$

Q)2. Suppose $V=U+W$. Let $\hat{V}$ be the external direct sum of $U$ and $W$. show that $V$ is isomorphic to $\hat{V}$ under the correspondence $v=u+w\leftrightarrow(u,w)$

I face some trouble to solve this problems help me for understanding.
thank you.

1 reply
We2592
May 21, 2025
Acridian9
Yesterday at 3:11 PM
How can I know the sequences's convergence value?
Madunglecha   5
N Yesterday at 10:50 AM by teomihai
What is the convergence value of the sequence??
(n^2)*ln(n+1/n)-n
5 replies
Madunglecha
Wednesday at 6:56 AM
teomihai
Yesterday at 10:50 AM
Prove the statement
Butterfly   12
N Yesterday at 9:44 AM by oty
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 $|x_n-x_m|<\frac{C}{|n-m|}$.
12 replies
Butterfly
May 7, 2025
oty
Yesterday at 9:44 AM
Classifying Math with Symbols Based on Behavior
Midevilgmer   2
N Yesterday at 4:06 AM by Midevilgmer
I have been working on a new math idea that prioritizes the behavior of numbers, equations, and expressions rather than their exact values. Numbers are described using symbols like P (Positive), N (Negative), Z (Zero), I (Imaginary), and D (Decimal). The goal is to create a system that uses symbols that allows you to perform operations like P*D and P+N and determine the behavioral outcomes based on the properties involved. For example, instead of identifying a number as 3, you would describe it as a positive, odd, whole, prime number, allowing you work with those traits individually or together. I would like to mention that I already have created an Addition, Subtraction, Multiplication, Division, Square Root, Exponent, and Factorial table that shows how these different behaviors work in basic operations. Finally, I want to mention that my current background includes a knowledge of geometry, algebra, and a very little amount of calculus. Any thoughts or ideas would be appreciated.
2 replies
Midevilgmer
Yesterday at 12:55 AM
Midevilgmer
Yesterday at 4:06 AM
36x⁴ + 12x² - 36x + 13 > 0
fxandi   3
N Yesterday at 1:48 AM by fxandi
Prove that for any real $x \geq 0$ holds inequality $36x^4 + 12x^2 - 36x + 13 > 0.$
3 replies
fxandi
May 5, 2025
fxandi
Yesterday at 1:48 AM
Weird integral
Martin.s   2
N Yesterday at 12:43 AM by ADus
\[
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 
\frac{1 - e^{-2} \cos\left(2\left(u + \tan u\right)\right)}
{1 - 2e^{-2} \cos\left(2\left(u + \tan u\right)\right) + e^{-4}} 
\, \mathrm{d}u
\]
2 replies
Martin.s
May 20, 2025
ADus
Yesterday at 12:43 AM
IMC 2018 P4
ThE-dArK-lOrD   18
N Wednesday at 8:50 PM by jonh_malkovich
Source: IMC 2018 P4
Find all differentiable functions $f:(0,\infty) \to \mathbb{R}$ such that
$$f(b)-f(a)=(b-a)f’(\sqrt{ab}) \qquad \text{for all}\qquad a,b>0.$$
Proposed by Orif Ibrogimov, National University of Uzbekistan
18 replies
ThE-dArK-lOrD
Jul 24, 2018
jonh_malkovich
Wednesday at 8:50 PM
convergence
Soupboy0   2
N Wednesday at 6:33 PM by fruitmonster97
If the function $\zeta(n) = \frac{1}{1^n}+\frac{1}{2^n}+\frac{1}{3^n}+....$ diverges for $n=1$ (harmonic sequence) but converges for $n=2$ because $\frac{\pi^2}{6}$, is there a value between $n=1$ and $n=2$ such that $\zeta(n)$ converges

(i dont know the answer could someone please help me)
2 replies
Soupboy0
Wednesday at 6:13 PM
fruitmonster97
Wednesday at 6:33 PM
a^2=3a+2imatrix 2*2
zolfmark   3
N Wednesday at 2:00 PM by Mathzeus1024
A
matrix 2*2

A^2=3A+2i
A^3=mA+Li


i means identity matrix,

find constant m ، L
3 replies
1 viewing
zolfmark
Feb 23, 2019
Mathzeus1024
Wednesday at 2:00 PM
polynomial having a simple root
FFA21   1
N Wednesday at 1:59 PM by Doru2718
Source: MSU algebra olympiad 2025 P4
$f(x)\in R[x]$ show that $f(x)+i$ has at least one root of multiplicity one
1 reply
FFA21
May 20, 2025
Doru2718
Wednesday at 1:59 PM
Funny function that there isn't exist
ItzsleepyXD   5
N Apr 26, 2025 by Hamzaachak
Source: Own, Modified from old problem
Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $m$ and $n$,
$$ m^{\phi(n)}+n^{\phi(m)} \mid f(m)^n + f(n)^m$$
5 replies
ItzsleepyXD
Apr 10, 2025
Hamzaachak
Apr 26, 2025
Funny function that there isn't exist
G H J
Source: Own, Modified from old problem
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ItzsleepyXD
147 posts
#1
Y by
Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $m$ and $n$,
$$ m^{\phi(n)}+n^{\phi(m)} \mid f(m)^n + f(n)^m$$
Z K Y
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ItzsleepyXD
147 posts
#2
Y by
Bump....
Z K Y
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EvansGressfield
5 posts
#3
Y by
any idea?
Z K Y
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Rayanelba
20 posts
#4
Y by
This problem is very hard pls share some hints
Z K Y
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Hamzaachak
61 posts
#5
Y by
No such function exist
Z K Y
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Hamzaachak
61 posts
#6 • 1 Y
Y by ItzsleepyXD
Hamzaachak wrote:
No such function exist

Small hint : try to prove f(2)=0
Z K Y
N Quick Reply
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