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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.

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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
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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
J
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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
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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
J
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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
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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
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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
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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
J
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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
J
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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
J
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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
J
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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
J
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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
J
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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
J
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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
J
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J
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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
J
Funny function that there isn't exist
G H J
Reveal topic tags
function
number theory
functional equation
Divisibility
Euler s Phi Function
L
G H BBookmark kLocked kLocked NReply
Source: Own, Modified from old problem
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ItzsleepyXD
147 posts
ItzsleepyXD
#1 h  Apr 10, 2025, 1:50 AM
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
ItzsleepyXD
#2 h  Apr 25, 2025, 4:06 AM
Y by
Bump....
Z K Y
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EvansGressfield
5 posts
EvansGressfield
#3 h  Apr 26, 2025, 4:17 PM
Y by
any idea?
Z K Y
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Rayanelba
20 posts
Rayanelba
#4 h  Apr 26, 2025, 6:39 PM
Y by
This problem is very hard pls share some hints
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Hamzaachak
61 posts
Hamzaachak
#5 h  Apr 26, 2025, 8:48 PM
Y by
No such function exist
Z K Y
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Hamzaachak
61 posts
Hamzaachak
#6 h  Apr 26, 2025, 9:26 PM • 1 Y
Y by ItzsleepyXD
Hamzaachak wrote:
No such function exist

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