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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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0 replies
jlacosta
May 1, 2025
0 replies
External Direct Sum
We2592   1
N 17 minutes ago 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
Yesterday at 2:45 AM
Acridian9
17 minutes ago
How can I know the sequences's convergence value?
Madunglecha   5
N 5 hours ago by teomihai
What is the convergence value of the sequence??
(n^2)*ln(n+1/n)-n
5 replies
Madunglecha
Yesterday at 6:56 AM
teomihai
5 hours ago
Confusion in derivative
Vulch   1
N 5 hours ago by Vulch
Solve the following problem:
1 reply
Vulch
5 hours ago
Vulch
5 hours ago
Prove the statement
Butterfly   12
N 6 hours ago 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
6 hours ago
Classifying Math with Symbols Based on Behavior
Midevilgmer   2
N Today 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
Today at 12:55 AM
Midevilgmer
Today at 4:06 AM
36x⁴ + 12x² - 36x + 13 > 0
fxandi   3
N Today 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
Today at 1:48 AM
Weird integral
Martin.s   2
N Today 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
Today at 12:43 AM
IMC 2018 P4
ThE-dArK-lOrD   18
N Yesterday 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
Yesterday at 8:50 PM
convergence
Soupboy0   2
N Yesterday 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
Yesterday at 6:13 PM
fruitmonster97
Yesterday at 6:33 PM
a^2=3a+2imatrix 2*2
zolfmark   3
N Yesterday 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
zolfmark
Feb 23, 2019
Mathzeus1024
Yesterday at 2:00 PM
polynomial having a simple root
FFA21   1
N Yesterday 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
Yesterday at 1:59 PM
non-solvable group has subgroup that is not isomorphic to any normal subgroup
FFA21   1
N Yesterday at 1:45 PM by Doru2718
Source: MSU algebra olympiad 2025 P7
Show that in every finite non-solvable group there is a subgroup that is not isomorphic to any normal subgroup
1 reply
FFA21
May 20, 2025
Doru2718
Yesterday at 1:45 PM
Reduction coefficient
zolfmark   1
N Yesterday at 1:26 PM by Mathzeus1024

find Reduction coefficient of x^10

in(1+x-x^2)^9
1 reply
zolfmark
Jul 17, 2016
Mathzeus1024
Yesterday at 1:26 PM
Metric space
wiseman   3
N Yesterday at 10:33 AM by alinazarboland
Source: IMS 2014 - Day1 - Problem4
Let $(X,d)$ be a metric space and $f:X \to X$ be a function such that $\forall x,y\in X : d(f(x),f(y))=d(x,y)$.
$\text{a})$ Prove that for all $x \in X$, $\lim_{n \rightarrow +\infty} \frac{d(x,f^n(x))}{n}$ exists, where $f^n(x)$ is $\underbrace{f(f(\cdots f(x)}_{n \text{times}} \cdots ))$.
$\text{b})$ Prove that the amount of the limit does not depend on choosing $x$.
3 replies
wiseman
Oct 2, 2014
alinazarboland
Yesterday at 10:33 AM
Two times derivable real function
Valentin Vornicu   13
N Apr 24, 2025 by solyaris
Source: RMO 2008, 11th Grade, Problem 3
Let $ f: \mathbb R \to \mathbb R$ be a function, two times derivable on $ \mathbb R$ for which there exist $ c\in\mathbb R$ such that
\[ \frac { f(b)-f(a) }{b-a} \neq f'(c) ,\] for all $ a\neq b \in \mathbb R$.

Prove that $ f''(c)=0$.
13 replies
Valentin Vornicu
Apr 30, 2008
solyaris
Apr 24, 2025
Two times derivable real function
G H J
Source: RMO 2008, 11th Grade, Problem 3
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Valentin Vornicu
7301 posts
#1 • 2 Y
Y by Adventure10, Mango247
Let $ f: \mathbb R \to \mathbb R$ be a function, two times derivable on $ \mathbb R$ for which there exist $ c\in\mathbb R$ such that
\[ \frac { f(b)-f(a) }{b-a} \neq f'(c) ,\] for all $ a\neq b \in \mathbb R$.

Prove that $ f''(c)=0$.
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harazi
5526 posts
#2 • 2 Y
Y by Adventure10, Mango247
The image of the function $ g(a,b)=\frac{f(a)-f(b)}{a-b}$ defined for $ a\ne b$ being an interval (connected subset of the real line) and $ f'(c)$ not being in this image, it follows that we may assume that $ f'(c)<g(a,b)$ for all $ a\ne b$. But then $ f'(c)\leq f'(x)$ for all $ x$ and so $ c$ is a minimum point of $ f'$. Of course, this can be written in 11-th grade vocabulary. :D
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Svejk
663 posts
#3 • 1 Y
Y by Adventure10
I tried to solve it harazi's way but I didn't get maximum since I didn't manage to prove that $ g(a,b)-f'(c)$ has the same sign for all $ a,b$.Can you be give me more detailes please :lol: ?The official solution is based on the fact that the function $ g(x)=f(x)-f'(c)\cdot x$ is injective ,hence monotonic.
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harazi
5526 posts
#4 • 1 Y
Y by Adventure10
The set of pairs $ (a,b)$ such that $ a\ne b$ is a connected subset of the plane and the function $ g$ is continuous on this domain, thus its image is a connected subset of the line, thus an interval. I agree however that this is not a solution of an 11-th grade student, but that's how life is. :D I will not be amazed if in a few years I see complex analysis, Lebesgue integration and other such stuff at RMO. It's quite à pity, it gives huges advantages to some people. :(
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enescu
741 posts
#5 • 2 Y
Y by Adventure10, Mango247
harazi wrote:
But then $ f'(c)\leq f'(x)$ for all $ x$ and so $ c$ is a minimum point of $ f'$.
Why for all $ x$?
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harazi
5526 posts
#6 • 2 Y
Y by Adventure10, Mango247
Well, $ f'(c)\leq g(a,x)$ for all $ a\ne x$ and now make $ a$ close to $ x$ and use the definition of derivative.
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harazi
5526 posts
#7 • 1 Y
Y by Adventure10
Well, I said however a very stupid thing: the function $ g$ should be defined on the set of pairs $ (a,b)$ such that $ a<b$ to ensure that its domain is connected. Of course, all the rest works with this modification, I don't know how I could write such a stupid thing. :D
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enescu
741 posts
#8 • 4 Y
Y by adityaguharoy, Adventure10, Mango247, RobertRogo
Well, when I created this problem, I was thinking to the obvious geometric meaning: if $ f''(c) \ne 0$, then $ f$ is strictly concave up or down on some neighbourhood of the point $ c$, thus one can draw a close enough parallel to the line tangent at $ c$ to the function's graph that intersects the graph in two points $ (a,f(a))$ and $ (b,f(b))$. The slope of that tangent would be $ \frac{f(b)-f(a)}{b-a}$, equal to the slope of the tangent at $ c$, that is,$ f'(c)$.
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subham1729
1479 posts
#9 • 1 Y
Y by Adventure10
One of above solutions uses that $C=\mathbb{R}^2-\{(a,a) \mid a \in \mathbb{R}\}$ is connected, but why $C$ is connected ? $C$ has clearly two connected components. However with this spirit we can also solve the problem, extend $g(a,b)=\frac{f(a)-f(b)}{a-b}$ to whole plane defining $g(a,a)=f'(a)$ and now $g$ is continuous on whole plane and do similar thing.
This post has been edited 1 time. Last edited by subham1729, Jun 14, 2016, 4:53 AM
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Raunii
28 posts
#10
Y by
Svejk wrote:
I tried to solve it harazi's way but I didn't get maximum since I didn't manage to prove that $ g(a,b)-f'(c)$ has the same sign for all $ a,b$.Can you be give me more detailes please :lol: ?The official solution is based on the fact that the function $ g(x)=f(x)-f'(c)\cdot x$ is injective ,hence monotonic.

where did you find the official solution?
This post has been edited 1 time. Last edited by Raunii, Mar 15, 2020, 6:05 PM
Reason: .
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Rohit-2006
245 posts
#11
Y by
Too easy for grade 11....
Attachments:
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solyaris
649 posts
#12
Y by
@above: This is not a valid argument. From the MVT you only get for every $(a,b)$ there exists an $m$ with the desired property. So you get $f'(m) \neq f'(c)$ only for values $m$ in some set $M$, which has to be dense in the reals, but $M$ need not be an interval, so the IVP you use later on in your proof does not give a contradiction. (See the solutions above for proofs that avoid this problem.)
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Rohit-2006
245 posts
#13
Y by
Solyaris....can you please elaborate what you are trying to say....I can't get it what you are trying to say....I am just interested that $f'$ is continuous on $\mathbb{R}$ and that is true because $f$ is twice differentiable.
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solyaris
649 posts
#15
Y by
@above. To elaborate: Let $M = \{x \in R : f'(x) \neq f'(c)\}$. In the first paragraph you show that for all $a < b$ $M$ has to contain some $m \in (a,b)$ (which means that $M$ is a dense subset of the real numbers). This part of your proof is fine. But in order to make the proof of the green claim work you need to show that $M$ contains all real numbers. This is missing in your proof (if I interpret you proof correctly).
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