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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
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How can I know the sequences's convergence value?
Madunglecha   2
N 4 minutes ago by Madunglecha
What is the convergence value of the sequence??
(n^2)*ln(n+1/n)-n
2 replies
Madunglecha
3 hours ago
Madunglecha
4 minutes ago
J
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Unsolving differential equation
Madunglecha   3
N 2 hours ago by solyaris
For parameter t
I made a differential equation :
y"=y*(x')^2
for here, '&" is derivate and second order derivate for t
could anyone tell me what is equation between y&x?
3 replies
Madunglecha
May 18, 2025
solyaris
2 hours ago
J
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Prove the statement
Butterfly   11
N 3 hours ago by solyaris
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 $|x_n-x_m|<\frac{C}{|n-m|}$.
11 replies
Butterfly
May 7, 2025
solyaris
3 hours ago
J
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External Direct Sum
We2592   0
Today at 2:45 AM
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.

0 replies
We2592
Today at 2:45 AM
0 replies
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Definite integration
girishpimoli   2
N Yesterday at 11:59 PM by Amkan2022
If $\displaystyle g(t)=\int^{t^{2}}_{2t}\cot^{-1}\bigg|\frac{1+x}{(1+t)^2-x}\bigg|dx.$ Then $\displaystyle \frac{g(5)}{g(3)}$ is
2 replies
girishpimoli
Apr 6, 2025
Amkan2022
Yesterday at 11:59 PM
J
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Putnam 1968 A6
sqrtX   11
N Yesterday at 11:47 PM by ohiorizzler1434
Source: Putnam 1968
Find all polynomials whose coefficients are all $\pm1$ and whose roots are all real.
11 replies
sqrtX
Feb 19, 2022
ohiorizzler1434
Yesterday at 11:47 PM
J
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Affine variety
YamoSky   1
N Yesterday at 9:01 PM by amplreneo
Let $A=\left\{z\in\mathbb{C}|Im(z)\geq0\right\}$. Is it possible to equip $A$ with a finitely generated k-algebra with one generator such that make $A$ be an affine variety?
1 reply
YamoSky
Jan 9, 2020
amplreneo
Yesterday at 9:01 PM
J
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Reducing the exponents for good
RobertRogo   0
Yesterday at 6:38 PM
Source: The national Algebra contest (Romania), 2025, Problem 3/Abstract Algebra (a bit generalized)
Let $A$ be a ring with unity such that for every $x \in A$ there exist $t_x, n_x \in \mathbb{N}^*$ such that $x^{t_x+n_x}=x^{n_x}$. Prove that
a) If $t_x \cdot 1 \in U(A), \forall x \in A$ then $x^{t_x+1}=x, \forall x \in A$
b) If there is an $x \in A$ such that $t_x \cdot 1 \notin U(A)$ then the result from a) may no longer hold.

Authors: Laurențiu Panaitopol, Dorel Miheț, Mihai Opincariu, me, Filip Munteanu
0 replies
RobertRogo
Yesterday at 6:38 PM
0 replies
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Differential equations , Matrix theory
c00lb0y   3
N Yesterday at 12:26 PM 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.
3 replies
c00lb0y
Apr 17, 2025
loup blanc
Yesterday at 12:26 PM
J
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The matrix in some degree is a scalar
FFA21   4
N Yesterday at 12:06 PM by FFA21
Source: MSU algebra olympiad 2025 P2
$A\in M_{3\times 3}$ invertible, for an infinite number of $k$:
$tr(A^k)=0$
Is it true that $\exists n$ such that $A^n$ is a scalar
4 replies
FFA21
Yesterday at 12:11 AM
FFA21
Yesterday at 12:06 PM
J
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Weird integral
Martin.s   0
Yesterday at 9:33 AM
\[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{1 - e^{-2} \cos\left(2\left(u + \tan u\right)\right)} {1 - 2e^{-2} \cos\left(2\left(u + \tan u\right)\right) + e^{-4}} \, \mathrm{d}u \]
0 replies
Martin.s
Yesterday at 9:33 AM
0 replies
J
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hard number theory problem
danilorj   4
N Yesterday at 9:01 AM by c00lb0y
Let \( a \) and \( b \) be positive integers. Prove that
\[ a^2 + \left\lceil \frac{4a^2}{b} \right\rceil \]is not a perfect square.
4 replies
1 viewing
danilorj
May 18, 2025
c00lb0y
Yesterday at 9:01 AM
J
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maximum dimention of non-singular subspace
FFA21   1
N Yesterday at 8:27 AM by alexheinis
Source: MSU algebra olympiad 2025 P1
We call a linear subspace in the space of square matrices non-singular if all matrices contained in it, except for the zero one, are non-singular. Find the maximum dimension of a non-singular subspace in the space of
a) complex $n\times n$ matrices
b) real $4\times 4$ matrices
c) rational $n\times n$ matrices
1 reply
FFA21
Yesterday at 12:02 AM
alexheinis
Yesterday at 8:27 AM
J
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functional equation
pratyush   4
N Yesterday at 8:00 AM by Mathzeus1024
For the functional equation $f(x-y)=\frac{f(x)}{f(y)}$, if f ' (0)=p and f ' (5)=q, then prove f ' (-5) = q
4 replies
pratyush
Apr 4, 2014
Mathzeus1024
Yesterday at 8:00 AM
J
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J
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Integrable function: + and - on every subinterval.
SPQ   3
N Apr 23, 2025 by solyaris
Provide a function integrable on [a, b] such that f takes on positive and negative values on every subinterval (c, d) of [a, b]. Prove your function satisfies both conditions.
3 replies
SPQ
Apr 23, 2025
solyaris
Apr 23, 2025
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Integrable function: + and - on every subinterval.
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Reveal topic tags
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SPQ
5 posts
SPQ
#1 h  Apr 23, 2025, 2:40 AM
Y by
Provide a function integrable on [a, b] such that f takes on positive and negative values on every subinterval (c, d) of [a, b]. Prove your function satisfies both conditions.
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greenturtle3141
3560 posts
greenturtle3141
#2 h  Apr 23, 2025, 4:26 AM
Y by
Really depends what you mean by "integrable", but for the lebesgue case the characteristic function on rationals works just fine
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SPQ
5 posts
SPQ
#3 h  Apr 23, 2025, 5:07 AM
Y by
Thanks for the suggestion. You are right that the characteristic function of the rationals in [a, b] is Lebesgue integrable. However, the characteristic function doesn't satisfy the second condition: it doesn't take on negative values anywhere; it only takes on the values 1 and 0.

Also, just to clarify, I was referring to Riemann integrability.
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solyaris
649 posts
solyaris
#4 h  Apr 23, 2025, 7:06 AM
Y by
@above: But of course you can slightly modify the charactersistic function (so that it takes values $1,-1$ instead of $1,0$) to give a counterexample.

For Riemann integrability you only have to introduce a further slight modification: Take $f(x) = 0$ for irrational $x$, and for rational $x = \frac k n$
with integer $k,n$ and $n > 0$ in reduced form, define $f(\frac k n) = -\frac 1 n$ for odd $n$ and $f(\frac k n) = \frac 1 n$ for even $n$. It is not hard to see that $f$ is Riemann-integrable and takes positive and negative values in any open interval.
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