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Help me this problem. Thank you
illybest   3
N 4 minutes ago by jasperE3
Find f: R->R such that
f( xy + f(z) ) = (( xf(y) + yf(x) )/2) + z
3 replies
illybest
Today at 11:05 AM
jasperE3
4 minutes ago
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AGM Problem(Turkey JBMO TST 2025)
HeshTarg   3
N 22 minutes ago by ehuseyinyigit
Source: Turkey JBMO TST Problem 6
Given that $x, y, z > 1$ are real numbers, find the smallest possible value of the expression:
$\frac{x^3 + 1}{(y-1)(z+1)} + \frac{y^3 + 1}{(z-1)(x+1)} + \frac{z^3 + 1}{(x-1)(y+1)}$
3 replies
HeshTarg
an hour ago
ehuseyinyigit
22 minutes ago
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Cute matrix equation
RobertRogo   3
N Today at 2:23 PM by loup blanc
Source: "Traian Lalescu" student contest 2025, Section A, Problem 2
Find all matrices $A \in \mathcal{M}_n(\mathbb{Z})$ such that $$2025A^{2025}=A^{2024}+A^{2023}+\ldots+A$$Edit: Proposed by Marian Vasile (congrats!).
3 replies
RobertRogo
May 9, 2025
loup blanc
Today at 2:23 PM
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Integration Bee Kaizo
Calcul8er   63
N Today at 1:50 PM by MS_asdfgzxcvb
Hey integration fans. I decided to collate some of my favourite and most evil integrals I've written into one big integration bee problem set. I've been entering integration bees since 2017 and I've been really getting hands on with the writing side of things over the last couple of years. I hope you'll enjoy!
63 replies
Calcul8er
Mar 2, 2025
MS_asdfgzxcvb
Today at 1:50 PM
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Japanese high school Olympiad.
parkjungmin   1
N Today at 1:31 PM by GreekIdiot
It's about the Japanese high school Olympiad.

If there are any students who are good at math, try solving it.
1 reply
parkjungmin
Yesterday at 5:25 AM
GreekIdiot
Today at 1:31 PM
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Already posted in HSO, too difficult
GreekIdiot   0
Today at 12:37 PM
Source: own
Find all integer triplets that satisfy the equation $5^x-2^y=z^3$.
0 replies
GreekIdiot
Today at 12:37 PM
0 replies
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Japanese Olympiad
parkjungmin   4
N Today at 8:55 AM by parkjungmin
It's about the Japanese Olympiad

I can't solve it no matter how much I think about it.

If there are people who are good at math

Please help me.
4 replies
parkjungmin
May 10, 2025
parkjungmin
Today at 8:55 AM
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ISI UGB 2025 P3
SomeonecoolLovesMaths   11
N Today at 8:21 AM by Levieee
Source: ISI UGB 2025 P3
Suppose $f : [0,1] \longrightarrow \mathbb{R}$ is differentiable with $f(0) = 0$. If $|f'(x) | \leq f(x)$ for all $x \in [0,1]$, then show that $f(x) = 0$ for all $x$.
11 replies
SomeonecoolLovesMaths
Yesterday at 11:32 AM
Levieee
Today at 8:21 AM
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D1020 : Special functional equation
Dattier   3
N Today at 7:57 AM by Dattier
Source: les dattes à Dattier
1) Are there any $(f,g) \in C(\mathbb R,\mathbb R_+)$ increasing with
$$\forall x \in \mathbb R, f(x)(\cos(x)+3/2)+g(x)(\sin(x)+3/2)=\exp(x)$$?

2) Are there any $(f,g) \in C(\mathbb R,\mathbb R_+)$ increasing with
$$\forall x \in \mathbb R, f(x)(\cos(x)+3/2)+g(x)(\sin(x)+3/2)=\exp(x/2)$$?
3 replies
1 viewing
Dattier
Apr 24, 2025
Dattier
Today at 7:57 AM
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Mathematical expectation 1
Tricky123   1
N Today at 6:57 AM by navier3072
X is continuous random variable having spectrum
$(-\infty,\infty) $ and the distribution function is $F(x)$ then
$E(X)=\int_{0}^{\infty}(1-F(x)-F(-x))dx$ and find the expression of $V(x)$

Ans:- $V(x)=\int_{0}^{\infty}(2x(1-F(x)+F(-x))dx-m^{2}$

How to solve help me
1 reply
Tricky123
Yesterday at 9:51 AM
navier3072
Today at 6:57 AM
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Tough integral
Martin.s   0
Today at 4:00 AM
$$\int_0^{\pi/2}\ln(\tan(\theta/2)) \;\frac{4\cos\theta\cos(2\theta)}{4\sin^4\theta+1}\,d\theta.$$
0 replies
Martin.s
Today at 4:00 AM
0 replies
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Minimum value
Martin.s   3
N Yesterday at 5:24 PM by Martin.s
What is the minimum value of
$$ \frac{|a + b + c + d| \left( |a - b| |b - c| |c - d| + |b - a| |c - a| |d - a| \right)}{|a - b| |b - c| |c - d| |d - a|} $$over all triples $a, b, c, d$ of distinct real numbers such that
$a^2 + b^2 + c^2 + d^2 = 3(ab + bc + cd + da).$

3 replies
Martin.s
Oct 17, 2024
Martin.s
Yesterday at 5:24 PM
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Polynomial
Z_.   1
N Apr 23, 2025 by rchokler
Let \( m \) be an integer greater than zero. Then, the value of the sum of the reciprocals of the cubes of the roots of the equation
\[ mx^4 + 8x^3 - 139x^2 - 18x + 9 = 0 \]is equal to:
1 reply
Z_.
Apr 23, 2025
rchokler
Apr 23, 2025
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Polynomial
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Z_.
28 posts
Z_.
#1 h  Apr 23, 2025, 9:21 PM
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Let \( m \) be an integer greater than zero. Then, the value of the sum of the reciprocals of the cubes of the roots of the equation
\[ mx^4 + 8x^3 - 139x^2 - 18x + 9 = 0 \]is equal to:
This post has been edited 1 time. Last edited by Z_., Apr 23, 2025, 9:21 PM
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rchokler
2975 posts
rchokler
#2 h  Apr 23, 2025, 10:22 PM
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Let $a,b,c,d$ be the reciprocals of the roots. Then they solve $9x^4-18x^3-139x^2+8x+m=0$.

Then by Newton's identities, where $p_n=a^n+b^n+c^n+d^n$ and $e_n$ are elementary symmetric polynomials give:

$p_3=e_1p_2-e_2p_1+3e_3=e_1(e_1p_1-2e_2)-e_2p_1+3e_3=e_1(e_1^2-2e_2)-e_1e_2+3e_3=e_1^3-3e_1e_2+3e_3=2^3+3\cdot 2\cdot\frac{139}{9}-3\cdot\frac{8}{9}=98$
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