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Mathematical expectation 1
Tricky123   3
N Yesterday at 1:13 PM by Tricky123
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
3 replies
Tricky123
May 11, 2025
Tricky123
Yesterday at 1:13 PM
J
H
Derivative of unknown continuous function
smartvong   2
N Yesterday at 12:43 PM by solyaris
Source: UM Mathematical Olympiad 2024
Let $f: \mathbb{R} \to \mathbb{R}$ be a function whose derivative is continuous on $[0,1]$. Show that
$$\lim_{n \to \infty} \sum^n_{k = 1}\left[f\left(\frac{k}{n}\right) - f\left(\frac{2k - 1}{2n}\right)\right] = \frac{f(1) - f(0)}{2}.$$
2 replies
smartvong
Yesterday at 1:05 AM
solyaris
Yesterday at 12:43 PM
J
H
Divisibility of cyclic sum
smartvong   1
N Yesterday at 12:06 PM by alexheinis
Source: UM Mathematical Olympiad 2024
Let $n$ be a positive integer greater than $1$. Show that
$$4 \mid (x_1x_2 + x_2x_3 + \cdots + x_{n-1}x_n + x_nx_1 - n)$$where each of $x_1, x_2, \dots, x_n$ is either $1$ or $-1$.
1 reply
smartvong
Yesterday at 9:49 AM
alexheinis
Yesterday at 12:06 PM
J
H
Polynomial with integer coefficients
smartvong   1
N Yesterday at 10:04 AM by alexheinis
Source: UM Mathematical Olympiad 2024
Prove that there is no polynomial $f(x)$ with integer coefficients, such that $f(p) = \dfrac{p + q}{2}$ and $f(q) = \dfrac{p - q}{2}$ for some distinct primes $p$ and $q$.
1 reply
smartvong
Yesterday at 9:46 AM
alexheinis
Yesterday at 10:04 AM
J
H
Existence of scalars
smartvong   0
Yesterday at 9:44 AM
Source: UM Mathematical Olympiad 2024
Let $U$ be a finite subset of $\mathbb{R}$ such that $U = -U$. Let $f,g : \mathbb{R} \to \mathbb{R}$ be functions satisfying
$$g(x) - g(y ) = (x - y)f(x + y)$$for all $x,y \in \mathbb{R} \backslash U$.
Show that there exist scalars $\alpha, \beta, \gamma \in \mathbb{R}$ such that
$$f(x) = \alpha x + \beta$$for all $x \in \mathbb{R}$,
$$g(x) = \alpha x^2 + \beta x + \gamma$$for all $x \in \mathbb{R} \backslash U$.
0 replies
smartvong
Yesterday at 9:44 AM
0 replies
J
H
Invertible matrices in F_2
smartvong   1
N Yesterday at 9:02 AM by alexheinis
Source: UM Mathematical Olympiad 2024
Let $n \ge 2$ be an integer and let $\mathcal{S}_n$ be the set of all $n \times n$ invertible matrices in which their entries are $0$ or $1$. Let $m_A$ be the number of $1$'s in the matrix $A$. Determine the minimum and maximum values of $m_A$ in terms of $n$, as $A$ varies over $S_n$.
1 reply
smartvong
Yesterday at 12:41 AM
alexheinis
Yesterday at 9:02 AM
J
H
ISI UGB 2025 P3
SomeonecoolLovesMaths   13
N Yesterday at 8:29 AM by iced_tea
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 $x$.
13 replies
SomeonecoolLovesMaths
May 11, 2025
iced_tea
Yesterday at 8:29 AM
J
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Group Theory
Stephen123980   3
N Monday at 9:01 PM by BadAtMath23
Let G be a group of order $45.$ If G has a normal subgroup of order $9,$ then prove that $G$ is abelian without using Sylow Theorems.
3 replies
Stephen123980
May 9, 2025
BadAtMath23
Monday at 9:01 PM
J
H
calculus
youochange   2
N Monday at 7:46 PM by tom-nowy
$\int_{\alpha}^{\theta} \frac{d\theta}{\sqrt{cos\theta-cos\alpha}}$
2 replies
youochange
Monday at 2:26 PM
tom-nowy
Monday at 7:46 PM
J
H
ISI UGB 2025 P1
SomeonecoolLovesMaths   6
N Monday at 5:10 PM by SomeonecoolLovesMaths
Source: ISI UGB 2025 P1
Suppose $f \colon \mathbb{R} \longrightarrow \mathbb{R}$ is differentiable and $| f'(x)| < \frac{1}{2}$ for all $x \in \mathbb{R}$. Show that for some $x_0 \in \mathbb{R}$, $f \left( x_0 \right) = x_0$.
6 replies
SomeonecoolLovesMaths
May 11, 2025
SomeonecoolLovesMaths
Monday at 5:10 PM
J
a