inequality

by Daytuz, Mar 23, 2025, 4:02 AM

Consider the function $f$ defined on $\mathbb{R}^2$ by
$f(x, y) = x^4 + y^4 - 2(x - y)^2.$
Show that there exist $(\alpha, \beta) \in \mathbb{R}^2$ (and determine them) such that
$\forall (x, y) \in \mathbb{R}^2, f(x, y) \geq \alpha \| (x, y) \|^2 + \beta,$ where $\| \cdot \|$ denotes the Euclidean norm.

inequalities

calculus

AMM 12481 (Neat Generalization of Maximum Modulus Principle)

by kgator, Mar 23, 2025, 3:49 AM

12481. Proposed by Bernhard Elsner, Université de Versailles Saint-Quentin-en-Yvelines, Versailles, France, and Eric Müller, Villingen-Schwenningen, Germany. Let $f_1, \ldots, f_n$ be holomorphic functions on $U$ , where $U$ is an open, connected subset of $\mathbb{C}$ . Suppose that the function $g : U \rightarrow \mathbb{R}$ given by $g(z) = |f_1(z)| + \cdots + |f_n(z)|$ takes a maximum value in $U$ . Must each function $f_k$ be constant on $U$ ?

AMM

complex analysis

maximum modulus principle

Maximum modulus theorem

Open mapping theorem

holomorphic

Constant term of minimal polynomial algebraic element

by M4tchash3l, Mar 22, 2025, 9:31 PM

Suppose $a \in \mathbb{R}$ and $a \neq 0$ and there exists a positive integer $n$ such that $a^n \in \mathbb{Q}$ . Let $p(x)$ be minimal polynomial $a$ over $\mathbb{Q}$ . Prove that $p(0) = \pm a^{\deg(p)}$

Do these have a closed form?

by Entrepreneur, Mar 22, 2025, 7:56 PM

$\int_0^\infty\frac{t^{n-1}}{(t+\alpha)^2+m^2}dt.$ $\int_0^\infty\frac{e^{nt}}{(t+\alpha)^2+m^2}dt.$ $\int_0^\infty\frac{dx}{(1+x^a)^m(1+x^b)^n}.$

calculus

integrals

Derivative of function R^2 to R^2

by Sifan.C.Maths, Mar 22, 2025, 7:09 AM

Give a function $f:\mathbb{R}^2 \to \mathbb{R}^2: f(x,y)=(x^2+xy,y^2+x)$ . Calculate the first and second derivative of the function at the point $(1,-1)$ .

Multivariable Calculus

Integrate the reciprocal of a geometric series

by IHaveNoIdea010, Mar 21, 2025, 2:31 PM

Determine the exact value of $\int_{0}^{\infty} \frac{1}{\sum_{n=0}^{10} x^n} \,dx$

calculus

integration

geometric series

Galois group

by ILOVEMYFAMILY, Mar 11, 2025, 5:19 AM

Let $K$ be a field. Find the Galois groups

$a) \text{Gal}(K(x), K)$

$b) \text{Gal}(K(x,y), K)$

This post has been edited 3 times. Last edited by ILOVEMYFAMILY, Mar 11, 2025, 5:21 AM

Galois Theory

Integrals problems and inequality

by tkd23112006, Feb 16, 2025, 1:32 PM

Let f be a continuous function on [0,1] such that f(x) ≥ 0 for all x ∈[0,1] and
$\int_x^1 f(t) dt \geq \frac{1-x^2}{2}$ , ∀x∈[0,1].
Prove that:
$\int_0^1 (f(x))^{2021} dx \geq \int_0^1 x^{2020} f(x) dx$

This post has been edited 1 time. Last edited by tkd23112006, Feb 16, 2025, 1:33 PM
Reason: Incorrect format

calculus

integration

inequalities

Initial Value Problem

by TheFlamingoHacker, Mar 5, 2020, 11:08 PM

Set up the IVP that will give the velocity of a $60$ kg sky diver that jumps out of a plane with no initial velocity and an air resistance of $0.8|v|$ . For this example assume that the positive direction is downward.

IVP

Miklos Schweitzer 1982_10

by ehsan2004, Jan 31, 2009, 2:23 PM

Let $p_0,p_1,\ldots$ be a probability distribution on the set of nonnegative integers. Select a number according to this distribution and repeat the selection independently until either a zero or an already selected number is obtained. Write the selected numbers in a row in order of selection without the last one. Below this line, write the numbers again in increasing order. Let $A_i$ denote the event that the number $i$ has been selected and that it is in the same place in both lines. Prove that the events $A_i \;(i=1,2,\ldots)$ are mutually independent, and $P(A_i)=p_i$ .

T. F. Mori

probability

probability and stats

Submit

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yo dawg i heard you imo
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Good job on 5 problems on USAMO. I know that is a pretty bad score for someone as awesome as you on a normal USAMO, but no one got all 6 this year so it's GREAT!

VICTOR WANG 42 ON IMO 2013 LET'S GO.

See you at MOP this year (probably ).
by yugrey, May 3, 2013, 10:32 PM
you can just write "Solution
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"

by math154, Feb 6, 2013, 6:33 PM
Hello. May I ask a stupid question: how to turn "Hidden Text" into hidden "Solution" tag?
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by math154, Nov 23, 2011, 10:26 PM
what i got uncontribbed for posting a nice geo problem?
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by Binomial-theorem, Oct 5, 2011, 12:39 AM
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Fine. I don't care.
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Math154: It was West Somethign Middle School from Columbia that got 2nd.
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by math154, Mar 6, 2009, 2:50 AM
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by AIME15, Mar 6, 2009, 1:06 AM
Thanks! Good luck to you too.
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good luck on saturday
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Me wants be contrib.
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by math154, Mar 2, 2009, 11:55 PM
third! contrib?
by nikeballa96, Mar 2, 2009, 10:40 PM
2nd Shout!
by gauss1181, Feb 28, 2009, 5:25 PM
1st Shout!
by mathemagician1729, Feb 28, 2009, 4:02 AM