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Σ to ∞

phiReKaLk6781 3

N Yesterday at 6:12 PM by Maxklark

Evaluate: $\sum\limits_{k=1}^\infty \frac{1}{k\sqrt{k+2}+(k+2)\sqrt{k}}$

3 replies

phiReKaLk6781

Mar 20, 2010

Maxklark

Yesterday at 6:12 PM

Geometric inequality

ReticulatedPython 0

Yesterday at 5:12 PM

Let $A$ and $B$ be points on a plane such that $AB=n$ , where $n$ is a positive integer. Let $S$ be the set of all points $P$ such that $\frac{AP^2+BP^2}{(AP)(BP)}=c$ , where $c$ is a real number. The path that $S$ traces is continuous, and the value of $c$ is minimized. Prove that $c$ is rational for all positive integers $n.$

0 replies

ReticulatedPython

Yesterday at 5:12 PM

0 replies

Inequalities

sqing 27

N Yesterday at 3:51 PM by Jackson0423

Let $a,b$ be reals such that $a^2+ab+b^2 =3$ . Prove that
$\frac{4}{ 3}\geq \frac{1}{ a^2+5 }+ \frac{1}{ b^2+5 }+ab \geq -\frac{11}{4 }$ $\frac{13}{ 4}\geq \frac{1}{ a^2+5 }+ \frac{1}{ b^2+5 }+ab \geq -\frac{2}{3 }$ $\frac{3}{ 2}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }+ab \geq -\frac{17}{6 }$ $\frac{19}{ 6}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }-ab \geq -\frac{1}{2}$ Let $a,b$ be reals such that $a^2-ab+b^2 =1$ . Prove that
$\frac{3}{ 2}\geq \frac{1}{ a^2+3 }+ \frac{1}{ b^2+3 }+ab \geq \frac{4}{15 }$ $\frac{14}{ 15}\geq \frac{1}{ a^2+3 }+ \frac{1}{ b^2+3 }-ab \geq -\frac{1}{2 }$ $\frac{3}{ 2}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }+ab \geq \frac{13}{42 }$ $\frac{41}{ 42}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }-ab \geq -\frac{1}{2 }$

27 replies

sqing

Apr 16, 2025

Jackson0423

Yesterday at 3:51 PM

Problem of the Week--The Sleeping Beauty Problem

FiestyTiger82 1

N Yesterday at 3:24 PM by martianrunner

Put your answers here and discuss!
The Problem

1 reply

FiestyTiger82

Yesterday at 2:30 PM

martianrunner

Yesterday at 3:24 PM

Inequalities

sqing 4

N Yesterday at 1:09 PM by sqing

Let $a,b,c$ be real numbers such that $a^2+b^2+c^2=1.$ Prove that $|a-b|+|b-2c|+|c-3a|\leq 5$ $|a-2b|+|b-3c|+|c-4a|\leq \sqrt{42}$ $|a-b|+|b-\frac{11}{10}c|+|c-a|\leq \frac{29}{10}$

4 replies

sqing

Yesterday at 5:05 AM

sqing

Yesterday at 1:09 PM

Inequalities

nhathhuyyp5c 2

N Yesterday at 12:38 PM by pooh123

Let $a, b, c$ be non-negative real numbers such that $a^2 + b^2 + c^2 = 3$ . Find the maximum and minimum values of the expression
$P = \frac{a}{a^2 + 2} + \frac{b}{b^2 + 2} + \frac{c}{c^2 + 2}.$

2 replies

nhathhuyyp5c

Apr 20, 2025

pooh123

Yesterday at 12:38 PM

Challenging Optimization Problem

Shiyul 5

N Yesterday at 12:28 PM by exoticc

Let $xyz = 1$ . Find the minimum and maximum values of $\frac{1}{1 + x + xy}$ + $\frac{1}{1 + y + yz}$ + $\frac{1}{1 + z + zx}$

Can anyone give me a hint? I got that either the minimum or maximum was 1, but I'm sure if I'm correct.

5 replies

Shiyul

Monday at 8:20 PM

exoticc

Yesterday at 12:28 PM

Radical Axes and circles

mathprodigy2011 4

N Yesterday at 7:53 AM by spiderman0

Can someone explain how to do this purely geometrically?

4 replies

mathprodigy2011

Yesterday at 1:58 AM

spiderman0

Yesterday at 7:53 AM

Combinatoric

spiderman0 0

Yesterday at 7:46 AM

Let $S = \{1, 2, 3, \ldots, 2024\}.$ Find the maximum positive integer $n \geq 2$ such that for every subset $T \subset S$ with n elements, there always exist two elements a, b in T such that:

$|\sqrt{a} - \sqrt{b}| < \frac{1}{2} \sqrt{a - b}$

0 replies

spiderman0

Yesterday at 7:46 AM

0 replies

BMT 2018 Algebra Round Problem 7

IsabeltheCat 5

N Yesterday at 6:56 AM by P162008

Let $h_n := \sum_{k=0}^n \binom{n}{k} \frac{2^{k+1}}{(k+1)}.$ Find $\sum_{n=0}^\infty \frac{h_n}{n!}.$

5 replies

IsabeltheCat

Dec 3, 2018

P162008

Yesterday at 6:56 AM

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