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Diophantine equation
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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
J
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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
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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
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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
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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
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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
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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
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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
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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
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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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a