Geometric inequality

by ReticulatedPython, Apr 22, 2025, 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 $n.$

This post has been edited 2 times. Last edited by ReticulatedPython, Yesterday at 8:06 PM

geometry

inequalities

geometric inequality

Problem of the Week--The Sleeping Beauty Problem

by FiestyTiger82, Apr 22, 2025, 2:30 PM

Put your answers here and discuss!
The Problem

Counting Probability

Problem of the Week

Combinatoric

by spiderman0, Apr 22, 2025, 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:

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

combinatoric

set

combinatorics

Geometry Angle Chasing

by Sid-darth-vater, Apr 21, 2025, 11:50 PM

Is there a way to do this without drawing obscure auxiliary lines? (the auxiliary lines might not be obscure I might just be calling them obscure)

For example I tried rotating triangle MBC 80 degrees around point C (so the BC line segment would now lie on segment AC) but I couldn't get any results. Any help would be appreciated!

Attachments:

geometry

Tetrahedrons and spheres

by ReticulatedPython, Apr 21, 2025, 6:39 PM

Let $OABC$ be a tetrahedron such that $\angle{AOB}=\angle{AOC}=\angle{BOC}=90^\circ.$ A sphere of radius $r$ is circumscribed about tetrahedron $OABC.$ Given that $OA=a$ , $OB=b$ , and $OC=c$ , prove that $r^2+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge \frac{9\sqrt[3]{4}}{4}$ with equality at $a=b=c=\sqrt[3]{2}.$

This post has been edited 19 times. Last edited by ReticulatedPython, Yesterday at 5:31 PM

Combinatorial proof

by MathBot101101, Apr 20, 2025, 7:37 AM

Is there a way to prove
\frac{1}{(1+1)!}+\frac{2}{(2+1)!}+...+\frac{n}{(n+1)!}=1-\frac{1}{{n+1)!}
without induction and using only combinatorial arguments?

Induction proof wasn't quite as pleasing for me.

This post has been edited 1 time. Last edited by MathBot101101, Apr 21, 2025, 6:45 AM
Reason: im dum

combinatorics

Combinatorial Proof

Inequalities

by sqing, Apr 16, 2025, 4:52 AM

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
$\frac{3}{ 2}\geq \frac{1}{ a^2+3 }+ \frac{1}{ b^2+3 }+ab \geq \frac{4}{15 }$ $\frac{14}{ 15}\geq \frac{1}{ a^2+3 }+ \frac{1}{ b^2+3 }-ab \geq -\frac{1}{2 }$ $\frac{3}{ 2}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }+ab \geq \frac{13}{42 }$ $\frac{41}{ 42}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }-ab \geq -\frac{1}{2 }$

inequalities

Σ to ∞

by phiReKaLk6781, Mar 20, 2010, 7:29 PM

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

Simiplifying a Complicated Expression

by phiReKaLk6781, Mar 15, 2010, 3:59 AM

Simplify: $\frac{a^3}{(a-b)(a-c)}+\frac{b^3}{(b-a)(b-c)}+\frac{c^3}{(c-a)(c-b)}$

Absolute value

by Silverfalcon, Jun 27, 2005, 1:17 AM

This problem seemed to be too obvious.. And I think I"m wrong..

Problem:

Consider the sequence $x_0, x_1, x_2,...x_{2000}$ of integers satisfying

$x_0 = 0, |x_n| = |x_{n-1} + 1|$

for $n = 1,2,...2000$ .

Find the minimum value of the expression $|x_1 + x_2 + ... x_{2000}|$ .

My idea

Pretty sure I'm wrong but where did I go wrong?

absolute value

Cracking the Nut Open

by ReticulatedPython, Apr 22, 2025, 5:12 PM

by FiestyTiger82, Apr 22, 2025, 2:30 PM

by spiderman0, Apr 22, 2025, 7:46 AM

by Sid-darth-vater, Apr 21, 2025, 11:50 PM

by ReticulatedPython, Apr 21, 2025, 6:39 PM

by MathBot101101, Apr 20, 2025, 7:37 AM

by sqing, Apr 16, 2025, 4:52 AM

by phiReKaLk6781, Mar 20, 2010, 7:29 PM

by phiReKaLk6781, Mar 15, 2010, 3:59 AM

by Silverfalcon, Jun 27, 2005, 1:17 AM