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Combinatoric
spiderman0 1
N
by MathBot101101
Let 
Find the maximum positive integer 
such that for every subset 
with n elements, there always exist two elements a, b in T such that:
Find the maximum positive integer 
such that for every subset 
with n elements, there always exist two elements a, b in T such that:
1 reply
Combinatorial proof
MathBot101101 10
N
by MathBot101101
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.
\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.
10 replies
Geometry Angle Chasing
Sid-darth-vater 2
N
by Sid-darth-vater
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!
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!
2 replies
Absolute value
Silverfalcon 8
N
by zhoujef000
This problem seemed to be too obvious.. And I think I"m wrong.. :D
Problem:
Consider the sequence
of integers satisfying
![\[x_0 = 0, |x_n| = |x_{n-1} + 1|\]](//latex.artofproblemsolving.com/4/8/e/48e7b6f98f2062879377a9a2e507a63eac9269a8.png)
for
.
Find the minimum value of the expression
.
Pretty sure I'm wrong but where did I go wrong?
Problem:
Consider the sequence
of integers satisfying![\[x_0 = 0, |x_n| = |x_{n-1} + 1|\]](http://latex.artofproblemsolving.com/4/8/e/48e7b6f98f2062879377a9a2e507a63eac9269a8.png)
for
.Find the minimum value of the expression
.
,
and 
and so forth.
.Pretty sure I'm wrong but where did I go wrong?
8 replies
Tetrahedrons and spheres
ReticulatedPython 3
N
by vanstraelen
Let 
be a tetrahedron such that 
A sphere of radius 
is circumscribed about tetrahedron 
Given that 
, 
, and 
, prove that ![$$r^2+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge \frac{9\sqrt[3]{4}}{4}$$](//latex.artofproblemsolving.com/6/c/6/6c665b4611f5fc8ebeba03ac649bfee4e2634950.png)
with equality at ![$a=b=c=\sqrt[3]{2}.$](//latex.artofproblemsolving.com/b/b/9/bb9c9671b23cb45e5f85ae01bf25d566c2e43e91.png)
be a tetrahedron such that 
A sphere of radius 
is circumscribed about tetrahedron 
Given that 
, 
, and 
, prove that ![$$r^2+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge \frac{9\sqrt[3]{4}}{4}$$](http://latex.artofproblemsolving.com/6/c/6/6c665b4611f5fc8ebeba03ac649bfee4e2634950.png)
with equality at ![$a=b=c=\sqrt[3]{2}.$](http://latex.artofproblemsolving.com/b/b/9/bb9c9671b23cb45e5f85ae01bf25d566c2e43e91.png)
3 replies
Geometric inequality
ReticulatedPython 0
Let 
and 
be points on a plane such that 
, where 
is a positive integer. Let 
be the set of all points 
such that 
, where 
is a real number. The path that traces is continuous, and the value of is minimized. Prove that is rational for all positive integers 
and 
be points on a plane such that 
, where 
is a positive integer. Let 
be the set of all points 
such that 
, where 
is a real number. The path that traces is continuous, and the value of is minimized. Prove that is rational for all positive integers 
0 replies
Inequalities
sqing 27
N
by Jackson0423
Let 
be reals such that 
. Prove that
Let be reals such that 
. Prove that
be reals such that 
. Prove that
Let be reals such that 
. Prove that
27 replies
Problem of the Week--The Sleeping Beauty Problem
FiestyTiger82 1
N
by martianrunner
Put your answers here and discuss!
1 reply
