[/quote]
,
be the set of non-negative integers and 
be the set of positive real numbers. Let 
be a function such that 
and 
for all integers 
,
for all integers 
.
.
and 
Prove that
Let 
and 
Prove that
which satisfy 
![$ \frac{a+b+c}{3} \ge \sqrt[10]{\frac{a^3+b^3+c^3}{3}}$](http://latex.artofproblemsolving.com/c/d/a/cdae05ad34239366b0dd3921243522c711b6ae58.png)
method, I can assum 
![$ \frac{2x+\frac{1}{x^2}}{3} \ge \sqrt[10]{ \frac{2x^3+ \frac{1}{x^6}}{3}} $](http://latex.artofproblemsolving.com/c/d/4/cd481672bee1e87d8c97ac623436e0ec75cfa73a.png)
are given such that 
.
in natural numbers m and n.
,
denote the smallest possible value of 
where 
are sets such that 
and 
whenever 
.
and 
intersect at 
.
of the two circles touches 
at 
respectively. Let 
be the symmetric points of 
intersects circles 
respectively which are distinct from 
and 
are equal.
,
.
intersects 
at 
.
cuts segment 
at 
respectively. If 
and 
,
equals 
.
are points on a circle 
,
,
.
at 
.
is on 
such that 
.
intersects 
.
is the reflection of 
.
lies on the same circle.
in its exterio
so that 
,
,
.
and
such that 
and 
.
be the points on 
and 
,
and 
.
and 
is on the circumference of triangle 
has a right angle at 
.
such that 
and 
lies between 
.
is chosen so that 
and 
.
is chosen so that 
and 
is the bisector of 
.
be the point such that 
is a parallelogram. Prove that 
and 
are concurrent.
such that the number of 
which satisfy the following condition is minimal:
has periodic 
and 
are tangent to the circumcircle of 
be the reflection of 
be the reflection of 
.
is the circumcenter of 
,
is perpendicular to 
Y by AlastorMoody, Adventure10, Mango247
Let 
be the set 
. Let 
be a subset of .
We arrange the elements of in increasing order, that is, 
with 
.
Define WSUM for this subset as
where the first term contains the odd numbered terms and the second the even numbered terms.
(For example, if
, WSUM is 
.)
Find the sum of WSUMs over all the subsets of S.
(Assume that WSUM for the null set is
.)
be the set 
. Let 
be a subset of .We arrange the elements of
in increasing order, that is, 
with 
.Define WSUM for this subset as
where the first term contains the odd numbered terms and the second the even numbered terms.(For example, if
, WSUM is 
.)Find the sum of WSUMs over all the subsets of S.
(Assume that WSUM for the null set is
.)
.
,
with 
.
denote the sum of all elements of the set 
,
denote the sum
,
is arranged in increasing order.![\[2\left(\sum_{X\subseteq [n]} f(X)\right) + \left(\sum_{X\subseteq [n]} g(X)\right)\]](http://latex.artofproblemsolving.com/e/b/8/eb8c47e256604b2f0bfa469733d123de36e95110.png)
![\(a\in[n]\)](http://latex.artofproblemsolving.com/c/6/4/c64228a3f4e9fde609161f673758d986a79411bf.png)
times; it is added once for each subset, and there are
numbers). Thus, this sum![\[\sum_{a=1}^n 2^{n-1}\cdot a = n(n+1)\cdot 2^{n-2}\]](http://latex.artofproblemsolving.com/4/4/3/443179a521479752dda70d7e0d1d9dd35f7df490.png)
and multiplying by 
.![\[\sum_{X\subseteq [n]} g(X)\]](http://latex.artofproblemsolving.com/a/f/0/af0b1f03d2110f67d389492fbeaa15eb2d23842d.png)
.
.
is at the 
-![\[2^{n-a}\cdot \binom{a-1}{2k}\]](http://latex.artofproblemsolving.com/6/4/3/643b35acdf4feb47fb78f334bb5cdbc09578f913.png)
where 
is the number of ways to add elements greater than 
is the number of ways to choose the first 
elements of the set.![\[2^{n-1} + \sum_{a=2}^{n} \sum_{k=0}^{\infty} a\binom{a-1}{2k}\cdot 2^{n-a}\]](http://latex.artofproblemsolving.com/7/b/2/7b2f8c832cfe12b49cf89893661c01f51e63248f.png)
(this is obviously finite assuming 
for 
)
![\[n(n+1)2^{n-3} + 2^{n-2}\]](http://latex.artofproblemsolving.com/3/4/9/349d103df3608f78219abb487e969d8f5fffe4d7.png)
![\[n(n+1)2^{n-1} + n(n+1)2^{n-3} + 2^{n-2}\]](http://latex.artofproblemsolving.com/b/4/6/b468c721df62f041bb8764876a429ae5d19daef1.png)
which does indeed equal![\[2^{n-3} (5n^2 + 5n + 2)\]](http://latex.artofproblemsolving.com/3/f/6/3f6dbd15b66b4f82075ee1bf494877d8a659ce98.png)
finishing the proof.
to find the answer to the original question.
.
,
.
and 
.
.
to appear as the oddth term of a subset 
which has an even number of elements, times the number of subsets 
has. This is just equal to 
.
appears in 
.
appears in 
is an exception, as it always appears as the first term of a subset 
appears 
times in 
never appears.
by Fn-1 i get a same answer
