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inscribed in circle 
with three altitudes 
intersecting at 
. Let 
be the points symmetric to 
through 
respectively. Circle 
intersects at 
, circle 
intersects at 
. 
is the intersection of 
and 
. Prove that 
is tangent to circle .
be two circles such that 
is inside 
. Suppose 
are six circles tangent internally to and externally to such that 
and 
intersect at 
(taken modulo six). Suppose further that the quadrilaterals 
and 
are all cyclic, then prove that their circumcircles meet at two points.
be a fixed integer. Find the least constant 
such the inequality![\[\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C
\left(\sum_{i}x_{i} \right)^4\]](http://latex.artofproblemsolving.com/d/d/e/dde2d1dd0c412d8ba640f1d4c044e2fa2dcccc82.png)
(the sum on the left consists of 
summands). For this constant , characterize the instances of equality.
, let 
be the excircle opposite to 
. Let 
and 
be the points where is tangent to 
, and , respectively. The circle 
intersects line at and . Let 
be the midpoint of 
. Prove that the circle 
is tangent to .
) defined by :
or
with 
and let 
.![\[P\left(\frac{p_1}{1-p_1}+\frac{p_2}{1-p_2}+\ldots+\frac{p_n}{1-p_n}\right).\]](http://latex.artofproblemsolving.com/9/8/f/98f9aaa76faf8ff093842f1cfe25040e9fa23da1.png)
be a vector space over and 
be a linear transformation such that 
and 
is a vector space over a field and that 
are subspaces of such that 
. The *projection onto 
along 
* is the map 
defined as follows: for each 
. be a vector space over a field and let be a linear map on . The following statements are equivalent:
is a projection
. Moreover, if this is the case, then 
.
irrational number or rational number？ Any comments or hints will welcome!
increasing with the limit to 
is .
converge ?
such that 
is rational ?
and a positive integer 
is given.
and for each 
![\[f_n(k)=\frac{a_n+\dots+a_{k+1}}{n-k}-\frac{a_k+\dots+a_1}{k}\]](http://latex.artofproblemsolving.com/0/4/1/041f80cf63f17c0f62c166021af474a5d305eb87.png)
Then for every integer 
the condition![\[a_{n+1}=\max_{0\leq k<n}(f_n(k))\]](http://latex.artofproblemsolving.com/b/1/f/b1f5aff787de8fc9c113cf8441a6b2900a19f24f.png)
is satisfied. Prove that this sequence must be eventually constant.
and a positive integer is given. and for each Then for every integer the conditionis satisfied. Prove that this sequence must be eventually constant.
, prove 
.
. Also 
. We have 
. We will proceed by using induction. Suppose that 
holds, we will prove 
. On the other hand, for all 
Thus, we obtain the following
implying that the sequence 
must be eventually constant as desired.
,![$\lim_{n \to \infty} [s_n(f,x)- 1/\pi\int_{\mathbb{T}}^{}f(t)\frac{\sin(n(x-t))}{x-t}d\lambda(t)]=0$](http://latex.artofproblemsolving.com/d/c/a/dca2f0cedb76d521f1c3254d64e62060bf30799e.png)
,
the nth fourier partial sum.
![\[a_{n+1}=\max_{0\leq k<n}(f_n(k))=f_n(g(n))\]](http://latex.artofproblemsolving.com/8/b/1/8b11c506616b015473f027f4874c01d88634da4e.png)
Step 
:
,
,
will hold.
: