AoPS Academy
+1 w
and 
, with 
. Prove that there exists and only exists one point 
such that 
.
, let 
be the 
-excenter. Points 
and 
are placed on line 
such that 
is between and 
, and is between and . Moreover, and are the contact points of with the -excircle of triangles 
and 
, respectively. Let 
be the -excenter of triangle 
, and let 
be the reflection of the orthocenter of triangle with respect to its circumcenter. Prove that , , and are collinear.
that are twice differentiable and satisfy 
is a continuous function on ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
such that 
, then there exists a 
such that:![\[
F(c) = \frac{1}{c} \int_0^c F(x) \,dx
\]](http://latex.artofproblemsolving.com/8/f/e/8fea4d84bc98b6f4953e19976cfa554945a37ea6.png)
such that 
for all 
.
, 
is the greatest prime divisor of 
. Find:
be a nonzero square matrix of order 
satisfying![\[
a_{ik} a_{jk} = a_{kk} a_{ij}, \quad \text{for all } i, j, k.
\]](http://latex.artofproblemsolving.com/7/5/2/752b16a277d9fd18bfda6c3be35d23d42d75ebaa.png)
Denote by 
the trace of , which is the sum of the diagonal elements of .
. in terms of .
be a ring. Let 
such that 
. Prove that if 
is nilpotent, so is 
.
id defined by![\[
f(x) = \frac{1}{\sqrt{2\pi} \sigma} e^{-\frac{(x-\mu)^2}{2\sigma^2}}
\]](http://latex.artofproblemsolving.com/0/2/c/02c7b4af3cac77da8e3cc102c79a7db2e95f437f.png)
now what is CDF or cumulative distribution function ![$F_X(x)=P[X\leq x]$](http://latex.artofproblemsolving.com/7/9/1/79150c7527581feec873c9c790f8aeddc8f8efb8.png)
is such that 
for any positive integers 
. Assume there exists a positive integer 
such that 
for all positive integers 
. Determine all possible values of 
.
is such that for any positive integers . Assume there exists a positive integer such that for all positive integers . Determine all possible values of .
and prove 
,
,
.
