AoPS Academy
intersect at 
, and there exist a circle 
tangent to the extensions of 
at 
respectively. Circle 
passes through points 
, and is externally tangent to circle at 
. Prove that 
.
let 
be the foot of the altitude from 
to the side 
and 
, 
, 
be the incenter, -excenter, and 
-excenter, respectively. Denote by 
and 
the other intersection points of the circle 
with the lines 
and 
, respectively. Prove that 
.
be positive real variables. Prove that
be a set that intersects all non-finite integer arithmetic progressions, 
be the set of prime divisors of 
and 
be the set of prime divisors of 
. Suppose 
. Prove that 
, 
.
.
1 (mod 7)
are of order 2. 
is of order 7? No elements of order 4 in in the Sylow 7-subgroup.
.
1 (mod 2)
, does this mean that there are no elements of order 2 in the Sylow-2 subgroup, but only elements of order 4. in Dih(28)
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)
, 
and 
, a triangle can be formed. Can it happen that from segments of lengths
an obtuse triangle can be formed?
implies 
and so 
, which is impossible.
,
,
.
