[/quote]
,
has a right angle at 
.
and 
.
lies on 
such that 
and point 
lies on 
such that 
= 
.
.
.
.
.
.
.
from base 
shares angles 
with 
through AA Postulate. 
,
and 
.
,
(as both are right angles), and 
,
through SAS postulate. 
,
.
.
.
?
?
be a positive integer. Define a sequence by setting 
and, for each 
,
be the unique integer in the range 
for which 
is divisible by 
.
the obtained sequence is 
.
eventually becomes constant.
,
lie on sides 
,
.
meet at orthocenter 
.
and 
lie on segment 
such that 
and 
.
and 
intersect at point 
.
.
be positive real numbers such that![\[
\log_x 64 + \log_{y^2} 16 = \frac{5}{3} \quad \text{and} \quad \log_y 64 + \log_{x^2} 16 = 1
\]](http://latex.artofproblemsolving.com/d/7/9/d79facd90d9c5b53cdc4df9db18923ad34f472df.png)
What is the value of 
?
.
is the midpoint of side 
And 
.
over 
intersects lines 
and 
and 
and 
.
lies on 
.
such that 
prove that the triangle must have a right angle.
such that for any real numbers 
and 
,![\[
f\left(x^2+2y f(x)\right) + (f(y))^2 \leq (f(x+y))^2
\]](http://latex.artofproblemsolving.com/8/4/3/8437449cd3f8f0da24d17a60b55dcb563e6b5655.png)
defined by 
and![\[
u_{n+1} = \frac{1}{2}u_n^2 - 4 \quad \text{for all } n \in \mathbb{N}.
\]](http://latex.artofproblemsolving.com/9/9/4/994aa754cc1288ce4f28a95a0276e64282fb5f66.png)
a) Prove that there exist infinitely many positive integers 
such that 
.![\[
\lim_{n \to \infty} \frac{2u_{n+1}}{u_0u_1\cdots u_n}.
\]](http://latex.artofproblemsolving.com/f/f/1/ff174e7431cbfbc17c650d109651241286756a1a.png)
is a cyclic quadrilateral. Let 
and 
.
and 
the orthocenters of triangles 
and 
,
,
be a triangle. A circle passing through 
and 
intersects the sides 
and 
again at 
and 
respectively. Prove that 
,
and 
are concurrent, where 
and 
are the orthocentres of triangles 
respectively.
.
and 
.
=
.
is the foot of the altitude from 
in triangle 
Y by
I was practicing problems and came across one as such:
"Find all integers
such that 
is a positive integral power of a prime positive integer."
I mean after factoring I don't really know where to go...
A hint would be appreciated, and if you want to solve it, please hide your solutions!
Thanks
"Find all integers
such that 
is a positive integral power of a prime positive integer."I mean after factoring I don't really know where to go...
A hint would be appreciated, and if you want to solve it, please hide your solutions!
Thanks
,
is prime and 
.
o
or 
or 
.
,
.
.
or 
or 
