[/quote]
,
by the expression
and 1.
Find the range of 
such that 
such that 
is an integer.
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
,![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
sports, each awarding one gold medal to a winner. 
such that the experts can make their predictions so that at least 
has exactly two real roots, one of which is 
where 
and 
are integers, 
and 
Find 
to the system of equations![\[\begin{aligned}
\begin{cases}
x^2+y^2-xy-3x+3=0,\\
x^2+y^2+z^2-xy-yz-2zx-3x+3=0.\\
\end{cases}
\end{aligned}\]](http://latex.artofproblemsolving.com/4/a/b/4ab8e94e2e39be219c80264aa5ea23456b74b3e8.png)
such that 
and 
,
.
by 
and, for 
,![\[a_n = a_{\lfloor n/2 \rfloor} + a_{\lfloor n/3 \rfloor} + \ldots + a_{\lfloor n/n \rfloor} + 1.\]](http://latex.artofproblemsolving.com/a/5/8/a58bbcbb429bcada079bb810b286a7c243f25c41.png)
.
be a triangle with orthic triangle 
and orthocenter 
and midpoint of 
as 
.
then prove that 
passes through pole of 
.
Prove that
,
,the bisector of 
meet 
at 
.
divides 
for all distinct positive integers 
,
.
such that 
for some positive integer 
.
be positive real numbers so that 
.![\[ \left( a - 1 + \frac 1b \right) \left( b - 1 + \frac 1c \right) \left( c - 1 + \frac 1a \right) \leq 1.
\]](http://latex.artofproblemsolving.com/7/6/e/76e794a62090c8c4b5f3a70ad4d036669418ca69.png)
Y by Adventure10, Mango247
Given a parallelogram with area 
and we will construct lines where this lines connect a vertex with a midpoint of the side no adjacent to this vertex; with the 
lines formed we have a octagon inside of the parallelogram. Determine the area of this octagon
and we will construct lines where this lines connect a vertex with a midpoint of the side no adjacent to this vertex; with the 
lines formed we have a octagon inside of the parallelogram. Determine the area of this octagonThis post has been edited 2 times. Last edited by mathisreal, Feb 20, 2018, 10:17 AM
