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and 
with 
satisfy 
What is the value of 
?![\[\prod^{63}_{k=4} \frac{\log_k (5^{k^2 - 1})}{\log_{k + 1} (5^{k^2 - 4})} = \frac{\log_4 (5^{15})}{\log_5 (5^{12})} \cdot \frac{\log_5 (5^{24})}{\log_6 (5^{21})}\cdot \frac{\log_6 (5^{35})}{\log_7 (5^{32})} \cdots \frac{\log_{63} (5^{3968})}{\log_{64} (5^{3965})}\]](http://latex.artofproblemsolving.com/1/8/a/18ad3a97e3df2da4f41cbdf62f07db5272a9948b.png)
is equal to 
where 
and 
are relatively prime positive integers. Find 
, , and 
be positive real numbers that satisfy ![\[ 2\log_x(2y) = 2\log_{2x}(4z) = \log_{2x^4}(8yz) \neq 0. \]](http://latex.artofproblemsolving.com/7/f/d/7fd6452d5e10482152d87a54846bc130ab957ab7.png)
The value of 
can be expressed in the form 
, where 
and 
are relatively prime integers. Find 
.
in the closed interval ![$[-500,500]$](http://latex.artofproblemsolving.com/7/6/8/768f4e11a324ee335c6a0689a310fdaa8c33c921.png)
for which the equation 
has exactly one real solution.
, 
, and 
satisfy 
and 
. Find 
.
![\[N= \sum_{k=1}^{1000}k(\lceil \log_{\sqrt{2}}k\rceil-\lfloor \log_{\sqrt{2}}k \rfloor).\]](http://latex.artofproblemsolving.com/a/6/c/a6cb741150d287ecd6bb40bb5734df9a63e68455.png)
Find the remainder when N is divided by 1000. (Here 
denotes the greatest integer that is less than or equal to x, and 
denotes the least integer that is greater than or equal to x.)
and 
have the property that![\[
\sqrt{\log{a}} + \sqrt{\log{b}} + \log \sqrt{a} + \log \sqrt{b} = 100
\]](http://latex.artofproblemsolving.com/0/e/2/0e2f29c24ae658aab1931faf129a293f0ba98493.png)
and all four terms on the left are positive integers, where 
denotes the base 10 logarithm. What is 
?
and satisfy the condition ![\[\log_2(\log_{2^a}(\log_{2^b}(2^{1000})))=0.\]](http://latex.artofproblemsolving.com/d/c/9/dc9cef809c8aabaa5bef3cfefd3062c60eaa600e.png)
Find the sum of all possible values of 
.
. Find the largest value of n for which 
.
is the greatest integer less than or equal to .
