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such that 
is a square.
such that there is a polynomial 
with integer coefficients for which 
and where 
is a prime number .
letters. A sequence of letters is a word, if there are no two equal letters between two other equal letters. Find the number of words of the maximum length.
defined on the positive reals such that 
for all 
.
such that 
is a perfect square, i.e., the square of an integer.
, without repetition. Find the largest multiple of 
. Justify your answer.
such that 
, for all 
.
satisfying the system of equations:
, compute the ratio 
. Prove that sum of these ratios is at least 
.
is given by 
.
without satisfying 
.
be an acute triangle and 
the altitude corresponding to vertex 
. If 
is the midpoint of 
and 
is the midpoint of 
, calculate 
knowing that 
and 
.
,![\[AB^2+CD^2+BD^2+CA^2=AD^2+BC^2+4MN^2.\]](http://latex.artofproblemsolving.com/5/9/6/5961d74530daf871c0a25a821972ddf232fa702c.png)
Spamming Pythagorean Theorem and using the givens, we eventually get that 
.