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geometry (In the World of Mathematics) (>>) Greek Aops Index Greek Aops Index - Άλγεβρα Greek Aops Index - Ανισότητες Greek Aops Index - Γεωμετρία Greek Aops Index - Θεωρία Αριθμών Greek Aops Index - Συνδυαστική
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 
.
, 
and 
is the midpoint of 
. Choose 
in segment 
such that 
and let 
be the other point of intersection between the circumcircles of triangles 
and 
. Prove that lies on the bisector of angle 
.
, and is the midpoint of . Choose in segment such that and let be the other point of intersection between the circumcircles of triangles and . Prove that lies on the bisector of angle .
