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, points 
are chosen such that when going around the triangle, the points occur in the order 
. It is given that![\[
AD_1 = AD_2 = BE_1 = BE_2 = CF_1 = CF_2.
\]](http://latex.artofproblemsolving.com/4/c/f/4cf9e7bc009736836bcd10c1d8ba46bd3a91cd7b.png)
Prove that the perimeters of the triangles formed by the lines 
and 
are equal.
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
, where![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
be a cyclic quadrilateral, and let 
, 
, 
, and 
be the midpoints of 
, 
, 
, and 
respectively. Let 
, 
, 
and 
be the orthocenters of triangles 
, 
, 
and 
, respectively. Prove that the quadrilaterals and 
have the same area.
for which the following statement is true: If positive integers 
and 
are such that 
is a multiple of 
and 
is a multiple of , then each of the numbers and is a multiple of .
for which the following statement is true: If positive integers and are such that is a multiple of and is a multiple of , then each of the numbers and is a multiple of .
for which the following statement is true: If positive integers and are such that is a multiple of and is a multiple of , then each of the numbers and is a multiple of .
