AoPS Academy
be a sequence of reals such that 
and 
for all positive integers 
. As grows large, the value of 
approaches the real number 
. What is the greatest integer less than ?
and let 
and 
be permutations of 
. Suppose there exists a permutation 
such that 
for all 
in .
is the number of possible pairs of permutations 
, find the remainder when is divided by 1000.
and 
Prove that 
be real numbers such that 
. Find the minimum of 
and 
Prove that 
print("this does not import turtle")
lies on the hypotenuse 
of the right-angled triangle 
, but does not coincide with its midpoint . Prove that there are no equal segments among the segments 
, 
and 
.
isn't the midpoint of 
, we know that 
. The only way one of them is equal is for 
to be isosceles and 
. Pretend this is true and draw altitude to 
, called 
. We now form similar triangles 
and 
. Because 
is half of 
, we know that 
. can't be the midpoint of , so this shows that there are no equal segments among the segments 
, 
, and 
. must be the midpoint of , but the problem states it isn't so there are no equal segments.
