AoPS Academy
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pairwise distinct complex numbers are written on a board. It turns out that there are exactly 760 ways to choose two numbers 
and 
from the board such that:![\[
a^2 + b^2 + 1 = 2ab
\]](http://latex.artofproblemsolving.com/8/f/c/8fc5df8dbc8e1647edfaa96dcfa8a0dfdac0c961.png)
Ways that differ by the order of selection are considered the same. Prove that there exist two numbers 
and 
from the board such that:![\[
c^2 + d^2 + 2025 = 2cd
\]](http://latex.artofproblemsolving.com/2/7/0/2706d72797c5e392b3f068a8174bd257e4d93fa1.png)
pairwise distinct complex numbers are written on a board. It turns out that there are exactly 760 ways to choose two numbers and from the board such that:Ways that differ by the order of selection are considered the same. Prove that there exist two numbers and from the board such that:
of nodes, one for each complex number and draw edges if the condition is held.
is form by exactly 
disconnected and simple path.
Condition is equivalent to 
meaning that 
can only be connected to 
and 
, this ensure that is a union of disconnected and simple path.
of these path, each of length 
. The we have that 
and 
, then 
.
nodes. So we have a complex number 
and 
and its easy to check that![\[
c^2+(c+45i)^2+2025 = 2c(c+45i)
\]](http://latex.artofproblemsolving.com/a/e/b/aeb34e72265ec8195327125c6953716ab047ab4b.png)
.
Prove that
and 
.
Let 
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