2019 MPFG Problems/Problem 15
Problem
How many ordered pairs of real numbers
and
are there such that
,
,
, and
?
Solution 1
According to the angle sum trigonometric identity,
The two equations and
create a set of Vieta's Formula for
, whose
is obviously greater than
. This indicates that there must be a constant value for the set
.
Assume that .
is represented by the upper line,
is represented by the lower line.
[insert picture]
As we can see, each value of matches a value of
on the other side of the
. Because
, which is approximately
, 6 values of x/y close to
cannot be taken.
There are values of (x,y) when
. Doubling this number, we get