View source for 2019 MPFG Problem 15

==Problem==
How many ordered pairs <math>(x, y)</math> of real numbers <math>x</math> and <math>y</math> are there such that <math>-100 \pi \leq x \leq 100 \pi</math>, <math>-100 \pi \leq y \leq 100 \pi</math>, <math>x + y = 20.19</math>, and <math>\tan x + \tan y = 20.19</math>?

==Solution 1==
According to the <imath>\tan</imath> angle sum trigonometric identity, 

<cmath>
\tan(x + y) = \frac{\tan x + \tan y}{1 + \tan x \cdot \tan y}
</cmath>

<cmath>
\tan 20.19 = \frac{20.19}{1 + \tan x \cdot \tan y}
</cmath>

<cmath>
\tan x \cdot \tan y = \frac{20.19}{\tan 20.19} - 1
</cmath>

The two equations <imath>\tan x \cdot \tan y = \frac{20.19}{\tan 20.19} - 1</imath> and <imath>\tan x + \tan y = 20.19</imath> create a set of [[Vieta's Formulas|Vieta's formulas]] for 

<cmath>
x^2 - 20.19x + \left( \frac{20.19}{\tan 20.19} - 1 \right) = 0,
</cmath>

whose discriminant <imath>\Delta</imath> is obviously greater than 0. This indicates that there must be a constant value for the set <imath>(\tan x, \tan y)</imath>.

Assume that <imath>\tan x > \tan y</imath>. <imath>\tan x</imath> is represented by the upper blue line, <imath>\tan y</imath> is represented by the lower red line.

[[File:Forgot_line.png|710px|center]]

As we can see, each value of <imath>x</imath> matches a value of <imath>y</imath> on the other side of the <imath>y</imath>-axis. Because <imath>x + y = 20.19</imath>, which is approximately <imath>6.42 \pi</imath>, 6 values of <imath>x/y</imath> close to <imath>-100 \pi</imath> cannot be taken.

There are <imath>200 - 6 = 194</imath> values of <imath>(x, y)</imath> when <imath>\tan x > \tan y</imath>. Doubling this number, we get <imath>\boxed{388}</imath>.

~cassphe

~edited by aoum