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| − | ==Problem==
| + | #REDIRECT [[2019 MPFG Problem 15]] |
| − | 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>?
| + | <br> |
| − | | + | {{delete|housekeeping}} |
| − | ==Solution 1==
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| − | According to the <math>\tan</math> angle sum trigonometric identity,
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| − | <cmath>
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| − | \tan(x + y) = \frac{\tan x + \tan y}{1 + \tan x \cdot \tan y}
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| − | </cmath>
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| − | <cmath>
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| − | \tan 20.19 = \frac{20.19}{1 + \tan x \cdot \tan y}
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| − | </cmath>
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| − | | |
| − | <cmath>
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| − | \tan x \cdot \tan y = \frac{20.19}{\tan 20.19} - 1
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| − | </cmath>
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| − | The two equations <math>\tan x \cdot \tan y = \frac{20.19}{\tan 20.19} - 1</math> and <math>\tan x + \tan y = 20.19</math> create a set of [[Vieta's Formulas|Vieta's formulas]] for
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| − | <cmath> | |
| − | x^2 - 20.19x + \left( \frac{20.19}{\tan 20.19} - 1 \right) = 0,
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| − | </cmath>
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| − | whose discriminant <math>\Delta</math> is obviously greater than 0. This indicates that there must be a constant value for the set <math>(\tan x, \tan y)</math>.
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| − | Assume that <math>\tan x > \tan y</math>. <math>\tan x</math> is represented by the upper blue line, <math>\tan y</math> is represented by the lower red line.
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| − | [[File:Forgot_line.png|710px|center]]
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| − | As we can see, each value of <math>x</math> matches a value of <math>y</math> on the other side of the <math>y</math>-axis. Because <math>x + y = 20.19</math>, which is approximately <math>6.42 \pi</math>, 6 values of <math>x/y</math> close to <math>-100 \pi</math> cannot be taken.
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| − | There are <math>200 - 6 = 194</math> values of <math>(x, y)</math> when <math>\tan x > \tan y</math>. Doubling this number, we get <math>\boxed{388}</math>.
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| − | ~cassphe
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| − | | |
| − | ~edited by aoum
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