Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2019 MPFG Problems/Problem 15"
 
(One intermediate revision by the same user not shown)
Line 23: Line 23:
 
</cmath>
 
</cmath>
  
−
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>.
+
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>.
  
−
Assume that <math>\tan x > \tan y</math>. <math>\tan x</math> is represented by the upper line, <math>\tan y</math> is represented by the lower line.
+
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.
  
−
{{image needed}}
+
[[File:Forgot_line.png|710px|center]]
  
 
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.
 
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.
  
 
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>.
 
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>.
 +
 +
~cassphe
 +
 +
~edited by aoum

Latest revision as of 06:54, 17 August 2025

Problem

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

Solution 1

According to the $\tan$ angle sum trigonometric identity,

\[\tan(x + y) = \frac{\tan x + \tan y}{1 + \tan x \cdot \tan y}\]

\[\tan 20.19 = \frac{20.19}{1 + \tan x \cdot \tan y}\]

\[\tan x \cdot \tan y = \frac{20.19}{\tan 20.19} - 1\]

The two equations $\tan x \cdot \tan y = \frac{20.19}{\tan 20.19} - 1$ and $\tan x + \tan y = 20.19$ create a set of Vieta's formulas for

\[x^2 - 20.19x + \left( \frac{20.19}{\tan 20.19} - 1 \right) = 0,\]

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

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

Forgot line.png

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

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

~cassphe

~edited by aoum

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2019_MPFG_Problems/Problem_15&oldid=257935"