Difference between revisions of "Arcticturn Prep"
Arcticturn (talk | contribs) (Created page with "Problem 5 Suppose that <math>x</math>, <math>y</math>, and <math>z</math> are complex numbers such that <math>xy = -80 - 320i</math>, <math>yz = 60</math>, and <math>zx = -96...") |
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Suppose that <math>x</math>, <math>y</math>, and <math>z</math> are complex numbers such that <math>xy = -80 - 320i</math>, <math>yz = 60</math>, and <math>zx = -96 + 24i</math>, where <math>i</math> <math>=</math> <math>\sqrt{-1}</math>. Then there are real numbers <math>a</math> and <math>b</math> such that <math>x + y + z = a + bi</math>. Find <math>a^2 + b^2</math>. | Suppose that <math>x</math>, <math>y</math>, and <math>z</math> are complex numbers such that <math>xy = -80 - 320i</math>, <math>yz = 60</math>, and <math>zx = -96 + 24i</math>, where <math>i</math> <math>=</math> <math>\sqrt{-1}</math>. Then there are real numbers <math>a</math> and <math>b</math> such that <math>x + y + z = a + bi</math>. Find <math>a^2 + b^2</math>. | ||
| − | Problem 6 | + | ==Problem 6== |
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A real number <math>a</math> is chosen randomly and uniformly from the interval <math>[-20, 18]</math>. The probability that the roots of the polynomial | A real number <math>a</math> is chosen randomly and uniformly from the interval <math>[-20, 18]</math>. The probability that the roots of the polynomial | ||
<cmath>x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2</cmath> | <cmath>x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2</cmath> | ||
are all real can be written in the form <math>\dfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. | are all real can be written in the form <math>\dfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. | ||
Revision as of 19:25, 15 June 2024
Problem 5
Suppose that 
, 
, and 
are complex numbers such that 
, 
, and 
, where 
. Then there are real numbers 
and 
such that 
. Find 
.
Problem 6
A real number is chosen randomly and uniformly from the interval ![$[-20, 18]$](http://latex.artofproblemsolving.com/5/e/d/5ed3fd299f7f005431bef3d44f28cf5f7494b169.png)
. The probability that the roots of the polynomial
![\[x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2\]](http://latex.artofproblemsolving.com/3/8/a/38a3a5e9eaaa5a957b5f806c1179ce9a38a46d20.png)
are all real can be written in the form 
, where 
and 
are relatively prime positive integers. Find 
.
