Difference between revisions of "1987 AIME Problems/Problem 14"
MRENTHUSIASM (talk | contribs) (→Solution 3 (Complex Numbers)) |
MRENTHUSIASM (talk | contribs) (→Solution 3.2 (Rectangular Form)) |
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Note that <math>N^2=\pm18i,</math> so there are two cases: | Note that <math>N^2=\pm18i,</math> so there are two cases: | ||
<ol style="margin-left: 1.5em;"> | <ol style="margin-left: 1.5em;"> | ||
− | <li><math>N^2=18i</math | + | <li><math>N^2=18i</math><p> |
We have | We have | ||
<cmath>\begin{align*} | <cmath>\begin{align*} | ||
Line 72: | Line 72: | ||
2ab &= 18 | 2ab &= 18 | ||
\end{aligned}, | \end{aligned}, | ||
− | \end{cases}</math> from which <math>(a,b)=(3,3),(-3,-3),</math> or <math>N=3+3i,-3-3i.</math> | + | \end{cases}</math> from which <math>(a,b)=(3,3),(-3,-3),</math> or <math>N=3+3i,-3-3i.</math></li><p> |
− | <li><math>N^2=-18i</math | + | <li><math>N^2=-18i</math><p> |
We have | We have | ||
<cmath>\begin{align*} | <cmath>\begin{align*} | ||
Line 84: | Line 84: | ||
2ab &= -18 | 2ab &= -18 | ||
\end{aligned}, | \end{aligned}, | ||
− | \end{cases}</math> from which <math>(a,b)=(3,-3),(-3,3),</math> or <math>N=3-3i,-3+3i.</math> | + | \end{cases}</math> from which <math>(a,b)=(3,-3),(-3,3),</math> or <math>N=3-3i,-3+3i.</math></li><p> |
</ol> | </ol> | ||
By the <b>Factor Theorem</b>, we get | By the <b>Factor Theorem</b>, we get |
Revision as of 17:34, 2 July 2021
Contents
Problem
Compute
Solution 1 (Sophie Germain Identity)
The Sophie Germain Identity states that can be factored as
. Each of the terms is in the form of
. Using Sophie Germain, we get that
so the original expression becomes
![$\frac{[(10(10-6)+18)(10(10+6)+18)][(22(22-6)+18)(22(22+6)+18)]\cdots[(58(58-6)+18)(58(58+6)+18)]}{[(4(4-6)+18)(4(4+6)+18)][(16(16-6)+18)(16(16+6)+18)]\cdots[(52(52-6)+18)(52(52+6)+18)]}$](http://latex.artofproblemsolving.com/4/d/f/4df1c85169d79080a1e4ad7cd3cb96f9c11ec0c7.png)
![$= \frac{(10(4)+18)(10(16)+18)(22(16)+18)(22(28)+18)\cdots(58(52)+18)(58(64)+18)}{(4(-2)+18)(4(10)+18)(16(10)+18)(16(22)+18)\cdots(52(46)+18)(52(58)+18)}$](http://latex.artofproblemsolving.com/4/b/d/4bdb1dff907fef36c98d06fdb044a05f81004d30.png)
Almost all of the terms cancel out! We are left with .
Solution 2 (Completing the Square and Difference of Squares)
In both the numerator and the denominator, each factor is of the form for some positive integer
We factor by completing the square, then applying the difference of squares:
The original expression now becomes
~MRENTHUSIASM
Solution 3 (Complex Numbers)
In both the numerator and the denominator, each factor is of the form for some positive integer
We factor by solving the equation
or
Two solutions follow from here:
Solution 3.1 (Polar Form)
We rewrite to the polar form
where
is the magnitude of
such that
and
is the argument of
such that
By De Moivre's Theorem, we have from which
so
so
By the Factor Theorem, we get
We continue with the last paragraph of Solution 2 to get the answer
~MRENTHUSIASM
Solution 3.2 (Rectangular Form)
We rewrite to the rectangular form
for some real numbers
and
Note that so there are two cases:
We have
We need
from which
or
We have
We need
from which
or
By the Factor Theorem, we get
We continue with the last paragraph of Solution 2 to get the answer
~MRENTHUSIASM
Video Solution
https://youtu.be/ZWqHxc0i7ro?t=1023
~ pi_is_3.14
See also
1987 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.