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 "University of South Carolina High School Math Contest/1993 Exam/Problem 4"

m
(→‎Solution)
 
(2 intermediate revisions by 2 users not shown)
Line 11: Line 11:
 
* [[University of South Carolina High School Math Contest/1993 Exam/Problem 3|Previous Problem]]
 
* [[University of South Carolina High School Math Contest/1993 Exam/Problem 3|Previous Problem]]
 
* [[University of South Carolina High School Math Contest/1993 Exam/Problem 5|Next Problem]]
 
* [[University of South Carolina High School Math Contest/1993 Exam/Problem 5|Next Problem]]
−
* [[University of South Carolina High School Math Contest/1993 Exam|Back to Exam]]
+
* [[University of South Carolina High School Math Contest/1993 Exam/Problems|Back to Exam]]
  
−
[[Category:Intermediate Complex Numbers Problems]]
 
 
[[Category:Intermediate Algebra Problems]]
 
[[Category:Intermediate Algebra Problems]]
−
[[Category:Intermediate Combinatorics Problems]]
 

Latest revision as of 13:36, 24 November 2016

Problem

If $(1 + i)^{100}$ is expanded and written in the form $a + bi$ where $a$ and $b$ are real numbers, then $a =$

$\mathrm{(A) \ } -2^{50} \qquad \mathrm{(B) \ } 20^{50} - \frac{100!}{50!50!} \qquad \mathrm{(C) \ } \frac{100!}{(25!)^2 50!} \qquad \mathrm{(D) \ } 100! \left(-\frac 1{50!50!} + \frac 1{25!75!}\right) \qquad \mathrm{(E) \ } 0$

Solution

Notice that $(1+i)^{2}=2i$. We then have $(2i)^{50}=-2^{50}$.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=University_of_South_Carolina_High_School_Math_Contest/1993_Exam/Problem_4&oldid=81573"