Application of Newton-Raphson: OMO Spring 2018
This year’s OMO included this problem: 13. Find the smallest positive integer for which the polynomial has a real root greater than 1.999. Solution. Equivalently, the polynomial is if is […]
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Sturm’s Theorem
Because it is fairly complicated, Sturm’s Theorem is best taught with an example. In the last post, we were investigating the roots of the function via Newton’s method, and we […]
Newton’s method
Say you’re in a competition and you need to calculate the square root of two to several decimal places. Or perhaps you need to approximate the roots to an unwieldy […]
Roots of Unity Filters
Problem 1: Find the sum of the coefficients of the even-power terms of the polynomial given that and To solve this problem, let’s write down a simple example: Now lets plug […]
Projective Geometry Practice Problem
Here is a nice problem from UGA 2004 Written: What is the largest area of an ellipse that can be inscribed in a triangle with sides 3, 4, and 5?
USAMTS Has Started
USAMTS started today. You can look at past problems on their website to see if you want to compete. If you do, you need to register on the site. Also, […]
Don’t Overcount Solution
There are 30 liberals and 30 conservatives. How many 8-person committees can be formed if at least one of them must be a liberal? If we choose a liberal and […]
Don’t Overcount
Sometimes, you may make a combinatorial argument which seems reasonable, but you find that your answer is larger than the correct answer. Here is a problem which may trip you […]
Combinatorics Solutions
Here are the solutions for the Combinatorics Problems handout. Make sure to read all the solutions, as many of the techniques discussed frequently show up in competition. Combinatorics Solutions
Combinatorics Problems
Click on the title of the post then on the following link to download the combinatorics problems given out on our second meeting. Combinatorics Problems