High School Math Contest University of South Carolina February 5, 2011 Solutions
... 29. (b) All together there are 27 ways to color the vertices garnet or black – we shall refer to them as “patterns”. Let us count how many patterns correspond to the same coloring. Assume first that the coloring is non-trivial (not all vertices black or all garnet). In this case, a rotation of a pat ...
... 29. (b) All together there are 27 ways to color the vertices garnet or black – we shall refer to them as “patterns”. Let us count how many patterns correspond to the same coloring. Assume first that the coloring is non-trivial (not all vertices black or all garnet). In this case, a rotation of a pat ...
Full text
... To complete the proof, we will evaluate (8) based on whether n = 0, 1, 2, or 3 (mod 4 ) . Odd Case: If n is odd, then [(n + 2)/2] = [(n + l)/2], so m = p and, applying (10) to (8), we have a sum involving every fourth Fibonacci number. ...
... To complete the proof, we will evaluate (8) based on whether n = 0, 1, 2, or 3 (mod 4 ) . Odd Case: If n is odd, then [(n + 2)/2] = [(n + l)/2], so m = p and, applying (10) to (8), we have a sum involving every fourth Fibonacci number. ...
H6
... 1. There is a rather surprising formula which counts the number of ways that a positive integer n can be written as a sum of two squares: ...
... 1. There is a rather surprising formula which counts the number of ways that a positive integer n can be written as a sum of two squares: ...
Mathematics of radio engineering
The mathematics of radio engineering is the mathematical description by complex analysis of the electromagnetic theory applied to radio. Waves have been studied since ancient times and many different techniques have developed of which the most useful idea is the superposition principle which apply to radio waves. The Huygen's principle, which says that each wavefront creates an infinite number of new wavefronts that can be added, is the base for this analysis.