Table of Content

Abstract:

The central issue we are investigating is based on a problem from The Hong Kong (China) Mathematical Olympiad. It is basically about whether a cosine ratio is expressible as a sum of rational numbers to powers of reciprocals of primes. In our project, we give the generalization of this problem by using some tricks in elementary Number Theory and Galois Theory.

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In this project, we shall introduce a new quantity associated with any given shape on the plane: “optimal curve”, which is defined as the shortest curve such that its convex hull fully covers a given shape S. Here curve can involve straight lines or union of straight lines. [See reviewer’s comment (2)] We shall investigate on some properties of this kind of curve and also prove a theorem that among shapes with a given fixed length of perimeter, the circle has the maximal optimal curve. [See reviewer’s comment (3)] Moreover, we will introduce an algorithm to find the shortest curve with convex hull equals a given shape in polynomial time.

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In this paper, we discuss the generalization of the orchard visibility problem – from that of grid shapes to that of the shapes of the trees. We will even take a look at the problem of the visibility problem on a spherical surface and 3-D space.

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Diseases are devastating. The SARS in 2003 and the swine in- fluenza in 2009 sparked myriad of questions in our minds. Our major concern is the spread of germs. Throughout the entire project, we investigate diseaserelated issues and try to study the impacts of a disease by mathematical modeling.

We first start with the simplest model followed by more complicated ones. We focus on different factors that affect the spread of diseases. Diagrams are included in each chapter to see how the values of different groups vary. Then we come up with possible ways to prevent epidemics. Altering the models by adding more conditions, we find one that fits the real life situation – the SEIRS model. The situation in Hong Kong (Swine Influenza from April 2009 to April 2010 in Hong Kong) is simulated by putting the data into the model and our goal is fulfilled.

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In this project we planned to study the division of a circle with the shortest curve. In a party, we often divide a circular cake into equal and unequal parts. Suppose that bacteria grow on the exposed surface area of a cake. In order to keep the cake hygienic, we should divide the cake with the shortest cut. We investigated this problem by using a simple mathematical model: dividing a circle into equal or unequal areas with the shortest curve.

The first possible solution was the radius method. It meant that we used radii to divide a circle into parts. But, were there any ways to divide a circle with a curve shorter than that of the radius method?

The results included:

1. Radius method is the solution of the problem for $n$ = 2, 3 and equal division.
2. Radius method is not a solution of the problem for $n$ = 4 and equal division.
3. Orthogonal circular arc is the solution of the problem for $n$ = 2 and unequal division.
4. We found a necessary condition of the problem for $n$ = 3 and unequal division by a “Y-shaped” curve.

[See reviewer’s comment (2) and (3)]

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In this report, we want to know whether there is a magic square whose entries are distinct perfect squares.

Firstly, we analyze the basic properties of a magic square and find that the magic sum of a magic square is equal to 3 times of the central entry and the 9 entries of a magic square contain 8 arithmetic progressions.

Secondly, we focus on our main target, magic square of squares. Investigating the properties of the prime factors of those 9 entries, we find that if the greatest common divisor of all entries is equal to 1, the prime factors of central entry are of the form \(p \equiv 1\) (mod 4), the central entry must not be a square of a prime number and the common prime factors of any two adjacent entries (if exist) are not of the form \(p \equiv 3\) (mod 4).

Thirdly, we find that this problem is equivalent to a system of Diophantine equations with ten variables. We provide a construction method of the solution to these partial equations:

$$a^2 + b^2 = c^2 + d^2 = e^2 + f^2 = g^2 + h^2 = 2M^2$$ ,

where these nine perfect squares are distinct.

Finally, based on the theorems obtained, we find that given a positive integer \(N\), there exists a positive integer \(M\) such that it has \(N\) essentially different representations of a sum of two perfect squares.

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In a given acute triangle, the inscribed triangle with minimum perimeter is the orthic triangle. This problem was proposed and solved using calculus by Fagnano in 1775.

Now we wonder, will the result remain unchange when the problem is discussed on a sphere? [See reviewer’s comment (2)] In this paper, we will first try to find the answer of the “spherical Fagnano’s problem”. Based on our results in spherical triangle cases, we will go further to generalize the problem to spherical quadrilateral and n-sided spherical polygon in spherical geometry.

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The survey [1] conducted by W. Morris and V. Soltan mentioned that in 1935 Erdős-Szekeres proved that for any integer \(n \geq 3\), there exists a smallest positive integer \(g(n)\) points in general position in the plane containing n points that are the vertices of a convex n-gon. [See reviewer’s comment (3)] They also conjectured that \(g(n) = 2n−2 + 1\) for any integer \(n \geq 3\). The conjecture is far from being solved for decades though many mathematicians had tried their very best on it. This paper is to investigate the Erdős-Szekeres conjecture by studying the greatest positive integer \(f(n)\) points in general position in the plane which contains no convex n-gons. We successfully prove the cases when \(n = 4\), 5 i.e. \(f(4) = 4\) and \(f(5) = 8\). For \(n = 6\), we arrive at the conclusion that \(f(6) \geq 16\) by creating an example of 16 points containing no convex hexagons. Moreover, we excitedly find an elegant proof for this example that one more point added to it will certainly give birth to a convex hexagon.

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