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Equilateral and Equiangular Polygons

A polygon is a 2 dimensional geometric figure bound with straight sides. A polygon is called Equilateral if all of its sides are congruent. Common examples of equilateral polygons are a rhombus and regular polygons such as equilateral triangles and squares.  Now, a polygon is equiangular if all of its internal angles are congruent.  Some important facts to consider The only equiangular triangle is the equilateral triangle If P is an equilateral polygon that has more than three sides, it does not have to be equiangular. A rhombus with no right angle is an example of an equilateral but non-equiangular polygon.  Rectangles, including squares, are the only equiangular quadrilaterals Equiangular polygon theorem. Each angle of an equiangular n-gon is  $$\Bigg(\frac{n-2}{n}\Bigg)180^{\circ} = 180^{\circ} -   \frac{360^{\circ}}{n} $$ Viviani's theorem   Vincenzo Viviani (1622 – 1703) was a famous Italian mathematician. With his exceptional intelligence in math...
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Polynomials: Key Vocabulary

Let us look at some key terms that pertain to Polynomials before we deep dive into the study of Polynomials.  Constant: A number having a fixed numerical value Example: $3, \cfrac{4}{5}, 4.2, 6.\overline{3}$ Variable:  A number which can take various numerical values Example:   $ x,  y,  z$ Algebraic Expression: A combination of constants and variables connected by arithmetic operators Example: $2x^2 + 7, 5x^3 + 4xy + 2xy^2 + 7,$ etc   Terms: Several parts of an algebraic expression separated  by $+$ or $-$ signs are called the terms of the expression.  Example: In the expression $9x + 7y + 5$, $9x$, $7y$, and $5$ are terms.   Coefficient of a Term:  In the term $8x^2$, $8$ is the numerical coefficient of $x^2$ and $x^2$ is said to be the literal coefficient of 8.  Like Terms: Terms having the same literal coefficients are called Like Terms.  Example: $8xy$, $9xy$, and $10xy$ are Like Terms ...
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Problems - 1

Let's look at some interesting applications of the basic arithmetic formulas in the problems below. Problem 1 - Evaluate $123456789 \cdot 999999999$.         Solution - $123456789 \cdot 999999999 = 123456789 \cdot (1000000000-1)$                         $=123456789000000000-123456789=\boxed{123456788876543211}$ Problem 2 - Find the value of $\frac{13579}{(-13579)^2+(-13578)(13580)}$.         Solution - Using $(a-b)(a+b) = a^2 - b^2$, we have $$\frac{13579}{(-13579)^2+(-13578)(13580)} = \frac{13579}{(13579)^2-(13579^2-1)} = \boxed{13579}$$ Problem 3 - Evaluate $\frac{83^3+17^3}{83 \cdot 66+17^2}$.         Solution - By the use of the formula $a^3+b^3=(a+b)(a^2-ab+b^2)$ $$\frac{83^3+17^3}{83 \cdot 66+17^2} = \frac{(83+17)(83^2-83 \cdot 17 + 17^2)}{83 \cdot 66 + 17^2} = \frac{100 \cdot (83 \cdot 66 + 17^2)}{83 \cdot 66 + 17^2} = \boxed{100}$$ Problem 4 ...
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Quadratic Equations

An equation, in which the highest power of the variable is $2$ is called a quadratic equation. The standard form of a quadratic equation is $ax^2+bx+c=0$, where $a, b,$ and $c$ are constants and $a \neq -$ Solution of Quadratic Equations: There are two main methods to find the solutions, or roots, of a quadratic equation. Factorization $\rightarrow$ Let $ax^2+bx+c = a(x - \alpha)(x - \beta) = 0$. Here, the roots of the equation are $x = \alpha$ and $x = \beta$. Hencec, factorizing the equation and equating each factor to zero is one method to find the roots. The second method is known as the Quadratic Equation, which is:  $$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$$ where the equation of the quadratic is $ax^2+bx+c = 0$.  Nature of the Roots:  In a quadratic equation in standard form, the term $b^2-4ac$ is known as the discriminant of the equation, which plays an important role in finding the nature of the roots. It is often denoted by $D$. If $a, b, c \in \ma...
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Equilateral and Equiangular Polygons

A polygon is a 2 dimensional geometric figure bound with straight sides. A polygon is called Equilateral if all of its sides are congruent. Common examples of equilateral polygons are a rhombus and regular polygons such as equilateral triangles and squares.  Now, a polygon is equiangular if all of its internal angles are congruent.  Some important facts to consider The only equiangular triangle is the equilateral triangle If P is an equilateral polygon that has more than three sides, it does not have to be equiangular. A rhombus with no right angle is an example of an equilateral but non-equiangular polygon.  Rectangles, including squares, are the only equiangular quadrilaterals Equiangular polygon theorem. Each angle of an equiangular n-gon is  $$\Bigg(\frac{n-2}{n}\Bigg)180^{\circ} = 180^{\circ} -   \frac{360^{\circ}}{n} $$ Viviani's theorem   Vincenzo Viviani (1622 – 1703) was a famous Italian mathematician. With his exceptional intelligence in math...
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Incenter/Excenter Lemma

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Given $\triangle ABC$ with incenter $I$, extend $AI$ to meet the circumcircle of $\triangle ABC$ at $L$. Then, reflect $I$ over $L$, and name this point $I_a$. Then:  $(i)$ $BICI_a$ is a cyclic quadrilateral with point $L$ as the center of the circumscribed circle.  $(ii)$ $BI_a$ and $CI_a$ bisect the exterior angles of $\angle B$ and $\angle C$,  respectively.  Proof $(i)$ Notice that the the question is equivalent to proving the distance from $L$ to points $B, I, C, I_a$ are equal. In other words, we need to show that  $$LB = LI = LC = LI_a$$ Firstly, it is obvious that $LI = LI_a$ because by definition, $I_a$ is the reflection of $I$ over $L$, and reflections preserve the length of the segment.  Now, we are left with proving that $LB=LI$, because similar calculations will provide $LI=LC$.  We see that most of our given information involves angles, so we focus on proving $\angle LBI = \angle LIB$. Firstly,  $$\angle LBC = \angl...
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Sequences and Series

 Let's take a look at some interesting sequences and series problems. 1. (USAMTS) Evaluate the value of  $$S = \sqrt{1+\cfrac{1}{1^2}+\cfrac{1}{2^2}} + \sqrt{1+\cfrac{1}{2^2}+\cfrac{1}{3^2}} + ... + \sqrt{1+\cfrac{1}{1999^2}+\cfrac{1}{2000^2}}$$ Solution Notice that $$1+\cfrac{1}{n^2}+\cfrac{1}{(n+1)^2} = \cfrac{n^4+2n^3+3n^2+2n+1}{n^2(n+1)^2} = \cfrac{n^2+n+1)^2}{n^2(n+1)^2} = \Bigg(1+\cfrac{1}{n(n+1)} \Bigg)^2$$ Thus, $$S= \Bigg(1+\cfrac{1}{1 \cdot 2} \Bigg) + \Bigg(1+\cfrac{1}{2 \cdot 3} \Bigg) + \Bigg(1+\cfrac{1}{3 \cdot 4} \Bigg) + ... + \Bigg(1+\cfrac{1}{1999 \cdot 2000} \Bigg)$$ $$= 1999 + \Bigg(1 - \cfrac{1}{2} \Bigg) + \Bigg(\cfrac{1}{2} - \cfrac{1}{3} \Bigg) + ... + \Bigg(\cfrac{1}{1999} - \cfrac{1}{2000} \Bigg) = 1999 + 1 - \cfrac{1}{2000} = \boxed{\cfrac{3,999,999}{2000}}$$ 2. (HMMT) Find the value of  $$S = \cfrac{1}{3^2+1} + \cfrac{1}{4^2+2} + \cfrac{1}{5^2+3} + ...$$ Solution For this problem, we can use partial fraction decomposition and try to find an alt...
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