Posts

Showing posts with the label Work

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...
Post a Comment
Read more

Time and Work Problems

Key Facts If a person can finish a job in n days, then the work done by the person in 1 day is $\frac{1}{n}$th of the total job If a person completes $\frac{1}{n}$th of the total job in 1 day, then the time taken by the person to finish the complete job is n days.  Another version of work is pool/tank problems where there is an inlet of water and an outlet as well.   If an inlet fills a tank in n hours, then it fills $\frac{1}{n}$th part of the tank in 1 hour. i.e. work done by it in 1 hour is $\frac{1}{n}$ If an outlet empties a full tank in m hours, then it will empty $\frac{1}{m}$th part of the tank in 1 hour. i.e. work done by it is $-\frac{1}{m}$ The concept is not hard to understand, however the application can be extremely tricky. So it would only help to solve as many problems as you can on this subject. Let's look at some examples. Ex 1. A copy machine can copy a paper in $36$ minutes. If a second copy machine were to be used at the same ...
3 comments
Read more

Distance and Work Problems: Basics

Distance and Work problems are another type of problems that you can almost be sure will appear on all math competitions at every level. It is imperative for young olympians to understand them well and get comfortable as early as you can. This post will touch upon the very basics of Speed/Distance/Time and Work problems. There will be future posts where I will discuss different variations and complexity levels within this category. Distance Problems The key in solving Speed/Distance/Time problems is to understand the relationship between each of these concepts. The basic formula to understand here is Distance = Rate $\cdot$ Time $D = R \cdot T$ It is highly recommended to make use of various tools like diagrams and charts to organize the given information leading up to the solution. The formula can be applied to different flavors of such problems. For example, if you know the time and rate a person is traveling on a bus, you can quickly calculate how far he traveled. And if...
Post a Comment
Read more

Popular posts from this blog

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...
Read more

Incenter/Excenter Lemma

Image
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...
Read more

Common Factorizations

Factorization is the process of writing a number or expression as a product of several factors, usually smaller or simpler. For example $8$ can be factorized as $2 \cdot 4$, and $(x-3)(x+3)$ is the factorization of the polynomial $x^2-9$. Here are some common factorizations worth memorizing: $$(a-b)(a+b)=a^2-b^2$$ $$(a \pm b)^2 = a^2 \pm 2ab + b^2$$ $$(a \pm b)(a^2 \mp ab + b^2) = a^3 \pm b^3$$ $\text{Proof }$ $$(a+b)(a^2 - ab + b^2) = a^3 - a^2b + ab^2 + a^2b - ab^2 +b^3 = a^3 + b^3$$ $$\text{Use ($-b$) to replace $b$ in the above, we obtain,}$$ $$(a-b)(a^2 + ab + b^2) = a^3 - b^3$$ $$(a \pm b)^3 = a^3 \pm 3a^2b + 3ab^2 \pm b^3$$ $\text{Proof }$ $$(a+b)^3 = (a + b) \cdot (a + b)^2 = (a+b)(a^2+2ab+b^2)$$ $$(a+b)^3 = a^3 + 2a^2b+ab^2+a^2b+2ab^2+b^3$$ $$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$$ $$\text{Use ($-b$) to replace $b$ in the above, we obtain,}$$ $$(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$$ $$(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca$$ $\te...
Read more