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Showing posts from June, 2019

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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Geometric Sequences

Let's take a look at geometric sequences today. In a way, they are similar to Arithmetic Sequences , but just think common ratio instead of common difference. Let's begin by taking a look at the geometric sequence $2, 6, 18, 54...$. The initial term , or the first term is $2$. The common ratio  is defined as the number you multiply to each term to get the next one. In this case, the common ratio would be $3$, because $2 \cdot 3 = 6, 6 \cdot 3 = 18, 18 \cdot 3 = 54$, and so on. Common problems include finding the $n$th term, or finding the sum of such sequences. To find the $n$th term of a geometric sequence, we use the formula: $$a_n = ar^{n-1}$$ Where $a_n =$ the $n$th term in the sequence $a =$ the initial term $r =$ the common ratio Your Turn! $1.$  Find the $7$th term in the sequence $5, 10, 20, 40...$ (Scroll down to the bottom of the page for the answer) Next, let's take a look at how to find the sum of such a sequence. Let's find the sum of the seque...
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Age Problems

Age problems is another commonly tested concept on competitive exams. Solving many practice problems is the key to mastering this topic. Interpreting the language of what is given and what is asked of is the biggest challenge you would need to overcome. Lets look at some commonly used phrases throughout these problems.  If the present age is $a$, then $n$ times the present age = $an$. If the present age is $a$, then $n$ years later, age = $a+n$. If the present age is $a$, then age $n$ years ago = $a-n$. The next step would be to organize the given information. What usually works for me is putting a chart together and to keep plugging pieces of data that are given. Let's walk through a few examples to see how this works.  Example 1:  Tim's age is three times his son, Alex's age. After $10$ years, he would only be twice Alex's age. What is Alex's present age?  Solution: Current Age Age $10$ years later Tim $3x$ $3x+10, 2(x+10)$ Alex $x$ $x+10$ ...
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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...
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Prime Factorization

What is the Prime Factorization of a Number? The prime factorization of a number is the unique set of prime numbers that, when multiplied, make up the number. For example, the prime factorization of the number $20 = 2^2 \cdot 5$, because both $2$ and $5$ are prime numbers.  Problems that ask you to find the number of factors a given number has are quite common. To solve such problems, first find the prime factorization of your number. Let's take $20$ as an example. As we saw above, $20 = 2^2 \cdot 5$. To find the number of factors that $20$ has, we will add $1$ to each exponent, and then multiply our results. So in this case, we would multiply $(2+1)(1+1) = 3 \cdot 2 = 6$. (Note: the $5$ has an exponent of $1$).  In general, to find the number of factors of a number, add one to each exponent and calculate the product. Less common, but still useful is the sum of the factors of a number. We start the same way as before and find the prime factorization of the number. ...
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Divisibility Rules

Today we will look at how to quickly tell if a number is divisible by $2$ - A number is divisible by $2$ if it's last digit is even $(0, 2, 4, 6,$ or $8)$. $3$ - A number is divisible by $3$ if the sum of it's digits is divisible by $3$. $4$ - A number is divisible by $4$ if it's last $2$ digits are divisible by $4$. $5$ - A number is divisible by $5$ if it's last digit is either a $0$ or $5$. $6$ - A number is divisible by $6$ if it is divisible by both $2$ and $3$. $7$ - A number is divisible by $7$ if you get a multiple of $7$ when you subtract twice the last digit of the number from the remaining number. (e.g. $665$ is divisible by $7$ because $66 - 2 \cdot 5 = 56$, which is a multiple of $7$.) $8$ -  A number is divisible by $8$ if the last $3$ digits are divisible by $8$. $9$ - A number is divisible by $9$ if the sum of it's digits adds up to a multiple of $9$. $10$ - A number is divisible by $10$ if the last digit of the number is $0$. $11$ - A nu...
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Arithmetic Sequences

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Let's take a look at Arithmetic Sequences today! What is an Arithmetic Sequence? A set of terms where every sequential term is created by adding or subtracting a common difference from the previous term. For example, let's look at the sequence $2, 5, 8, 11, ...$ The initial term , or the first term of the sequence here is $2$. The common difference is the number that is added to each term to get to the next term. So in this case, we add $3$ to get from $2 \to 5, 5 \to 8, 8 \to 11$ and so on. To find the fifth term, I could keep adding $3$ to the first term $4$ times. Sounds easy, right! Now try finding the $25$th term. You would be here all day long trying to add $3$, $24$ times. Or, you can think smart and add $(3 \cdot 24)$ to the first term, $2$ which brings us to our first formula. $$a_n = a_1 + (n-1)d$$ Where $a_n =$ the $n$th term in the sequence $a_1 =$ the first term in the sequence $d =$ the common difference So, going back to our example, we know that we need...
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Basics to get you started

Lets start with some basics that are foundational to some early math competitions.  Know your Number Line well.             Natural (Counting) Numbers:                   1, 2, 3, 4, ...             Whole Numbers (All Natural Numbers and Zero):                   0, 1, 2, 3, 4, ...             Opposites are the same distance from 0, but on opposite sides of zero             Integers (Whole Numbers and their opposites):                   ..., -3, -2, -1, 0, 1, 2, 3, ...             Negative Integers: Integers that are less than 0.             Positive Integers: Integers that are greater than 0. Consecutive Numbers: A sequence of numbers that differ onl...
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Why this Blog?

About Me I am no one special, just a 14 year old who just does what most kids my age do. I play a lot and goof a little. I love to hike, ski, and play a ton of soccer. I am one of the lucky few to get several vacations every year and memories of each of them have a very special place in my heart. I know Switzerland not from the Titlis mountain or the Jungfrau region, but from the miles and miles of hiking through the rolling hills, lush greenness, and the magnificence of the Matterhorn. Norway blew my mind away when I experienced the 24 hour sun and the vastness of the Norwegian fjords. The Jurassic coast and the Roman baths in England replaced the London Eye for our to do list! I am thankful to my parents who have taught me to love to travel and hopefully, I can bring that forward. Somewhere along my late elementary years, I was exposed to competition math. I won't say I am a born genius, however, I did enjoy pursuing some of the challenges. Its been over 5 years of thes...
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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...
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Irrational Numbers

A number which cannot be written in the form $ \cfrac {p}{q}$  where $p$ and $q$ are integers and $ q $  $ \neq 0$. Example: $ \sqrt {2}$, $ \sqrt {3}$, $ \sqrt {6}$, $ \sqrt {7}$, $ \sqrt {8}$, $ \sqrt {10}$ Theorem: If $p$ divides $x^3$, then $p$ divides $x$, where $x$ is a positive integer and $p$ is a prime number.  Proof :  Let $x = p_{1} p_{2}...p_{n}$ where $p_{1}, p_{2}, p_{3}.....p_{n}$  are primes, not necessarily distinct.  $ \rightarrow x^3 = p_{1}^3 p_{2}^3.....p_{n}^3$ Given that $p$ divides $x^3$ By fundamental theorem, $p$ is one of the primes of $x^3$.  By the uniqueness of fundamental theorem, the distinct primes of $x^3$ are same as the distinct primes of $x$.  $ \rightarrow p$ divides $x$ Similarly if $p$ divides $x^2$, then $p$ divides $x$, where $p$ is a prime number and $x$ is a positive integer. Example: Prove that  $ \sqrt {2}$ is irrational.  Solution : Let us assume that  $ \sqrt ...
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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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