Competitive Math Made Easy

Probability is a quantity that expresses the chance, or likelihood, of an event. It is most helpful to think of probability as a fraction. The literal definition of probability is the chance of occurrence of an event. For example, if a person is standing at the intersection of two roads which direct towards North, South, East, and West. Thus, he has a total of $4$ choices (four different directions) to proceed. Now, if he wished to go towards a particular direction, then the probability of completing his wish is $\frac{1}{4}$ since he can only choose one out of the four directions. Consider another example: A person has two different cars, a Toyota and Honda, which he uses randomly. It can then be said that the probability of using the Toyota is $\frac{1}{2}$ because out of his total of $2$ cars, he can randomly pick $1$ of them. Hence, from the above examples, we can conclude that the probability of an event occurring is $$ = \frac{\text{Number of desired or successful outcomes...

Let $A, B, C$ be three letters , then we can combine any two of them in the following ways: $$ AB, BC, AC$$ Similarly, if $A, B, C, D$ are four letters, then we can combine any two of them in the following manner: $$ AB, AC, AD, BC, BD, CD$$ Similarly, we can combine any $3$ of $A, B, C, D$ as : $$ ABC, ABD, ACD, BCD$$ We can generalize by saying that the number of all combinations of $n$ distinct things taken $r$ at a time $(r \le n)$ is ${n}\choose{r}$  $= \frac{n!}{(n-r)!r!}$ Note In combinations, the order of the letters (or things) is not considered. Here, AB and BA are the same, so they are only counted once, unlike permutations. The term "combination" is generally used for selection of things and "permutations" are used for rearrangements. Combinations with Restrictions Number of combinations of $n$ things taken $r$ at a time in which $x$ particular things always occur is $${n-x}\choose{r-x}$$ Number of combinations of $n$ things t...

In circular permutations, things are to be arranged in the form of a ring or a circle, e.g. arrangements of people around a circular table. In circular permutation there are no end points, i.e. there are no beginning or ending positions. So, the number of circular permutations of n objects are $$\frac{n!}{n}=(n-1)!$$ Thus in a circular permutation, one thing is kept fixed and the remaining  $(n-1)$ things are arranged in $(n-1)!$ ways. If the clockwise and counter clockwise orders are not distinguishable, then the number of ways = $\frac{1}{2}(n-1)!$ Let us look at a few examples.  Ex:   In how many ways can $6$ boys be seated at a circular table? Sol:   We keep one boy in a fixed position and to find the number of permutations to arrange the remaining 5 boys, we simply take $5!$, which gets us to our answer $\boxed {120} $.  Ex:  In how many ways can $6$ boys be arranged at a round table so that $2$ particular boys can be seat...

Each of the different arrangements that can be made out of a given number of things by taking some or all of them at a time is called a permutation. Thus the permutation of the three letters $a, b, c$ taken two at a time are: $$ab, ba, bc, cb, ac, ca$$ Therefore the number of permutations of three different things taken two a time is $_{3}P_{2}$ or $P(3, 2) = 6$. We can generalize this to say $_{n}P_{r} =$ Total number of permutations of $n$ distinct things taken $r$ at a time. $$_{n}P_{r} = \frac{n!}{(n-r)!}$$ Below are some extensions of the above concept. Permutations of $n$ different things taken all at a time $$ = _{n}P_{n} = n!$$ Ex . Find the number of ways the letters of the word RAINBOW can be rearranged. Sol . Every letter in the word RAINBOW is different, so to get the number of permutations, we would simply use $n!$. $$n = 7 \Rightarrow n! = \boxed{5040}$$ Permutations of $n$ different things taken $r$ at a time, when $k$ things never occurs $$ = _{n-k}P_{...

In this blog post, we will study the very  Fundamental Principles of Counting (i) Multiplication  If one operation can be performed in $m$ ways and corresponding to each way of performing the first operation, a second operation can be performed in $n$ ways then the two operations can be performed in $m \cdot n$ ways. In other words, if there are $m$ ways to do one thing and $n$ ways to do another, then there are $m \cdot n$ ways of doing both.  Here the different jobs/operations are mutually inclusive. It implies that all the jobs are being done in succession. In this case we use the ' and ' operator to account for all scenarios, and remember ' and ' refers to multiplication. Example :  A student has to select a letter from vowels and another letter from consonants, then in how many ways can he make this selection? Solution : Out of $5$ vowels he can select one vowel in $5$ ways and out of $21$ consonants he can select one consonant i...

A Cevian is a line segment which joins a vertex of a triangle with a point on the opposite side (or its extension). There are 3 special types of cevians and their properties that we will be looking at today. 1. Median  A line joining the midpoint of a side of a triangle to the opposite vertex is called a median. Properties A median divides a triangle into two parts of equal area. $[ABD] = [ACD]$. The point where the three medians of a triangle meet is called the centroid of the triangle. Point $G$ is the centroid of $\triangle ABC$ The centroid of a triangle divides each median in the ratio $2:1$. $\frac{AG}{GD} = \frac{BG}{GE} = \frac{CG}{GF} = 2$ The three medians divide the triangle into $6$ triangles with equal area. $[AFG] = [BFG] = [BDG] = [CDG] = [CEG] = [AEG]$ An Important Result  $2(AD)^2 + 2 \cdot \big( \frac{BC}{2} \big) = (AB)^2 + (AC)^2$ 2. Angle Bisector An angle bisector of a triangle is a segment or ray that bisects an a...

We have looked at some basics of coordinate geometry. Today, let's look at some intermediate applications of this topic. 1. Shoelace Theorem The Shoelace Theorem is an algorithm used to find the area of a polygon in the coordinate plane when the coordinates are known. If the coordinates are $(x_1, y_1), (x_2, y_2) ... (x_n, y_n), $ $\text{Area}=\frac{1}{2}\Big[(x_1y_2 - x_2y_1)+(x_2y_3 - x_3y_2) + ... + (x_ny_1 - x_1y_n)\Big].$ 2. Centroid The centroid is the point where the three medians intersect. It is also sometimes called the center of gravity for the triangle. Note: A median of a triangle is the line segment joining the vertex to the midpoint of the opposite side. If $(x_1,y_1), (x_2, y_2),$ and $(x_3,y_3)$ are the vertices of a triangle, then the coordinates of its centroid are $$\Bigg(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3} \Bigg).$$ 3. Incenter The incenter is the point where the three angle bisectors intersect. Note: An angle bisector of a triangle...

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 ...

Units digit of a number is the digit in the one's place of the number. For example, the units digit of $243$ is $3$. In competition math, you might come across occasional problems that ask you to find the units digit of an expression. These problems are more commonly found on Mathcounts Countdown Round. 1. The units digit of any number expressed as a power of $2$ repeats in the cycle $2, 4, 8, 6.$        $2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 16, 2^5 = 32, ....$ etc. 2. The units digit of any number expressed as a power of $3$ repeats in the cycle $3, 9, 7, 1.$        $3^1 = 3, 3^2 = 9, 4^3 = 27, 3^4 = 81, 3^5 = 243, ....$ etc. 3. The units digit of any number expressed as a power of $4$ is $4$ if the power is odd and $6$ if the power is even.        $4^1 = 4, 4^2 = 16, 4^3=64, 4^4=256, ....$ etc. 4. The units digit of any number expressed as a power of $5$ is always $5$.        $5^1...

Key Facts 1. Numbers of the form $\frac{p}{q}, q \neq 0$, where $p$ and $q$ are integers and those that can be expressed in the form of terminating or repeating decimals are called rational numbers .             Ex. $\frac{7}{32} = 0.21875, \frac{8}{15} = 0.5\bar{3}$ are rational numbers. 2. Properties of operations of rational numbers     For any rational numbers $a, b, c,$     (i) Rational numbers are closed under addition, multiplication, and subtraction.          $i.e., (a+b), (a-b),$ and $(a \cdot b)$ are all rational     (ii) Rational numbers follow the commutative law  of addition and multiplication.          $i.e., a + b = b + a$ and $a \cdot b = b \cdot a.$     (iii) Rational numbers follow the associative law of addition and multiplication,          $i.e., (a + b) + c = a + (b + c)$ and $(a \cdot b) \cdot c = ...

Coordinate Geometry is the study of geometry using algebraic tools like equations, relations, and operations. Before we start our deep dive, let's go over some basics of coordinate geometry. A coordinate grid consists of 2 perpendicular lines, which are called axes , and they are labeled like a number line would be. The horizontal axis is known as the x-axis , while the vertical axis is called the y-axis . The point of intersection of the two axes is known as the origin . The coordinate of the origin will always be $(0, 0)$. The numbers along the axes are used to locate points in the plane. The point at which a graph intersects the x-axis is known as the x - intercept  and the point of intersection with the y-axis is known as the y - intercept . The coordinates  also known as an Ordered Pair, is a set of values that expresses the distance a point lies from the origin. It is written in the format $(x, y)$ where $x$ is the value on the x-axis and the $y$ is the value on the ...

Now that you know the fundamentals of circles, let's take a look at some important circle theorems.  Inscribed angle is half the measure of the intercepted arc. An inscribed angle is the angle formed in the interior of a circle when two secant lines intersect on the circle.  The above theorem leads to an important result that comes handy in solving many geometry problems. The angle at the circumference subtended by a diameter is a right angle.  Simply put, the angle in a semicircle is a right angle.    All inscribed angles that intercept the same arc have the same measure.  The angle at the center is twice the angle at the circumference subtended by the same arc. The measure of the angle formed by the intersection of two chords is equal to the average of it's two corresponding arcs. Simply put, m$\angle \alpha = \frac{\overset{\LARGE{\frown}}{NP} + \overset{\LARGE{\frown}}{QO}}{2}$ Similar to the ...

The past 2 weeks that I have been away, were quite the adventure in the Grand Teton and the Yellowstone National Parks. As a family we have always loved spending time outdoors and this was the perfect trip right before the onset of my sophomore year. Getting back to life always seems so hard after a trip like that. Before I digress too much, lets get back to work. Today I would like to review the fundamentals of Circles. Let's first understand the different components of a Circle. Here are the key terms you should know. Radius - Line Segment from the center of the Circle to a point on the circumference. Diameter - Line Segment from one point on the circumference to another point on the circumference that passes through the center of the Circle. Needless to say, the length of the diameter is twice that of the radius. Area - The set of all points contained inside the circumference. Can be found using the formula $ \pi r^2$ where $r$ is the radius. ...

Let's take a look at right triangles and some of their special properties today. As a quick refresher, a right triangle is a triangle with a right angle. We can call the side opposite the right angle as the  hypotenuse , and the two sides adjacent to the $90^{\circ}$ angle as the  legs . In the image to the right, $AC$ is the hypotenuse, and the two legs are $AB$ and $BC$. Pythagorean Theorem One of the most famous and useful theorems in geometry is the Pythagorean Theorem. The theorem states that in right triangle $\triangle ABC$ with hypotenuse $c$ and legs $a$ and $b$, $$a^2+b^2=c^2$$ Lets apply this in a problem to see how it works.  Problem 1 : Let right triangle $\triangle ABC$ with right angle at $B$ have hypotenuse of $r+1$, and legs of length $7$ and $r$. Find $r$.  Solution:  We know that the sum of the squares of the legs is equal to the square of the hypotenuse. In other words, we have $$7^2 + r^2 = (r+1)^2.$$ Simplifying this a...