1: Algebraic Manipulations
BFL Competition Math 2022
Class Introduction
This class covers AMC 10 level content, with a focus on developing thinking skills and problem solving.
Weekly lectures supplemented by problem sets
Happy to answer questions at any time! Feel free to say something or message me on Zoom, or email dengr557@gmail.com with any questions.
Warmup: Recap Of Some Tactics
You may recall common methods to solve linear equations like substitution and elimination, and methods like factoring to solve quadratic equations.
Today we’ll take these ideas a step further and see the big picture with manipulating equations and expressions.
Substitutions
Problem (classic): What is the value of ?
Substitutions
Problem (classic): What is the value of ?
Notice that this expression repeats itself!
So if we set it equal to x, then we have
Rearranging, this is equivalent to x2 + x - 1 = 0, which we can solve to obtain that x = . Clearly, since everything here is positive, we ignore the negative solution as extraneous so the answer is
This idea will come in handy later!
Combining Equations
(AoPS) Solve the system a(a+b) = 108, b(a+b) = 297/4.
If we add the two equations together, we get (a+b)2 = 729/4, so a+b = ±27/4.
Substituting this into the original equations, we get that either:
a(27/2) = 108, so a = 8, b = 11/2, or
a(-27/2) = 108, so a = -8, b = -11/2.
Symmetric Systems
Problem (classic): If a+b = 17, b+c = 21, and a+c = 20, then what are the values of a, b, and c?
The key here is that the three equations are of a similar form - there are three variables and each equation involves two being added together.
We can refer to this idea as symmetry.
One thing we can do here is combine all the equations to get a fourth equation.
Symmetric Systems
Problem (classic): If a+b = 17, b+c = 21, and a+c = 20, then what are the values of a, b, and c?
Combining, we get that 2a + 2b + 2c = 58, so a+b+c = 29.
Now, we can just do elimination with our original equations to get a, b, and c!
Answer: a=8, b=9, c=12
Symmetry Continued
(classic) If , then what is
While it’s possible to solve for x, we’ll only get a value that’s not fun to work with, so it’s better to do some manipulation.
Symmetry Continued
(classic) If , then what is
We know that does not equal , but it helps us get a bit closer so we’ll give it a try.
Cubing the equation, we obtain
Notice that we can get rid of the part in the middle by substituting in
Thus we get , so
Problem (2021 Spring AMC 10B)
Let’s look at this idea in action.
The real number x satisfies the equation . What is the value of x11 - 7x7 + x3?
Hint 1: Wouldn’t it be nice to get this to look like xn + 1/xn?
Hint 2: Try factoring out a x7.
Problem (2021 Spring AMC 10B)
Let’s look at this idea in action.
The real number x satisfies the equation . What is the value of x11 - 7x7 + x3?
We create symmetry by factoring out an x7 to get that our desired expression is equal to
Using the same idea as slide 10, we get that , so the quantity inside the parentheses is equal to 0.
Hence, it doesn’t matter what x equals, so we have that the answer is 0x7 = 0.
Symmetry Tips
If you’re given something like (x+3)(x+7), try substituting in y = x+5 to turn it into (y-2)(y+2).
To handle some equations with something in common, try adding them together or making a substitution to see what you can find out.
A lot of times it’s even worth looking for a way to modify the expression to create symmetry and make it easier to work with.
Simplifying Radicals - A Reprise
Simplifying Radicals - A Reprise
Warmup: Simplify
Problem (classic): Simplify
Again, we can do some wishful thinking. Set this expression equal to x and square it. Remember that since (a+b)2 = a2 + 2ab + b2, we have that x2 is equal to , which simplifies to 36.
Since x2 = 36, we have that x = 6. Be careful not to miss this last step!
Remark: this can also be done by solving by guessing values for a and b, but this is left as an exercise.
Special Factorizations
You’ve probably seen a2 - b2 = (a-b)(a+b) and (a+b)2 = a2 + 2ab + b2 before.
It’s also common to see a3-b3 = (a-b)(a2+ab+b2) and a3+b3 = (a+b)(a2- ab+b2)
In general, an - bn = (a-b)(an-1 + an-2b + an-3b2 … + a2bn-3 + abn-2 + bn-1). We’ll look at this one more in depth in a bit.
SFFT: (x+b)(y+a) = xy + ax + by + ab, use this and “complete the rectangle”
And other times, we might want to get creative with factoring, like with grouping or by cleverly using the Factor Theorem (next week!)
Problem (2021 AMC 10A)
Which of the following is equal to
Hint: Wouldn’t it be nice to have something like 32 - 22?
Solution: We multiply the whole thing by 3-2, which is 1. Then we apply a2-b2 = (a-b)(a+b) a bunch of times to get it to simplify down to C) 3128 - 2128.
Geometric Series
Let’s start by proving that a + ar + ar2 + … + arn-1 = (a-arn)/(1-r).
This is the same idea as one of the factorizations!
In an infinite geometric series where the terms get smaller and smaller, the sum is equal to a/(1-r).
So why is it called a geometric series?
We can also use geometry to prove the result for infinite geometric series!
Credit: https://youtu.be/T7qAQ3MLV-o?t=263
Telescoping Series
(classic) What is the value of (1-½)(1-⅓)(1-¼)(1-⅕)........(1-1/2020)?
(classic) Evaluate 1/1*2 + 1/2*3 + 1/3*4 + … + 1/99*100.
This idea is called telescoping because it makes something long get collapsed into something manageable like a telescope.
Recap
Look for substitutions you can make to make an expression more comfortable to work with
Find ways to combine the information you are given
Try to take advantage of symmetry whenever you see it, and even look for ways to create symmetry
Use wishful thinking! Even if you know something doesn’t quite work but gets you closer to what you want, give it a try to see what you find out.
Thank you for coming!
Slides and problems will be posted in the next few days, reach out to dengr557@gmail.com with any questions