The Art of Algebraic Manipulations
by shiningsunnyday, Apr 2, 2016, 9:23 AM
"Wow. How would anyone think of that?" That's essentially how I feel everytime I read the solution to a hard problem in 108 Alg.
I've noticed it becoming a theme for hard problems in 108 to being highly-contrived. Below I included 4 problems I solved in the book, each of which are instructive and contain a good insight. I encourage you guys to play with these! If stuck, you can read the first line of my solution to get a piece of motivation. Onto the problems!
Two slightly harder problems:
Here're two problems I'm still stuck on. I would appreciate any ideas/hints/solutions.
Whew. These problems sure can get difficult for me. I'm curious, do you guys think solving such problems, like the ones below, are the result of experience or ingenious insights?
I've noticed it becoming a theme for hard problems in 108 to being highly-contrived. Below I included 4 problems I solved in the book, each of which are instructive and contain a good insight. I encourage you guys to play with these! If stuck, you can read the first line of my solution to get a piece of motivation. Onto the problems!
be real numbers such that 
imply the third.
Initially, it appears that squaring is a bad idea cause we're going to generate 
terms but look again! Does 
remind of something? It should!
.
and the squares cancel out, to obtain: 
which is true by the given. Looking at equation 
it's obvious that if the third equality must be true given that we have two equalities.
as the denominator, but since we have an annoying third power, we should try to write the numerator as something involving cubes as well. Specifically, note how the 
may remind you of the coefficients upon expanding 
Yes!
Two slightly harder problems:
that satisfy 
and 
for all real 
since the first is nonnegative and the second factor has a negative discriminant.
but by the given conditions, it must be equal to zero! So we can quickly find that the only possibility is 
be positive real numbers such that 
and let 
Prove that 
are extremely hard to work with, so are 
.
yielding 
Note that 
Similarly, we get 
and what we want to prove reduces to 
the given condition.Here're two problems I'm still stuck on. I would appreciate any ideas/hints/solutions.
Prove that 
be a positive integer. Simplify
Whew. These problems sure can get difficult for me. I'm curious, do you guys think solving such problems, like the ones below, are the result of experience or ingenious insights?
May 2018
April 2018