I'm Feeling Particularly Wishful Today
by shiningsunnyday, Apr 12, 2016, 10:52 AM
Today I solved 3 problems from 108 Alg, at the very end of the introductory section (first 54 problems), leaving me with only ONE more problem to go before I move on to the advanced section. The common theme of this blog post is wishful thinking, which, as Richard taught, is a useful powerful problem solving mindset.
For those who never heard of this term before, ask:
What's the key obstacle preventing me from solving this problem?
What could you "wish" to happen that would remove this obstacle?
Then make it happen!
Let
and 
be nonnegative real numbers such that 
Prove that 
Let
be positive real numbers. Prove that 
Solve the system in reals:
So the problem below (P54 of 108) is now the only problem remaining in this section. I'm still stuck on it. Any hints/solutions will be appreciated...
Let
be a positive integer. Simplify
For those who never heard of this term before, ask:
What's the key obstacle preventing me from solving this problem?
What could you "wish" to happen that would remove this obstacle?
Then make it happen!
Let
and 
be nonnegative real numbers such that 
Prove that 
How can we use the fact that 
Well, if it is always less than or equal to zero, I wish we can somehow show that with our assumption it must be always bigger than zero. Also, the two variables are irksome. I wish we could reduce this to a quadratic...
leading to the contradiction we desired.Let
be positive real numbers. Prove that 
What's interesting about 
is that both the numerator and denominator contain 
Cancellation is futile, but both the numerator and denominator are in the form 
This should immediately trigger Cauchy. Plus, the denominator is a square, an even more obvious trigger.
by Cauchy. Aha! So it suffices to prove that 
,Solve the system in reals:
is so random, sitting there on the right. If only we can somehow combine 
and 
Now we really need to subtract the two quantities in the roots! So we make this happen by squaring both sides, leading to:
Subtracting yields 
Note that 
satisfies both equations, while 
doesn't as it makes the first and second radicals imaginary, so 
is our solution.So the problem below (P54 of 108) is now the only problem remaining in this section. I'm still stuck on it. Any hints/solutions will be appreciated...
Let
be a positive integer. Simplify
then
This post has been edited 2 times. Last edited by shiningsunnyday, Apr 12, 2016, 10:55 AM
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