Try this beautiful Problem on Algebra based on finding greatest integer from AMC 10 A, 2018. You may use sequential hints to solve the problem.
What is the greatest integer less than or equal to $\frac{3^{100}+2^{100}}{3^{96}+2^{96}} ?$
Algebra
greatest integer
Pre College Mathematics
AMC-10A, 2018 Problem-14
$80$
The given expression is $\frac{3^{100}+2^{100}}{3^{96}+2^{96}} ?$
We have to find out the greatest integer which is less than or equal to the given expression .
Let us assaume that $x=3^{96}$ and $y=2^{96}$
Therefore the given expression becoms $\frac{81 x+16 y}{x+y}$
Now can you finish the problem?
Now $\frac{81 x+16 y}{x+y}$
=$\frac{16 x+16 y}{x+y}+\frac{65 x}{x+y}$
$=16+\frac{65 x}{x+y}$
Now if we look very carefully we see that $\frac{65 x}{x+y}<\frac{65 x}{x}=65$
Therefore $16+\frac{65 x}{x+y}<16+65=81$
Now Can you finish the Problem?
Therefore less than \(81\) , the answer will be \(80\)
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Can you explain to me :
$\frac{3^{100}+2^{100}}{3^{96}+2^{96}} ?$
These symbols are All Greek to me!