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2018-06-18 Advanced


What's the minimum number of straight cuts required to cut a pizza into 77 pieces of equal area?

Details and Assumptions:

  • A straight cut may start and end anywhere on the pizza—not just on the circumference.
  • The pizza comes uncut and round in the box.
  • You cannot take the pizza out of the box or move any part of it.

At a party with 100 people, there is a pie. The first guest gets 1% of the pie, the second gets 2% of what is left, the third gets 3% of what is left and so on.

If the nnth guest gets the largest piece of all 100, what is nn?

This is the graph of r=sin4θsinθr=\frac{\sin4\theta}{\sin\theta} in polar coordinates.

Rotating the graph about the xx-axis produces a volume of abπ\frac{a}{b}\pi, where aa and bb are coprime positive integers.

What is a+b?a+b?

0π2tanx dx\large \int_0^\frac \pi 2\sqrt{\tan{x}}\ dx

Evaluate this integral.

Let FnF_n denote the nthn^\text{th} Fibonacci number, where F1=1,F2=1,F_1 = 1, F_2 = 1, and Fn+2=Fn+1+FnF_{n+2} = F_{n+1} + F_{n} for n=1,2,3,.n=1,2,3,\ldots.

Define fn(x)f_n (x) as a least-degree polynomial that passes through the coordinates (x,y)=(1,F1),(2,F2),(3,F3),,(n1,Fn1).(x,y)= (1, F_1), (2,F_2) , (3,F_3) , \ldots , (n-1, F_{n-1}). Hence, we define the expected Fibonacci sequence eFn^e F_n to be equal to fn(n).f_n (n).

For example, with F1=F2=1F_1 = F_2 = 1 and F3=2,F_3 = 2, we have f4(x)=12(x23x+4),f_4 (x) = \frac12\big(x^2-3x+4\big), so eF4=f4(4)=4.^e F_4 = f_4 (4) = 4.

Find the closed form of the limit below (submit your answer to three decimal places): limneFnFnFn. \lim_{n\to\infty} \dfrac{ \big|^e F_n - F_n \big|}{F_n}.

Notation: | \cdot | denotes the absolute value function.


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