Why do I love 2016 so much?

Geometry Level 5

∑ n = 1 201 6 2 + 4032 4034 + 2017 + n ∑ n = 1 201 6 2 + 4032 4034 − 2017 + n \dfrac{\displaystyle \sum_{n=1}^{2016^2+4032} \sqrt{\sqrt{4034}+\sqrt{2017+\sqrt{n}}}}{\displaystyle \sum_{n=1}^{2016^2+4032} \sqrt{\sqrt{4034}-\sqrt{2017+\sqrt{n}}}}

If the value of above expression is in the form cot ⁡ π a b \cot\dfrac{\pi^a}{b} , find a + b a+b .


The answer is 17.

Rohit Udaiwal by Rohit Udaiwal , 20, India

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1 solution

Fletcher Mattox
May 20, 2020

Since the author is no longer active and since none of the 10 solvers so far have provided a solution, I'll offer some code just for grins: 

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from mpmath import *

def f(n):
    return sqrt(sqrt(mpf(4034)) + sqrt((mpf(2017) + sqrt(mpf(n)))))

def g(n):
    return sqrt(sqrt(mpf(4034)) - sqrt((mpf(2017) + sqrt(mpf(n)))))

mp.dps = 100

hi = 2016**2 + 4032
fsum = 0    
gsum = 0

for n in range(1, hi+1):
    fsum += f(n)
    gsum += g(n)

result = fsum/gsum
for a in range(50):
    for b in range(50):
        try:
            c = cot((pi**a)/b)
        except :
            continue
        if abs(c - result) < 10**-9:
            print(a + b)

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