057.py

#!/usr/bin/python
# -*- coding: utf-8 -*-

#It is possible to show that the square root of two can be expressed as an infinite continued fraction.

#√ 2 = 1 + 1/(2 + 1/(2 + 1/(2 + ... ))) = 1.414213...

#By expanding this for the first four iterations, we get:

#1 + 1/2 = 3/2 = 1.5
#1 + 1/(2 + 1/2) = 7/5 = 1.4
#1 + 1/(2 + 1/(2 + 1/2)) = 17/12 = 1.41666...
#1 + 1/(2 + 1/(2 + 1/(2 + 1/2))) = 41/29 = 1.41379...

#The next three expansions are 99/70, 239/169, and 577/408, but the eighth expansion, 1393/985, is the first example where the number of digits in the numerator exceeds the number of digits in the denominator.

#In the first one-thousand expansions, how many fractions contain a numerator with more digits than denominator?


#Answer:
	#153

#from time import time; t=time()
from math import log10

M = 1000
a, b = 1, 1
c = 0
for i in range(M):
    a, b = a+2*b, a+b
    if int(log10(a)) > int(log10(b)): c += 1

print(c)#, time()-t