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1. [ 10 points ] How many positive integers smaller than 2018000 have an odd number of factors?


2. [ 25 points ] Alice has a large number of 12c, 17c and 19c coins. What is the largest price for which she cannot pay the exact amount?


3. Consider the four numbers: $$2^{2^{2^{2^2}}},3^{3^{3^3}},4^{4^4},10^{1000}$$
   [ 15 points ] How many digits does the smallest number have?
   [ 25 points ] How many digits does the largest number have?


4. [ 25 points ] Suppose a keyboard has all the letter keys, along with spacebar, question mark, comma and full stop. What are the first five significant digits of the chance that a monkey randomly hitting this keyboard would type out this question?
   Note: Ignore the point tally, this note and case (e.g lowercase/uppercase) in your answer.