1. For a certain base b, the product (12 subscript b)(15 subscript b)(16 subscript b) is equal to 3146 subscript b. Let S = 12 subscript b + 15 subscript b + 16 subscript b. What is S in base b?
2. Let b be an integer greater than 2, and let N subscript b = 1 subscript b + 2 subscript b + .... + 100 subscript b (the sum contains all valid base b numbers up to 100 subscript b). Compute the number of values of b for which the sum of the squares of the base b digits of N subscript b is at most 512.
3. Let a subscript 2, a subscript 1, and a subscript 0 be three digits. When the 3-digit number a-sub2,a-sub1,a-sub0 is read in base b and converted to decimal, the result is 254. When the 3-digit number a-sub2,a-sub1,a-sub0 is read in base b+1 and converted to decimal, the result is 330. Finally, when the 3-digit number a-sub2,a-sub1,a-sub0 is read in base b+2 and converted to decimal, the result is 416. Find the 3-digit number a-sub2,a-sub1,a-sub0. (Express your answer in decimal.)
4. Consider the set S = {0, 1, 2, ... , 3^k-1}. Prove that one can choose T to be a 2^k-element subset of S such that none of the elements of T can be represented as the arithmetic mean of two distinct elements of T.
