+0

-2
107
8

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.

Mar 26, 2020
edited by Guest  Mar 26, 2020

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Mar 26, 2020
#2
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convert it into latex

Guest Mar 26, 2020
#3
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The latex for me doesn't work. I can't copy and paste the questions into the latex bar. It just makes more junk.

inomath  Mar 26, 2020
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the problem has been edited so it is readable now

Guest Mar 26, 2020
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For problem #1, b = 9. Can you finish the problem?

Mar 26, 2020
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i know that b is 9 but I plugged in the values and got 40 for s

Guest Mar 26, 2020
#8
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129 + 159 + 169  =  11 + 14 + 15  =  40

geno3141  Mar 26, 2020
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3)  base b is 7; base b + 1 is 8; base b + 2 is 9

Mar 26, 2020