2020 AMC 10B Problems/Problem 15

Problem

Steve wrote the digits , , , , and in order repeatedly from left to right, forming a list of digits, beginning He then erased every third digit from his list (that is, the rd, th, th, digits from the left), then erased every fourth digit from the resulting list (that is, the th, th, th, digits from the left in what remained), and then erased every fifth digit from what remained at that point. What is the sum of the three digits that were then in the positions ?

Solution 1

After erasing every third digit, the list becomes repeated. After erasing every fourth digit from this list, the list becomes repeated. Finally, after erasing every fifth digit from this list, the list becomes repeated. Since this list repeats every digits and since are respectively in we have that the th, th, and st digits are the rd, th, and th digits respectively. It follows that the answer is

Solution 2

As the LCM of 3, 4, and 5 is , let's look at a 60-digit block of original numbers (many will be erased by Steve). After he erases every third number (1/3), then every fourth number of what remains (1/4), then every fifth number of what remains (1/5), we are left with digits from this 60-digit block. . Writing out the first few digits of this sequence, we have . Therefore, our answer is . ~BakedPotato66

Solution 3 (Illustrations of Solutions 1 and 2)

Note that cycles exist initially and after each round of erasing digits.

Two solutions follow from here:

Solution 3.1 (Stepwise: Similar to Solution 1)

We will use parentheses to denote a cycle. Initially, the list has a cycle of length

I will finish shortly. No edit please.

~MRENTHUSIASM

Solution 3.2 (Direct: Similar to Solution 2)

I will finish shortly. No edit please.

~MRENTHUSIASM

Video Solution

https://youtu.be/t6yjfKXpwDs 16:40 ~IceMatrix

See Also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.