Difference between revisions of "2019 AMC 10A Problems/Problem 23"
|Line 64:||Line 64:|
Revision as of 17:08, 27 January 2021
Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for them. First Tadd will say the number , then Todd must say the next two numbers ( and ), then Tucker must say the next three numbers (, , ), then Tadd must say the next four numbers (, , , ), and the process continues to rotate through the three children in order, each saying one more number than the previous child did, until the number is reached. What is the th number said by Tadd?
Define a round as one complete rotation through each of the three children.
We create a table to keep track of what numbers each child says for each round.
Tadd says number in round 1, numbers in round 2, numbers in round 3, and in general numbers in round . At the end of round , the number of numbers Tadd has said so far is , by the arithmetic series sum formula.
Therefore substituting as the turn, we get the term or 109th number, which means that it's , which is the 2035th term, and -=, and thus the answer is .
Firstly, as in Solution 1, we list how many numbers Tadd says, Todd says, and Tucker says in each round.
We can find a general formula for the number of numbers each of the kids say after the th round. For Tadd, we can either use the arithmetic series sum formula (like in Solution 1) or standard summation results to get .
Now, to find the number of rotations Tadd and his siblings go through before Tadd says his th number, we know the inequality must be satisfied, and testing numbers gives the maximum integer value of as .
The next main insight, in order to simplify the computation process, is to notice that the th number Tadd says is simply the number of numbers Todd and Tucker say plus the Tadd says, which will be the answer since Tadd goes first.
Carrying out the calculation thus becomes quite simple:
At this point, we can note that the last digit of the answer is , which gives . (Completing the calculation will confirm the answer, if you have time.)
Video Solution by TheBeautyofMath
|2019 AMC 10A (Problems • Answer Key • Resources)|
|1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25|
|All AMC 10 Problems and Solutions|