2013 AMC 10B Problem 13

Question

Jo and Blair take turns counting from

1

to one more than the last number said by the other person. Jo starts by saying “

1

”, so Blair follows by saying “

1,

2

”. Jo then says “

1,

2,

3

”, and so on. What number is spoken in position

53

?

2

3

5

6

8

Find the triangular number just before

53

The next spoken list starts again at

1

Through the turn ending with

n

, the total number of numbers spoken is

1+2+\cdots+n=\frac{n(n+1)}2.

After the turn ending with

9

, exactly

\frac{9\cdot10}{2}=45

numbers have been spoken. The next turn is the list

1,2,\ldots,10

, so its eighth entry is the

53

rd number spoken. That entry is

8

.

Thus, the correct answer is E .

Problem 13 in other years