2013 AMC 12B Problem 14

Question

Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is

N.

What is the smallest possible value of

N?

55

89

104

144

273

Answer: C

Concepts:Fibonacci Diophantine Equation greatest common divisor

Difficulty rating: 1750

Small hint:

In terms of the first two terms

a_1, a_2,

the seventh term equals

5a_1 + 8a_2

Big hint:

Setting the two seventh terms equal gives

5(b_1 - a_1) = 8(a_2 - b_2);

coprimality of

5

and

8

forces

b_1 - a_1

to be a multiple of

8

Solution:

A sequence starting

a_1, a_2

has seventh term

5a_1 + 8a_2.

For the two sequences,

5a_1 + 8a_2 = 5b_1 + 8b_2,

so

5(b_1 - a_1) = 8(a_2 - b_2).

Since

\gcd(5, 8) = 1,

we need

b_1 - a_1

to be a multiple of

8

and

a_2 - b_2

to be a multiple of

5.

Taking

a_1 \lt b_1

with nondecreasing terms gives

a_1 \le b_1 - 8 \le b_2 - 8 \le a_2 - 13.

Choosing

a_1 = 0,

b_1 = b_2 = 8,

a_2 = 13

yields

N = 5\cdot 0 + 8\cdot 13 = 104.

Thus, the correct answer is C.

2013 AMC 12B Problem 14

Answer: C

Difficulty rating: 1750

Small hint:

Big hint:

Solution:

Problem 14 in other years