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

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 $8 \mid b_1 - a_1$ and $5 \mid a_2 - b_2.$ 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

Solution:

Problem 14 in Other Years