Fibonacci sequence

The Fibonacci sequence is a sequence of integers in which the first and second term are both equal to 1, and each subsequent term is the sum of the two preceding it. The first few terms are
$1, 1, 2, 3, 5, 8, 13, 21, 34, 55,...$ .

The Fibonacci sequence can be written recursively as $F_n=F_{n-1}+F_{n-2}$ .

Phi

Ratios between successive terms, $\frac{1}{1}$ , $\frac{2}{1}$ , $\frac{3}{2}$ , $\frac{5}{3}$ , $\frac{8}{5}$ , tend towards the limit phi.

Binet's formula

Binet's formula is an explicit formula used to find any nth term. It is $\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^n-\left(\frac{1-\sqrt{5}}{2}\right)^n\right)$ .

Problems

Introductory

The Fibonacci sequence $1,1,2,3,5,8,13,21,\ldots$ starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?

$\mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 4 } \qquad \mathrm{(C) \ 6 } \qquad \mathrm{(D) \ 7 } \qquad \mathrm{(E) \ 9 }$
(2000 AMC 12, Problem 4)

Intermediate

Seven line segments, with lengths no greater than 10 inches, and no shorter than 1 inch, are given. Show that one can choose three of them to represent the sides of a triangle.
(Manhattan Mathematical Olympiad 2004)
A fair coin is to be tossed $10_{}^{}$ times. Let $i/j^{}_{}$ , in lowest terms, be the probability that heads never occur on consecutive tosses. Find $i+j_{}^{}$ .
(1990 AIME, Problem 9)

Olympiad

Determine the maximum value of $\displaystyle m^2 + n^2$ , where $\displaystyle m$ and $\displaystyle n$ are integers satisfying $m, n \in \{ 1,2, \ldots , 1981 \}$ and $\displaystyle ( n^2 - mn - m^2 )^2 = 1$ .
(1981 IMO, Problem 3)

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