PROBLEMS from volume 32 (1994) issue #5 of The Fibonacci Quarterly.

Copyright ◯^{c} 1994 by The Fibonacci Association.

                         Inquiries to: Fibonacci@MathPro.com

ELEMENTARY PROBLEMS AND SOLUTIONS Edited by Stanley Rabinowitz

Please send all material for ELEMENTARY PROBLEMS AND SOLUTIONS to Dr. STAN-LEY RABINOWITZ; 12 VINE BROOK RD; WESTFORD, MA 01886-4212 USA. Correspondence may also be sent to the problem editor by electronic mail to Fibonacci@MathPro.com on Internet. All correspondence will be acknowledged.

Each solution should be on a separate sheet (or sheets) and must be received within six months of publication of the problem. Solutions typed in the format used by The Fibonacci Quarterly will be given preference. Proposers of problems should normally include solutions. Proposers should inform us of the history of the problem, if it is not original. A problem should not be submitted elsewhere while it is under consideration for publication in this column.

BASIC FORMULAS

The Fibonacci numbers F_{n} and the Lucas numbers L_{n} satisfy

F_{n+2} = F_{n+1} + F_{n}, F_{0} = 0, F_{1} = 1;
L_{n+2} = L_{n+1} + L_{n}, L_{0} = 2, L_{1} = 1.

Also, α = (1 + \sqrt{5})/2, β = (1 − \sqrt{5})/2, F_{n} = (α^{n} − β^{n})/\sqrt{5}, and L_{n} = α^{n} + β^{n}.

PROBLEMS PROPOSED IN THIS ISSUE

Nary a month goes by without my receiving a problem proposal from the inveterate problemist, Herta Taussig Freitag. So, as a tribute to Herta, and to reduce my backlog, all of the problems in this issue come from her. As usual, generalizations are always welcome.

\underline{B–772} Proposed by Herta T. Freitag, Roanoke, VA Prove that

\frac{L_{n}^{2} + L_{n+a}^{2}}{F_{n}^{2} + F_{n+a}^{2}}

is always an integer if a is odd. How should this problem be modified if a is even?

\underline{B–773} Proposed by Herta T. Freitag, Roanoke, VA

Find the number of terms in the Zeckendorf representation of ∑_{i=1}^{n} F_{i}^{2}.

[The Zeckendorf representation of an integer expresses that integer as a sum of distinct nonconsecutive Fibonacci numbers.]

\underline{B–774} Proposed by Herta T. Freitag, Roanoke, VA

Let ⟨H_{n}⟩ be any sequence of integers such that H_{n+2} = H_{n+1} + H_{n} for all n. Let p and m be positive integers such that H_{n+p} ≡ H_{n} (mod m) for all integers n. Prove that the sum of any p consecutive terms of the sequence is divisible by m.

\underline{B–775} Proposed by Herta T. Freitag, Roanoke, VA Let g = α + 2. Express g^{17} in the form pα + q where p and q are integers.

\underline{B–776} Proposed by Herta T. Freitag, Roanoke, VA

Find all values of n for which ∑_{k=1}^{n} kF_{k} is even.

\underline{B–777} Proposed by Herta T. Freitag, Roanoke, VA Find all integers a such that n ≡ a (mod 4) if and only if L_{n} ≡ a (mod 5).