PROBLEMS from volume 31 (1993) issue #2 of The Fibonacci Quarterly.

Copyright ◯^{c} 1993 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 below 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

\underline{B–736} Proposed by Herta Freitag, Roanoke, VA Prove that (2L_{n} + L_{n−3})/5 is a Fibonacci number for all n.

\underline{B–737} Proposed by Herta T. Freitag, Roanoke, VA

A right triangle, one of whose legs is twice as long as the other leg, has a hypotenuse that is one unit longer than the longer leg. Let r be the inradius of this triangle (radius of inscribed circle) and let r_{a}, r_{b}, r_{c} be the exradii (radii of circles outside the triangle that are tangent to all three sides).

Express r, r_{a}, r_{b}, and r_{c} in terms of the golden ratio, α.

\underline{B–738} Proposed by Daniel C. Fielder and Cecil O. Alford, Georgia Institute of Technology, Atlanta, GA

Find a polynomial f(w, x, y, z) such that

f(L_{n}, L_{n+1}, L_{n+2}, L_{n+3}) = 25f(F_{n}, F_{n+1}, F_{n+2}, F_{n+3})

is an identity.

\underline{B–739} Proposed by Ralph Thomas, University of Chicago, Dundee, IL Let S = \{\frac{F_{i}}{F_{j}} | i ≥ 0, j > 0\}. Is S dense in the set of nonnegative real numbers?

\underline{B–740} Proposed by Thomas Martin, Phoenix, AZ

Find all positive integers x such that 10 is the smallest integer, n, such that n! is divisible by x.

\underline{B–741} Proposed by Jayantibhai M. Patel, Bhavan’s R. A. College of Science, Gujarat, India

Prove that F_{n+8}^{4} + 331F_{n+4}^{4} + F_{n}^{4} is always divisible by 54.