ELEMENTARY PROBLEMS AND SOLUTIONS

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1 ELEMENTARY PROBLEMS AND SOLUTIONS Edited by A. P. HILLMAN Uiversity of Sata Clara, Sata Clara, Califoria Sed a l l c o m m u i c a t i o s r e g a r d i g E l e m e t a r y P r o b l e m s a d S o l u t i o s to P r o f e s s o r A. P. H i l l m a, D e p a r t m e t of M a t h e m a t i c s a d S t a t i s t i c s, U i v e r s i t y of N e w M e x i c o, A l b u q u e r q u e, New M e x i c o. E a c h p r o b l e m o r s o l u t i o s h o u l d be s u b m i t t e d i l e g i b l e f o r m, p r e f e r - a b l y t y p e d i d o u b l e s p a c i g, o a s e p a r a t e s h e e t o r s h e e t s i t h e f o r - m a t u s e d b e l o w. S o l u t i o s s h o u l d be r e c e i v e d w i t h i two m o t h s of p u b l i c a t i o. B Proposed by Douglas Lid, Uiversity of Virgiia, Charlottesville, Va. D e o t e x by e x ( a ). Show t h a t t h e followig e x p r e s s i o, c o - t a i i g i t e g r a l s, I e x ( J ex( J e x (... / ex( f x d x ) d x )... dx)dx)dx ^0 ^0 JO JO JO e q u a l s F,, / F, _, w h e r e F i s t h e - t h F i b o a c c i u m b e r. +1 ' +2 B Proposed by Douglas Lid, Uiversity of Virgiia, Charlottesville, Va. F i d a " 2 + a ~ 3 + a , w h e r e a = (1 + 5 ) / 2. B Proposed by ]. A. H. Huter, Toroto, Caada E a c h d i s t i c t l e t t e r i t h i s s i m p l e a l p h a m e t i c s t a d s for a p a r t i c - u l a r ad d i f f e r e t d i g i t. We a l l kow h o w r a b b i t s lik up w i t h t h e F i b o a c c i s e r i e s, s o ow e v a l u a t e o u r R A B B I T S. R A B B I T S B E A R R A B B I T S A A S S E R I E S B Proposed by Douglas Lid, Uiversity of Virgiia, Charlottesville, Va. P r o v e t h a t i i 2 + r - 2 m _ 2 2 (?) ( k+t T l ) = l ( m T X ). k=0 j=0 J 235 m = 0 p=0 p

2 236 ELEMENTARY PROBLEMS AND SOLUTIONS Oct. where ( ) = 0 for < r. V B Proposed by M. N. S. Swamy, Uiversity of Saskatchewa, Regia, Caada The Fiboacci polyomial f (x) is defied by 1 = 1, f? = x, ad f (x) = xf. (x) + f _(x) for > 2. Show the followig: (a) x 2 f (x) = f,. + f -1. N ', r +1 r=l (b) f,,. = f, 1 f,. + f f. * m++1 m+1 +1 m (c) f (x) S ( V j=0 J ZJ l, where [kj is the greatest iteger ot exceedig k. the -th Fiboacci umber Hece show that F O - D / 2 ] = s (.j_i! ). i=o J B Proposed by M. N. S. Swamy, Uiversity of Saskatchewa, Regia, Caada Let f (x) be as defied i B-74. Show that the derivative, -1 f x (x) ' = X, f r x (x) ' f -r x (x) ' for > 1. r=l SOLUTIONS ONE, TWO, THREE OUT B Proposed by Sidey Kravitz, Dover, New ]ersey Show that o Fiboacci umber other tha 1, 2, or 3 is equal to a Lucas umber. Solutio by Douglas Lid, Uiversity of Virgiia, Charlottesville, Va. Sice L, = F,, + F,,,, the a s s e r t i o is equivalet to (I) F = F + F {L) * k - l k+1*

3 1965 ELEMENTARY PROBLEMS AND SOLUTIONS 237 If k > 3, the > k+1 ad (1) is clearly impossible sice F l 1 + F l XI < F l + F l _L1 = F T XO - k-1 k+1 k k+1 k+2 Impossibility for k ^ 3 implies impossibility for k i -3 sice oly sigs are differet. For -3 < k < 3 we fid F 9 = L = 1, & l F~ = L = 2, F, = L, = 1, ad F. = L~ = 3, correspodig to k = - 1, 0, 1, ad 2 respectively. Hece these are the oly solutios. (The crux of this problem is solved i the discussio of equatio (12) i Carlitz' "A Note o Fiboacci Numbers, M this Quarterly 1 (1964) No. 2 pp ). Also solved by J. L. Brow, Jr.; Gary C. McDoald; C. B. A. Peck; ad the proposer. F - B Proposed by Brother U. Alfred, St. Mary's College, Califoria Show that the volume of a trucated right circular coe of slat height F with F, ad F i the diameters of the bases is & ^ " < F + l - F - l ) / 2 4. Solutio by Douglas Lid, Uiversity of Virgiia, Charlottesville, Va. It is we 11-kow that if h is the height of the frustrum of a right circular coe, s the slat height, ad r ad r the radii of the bases, the the volume V is V = (rrh/3)(rj + r } r 2 + r^) = (rr/3) \ / 7 ^ ( r 2 - r 1 ) 2 ( r J + r ^ + r 2 ). For this problem r, = F. / 2, r 0 = F,, / 2 ad s = F, so that r 1-1 ' 2 +1' V = 5 \ / F 2 - (F,, - F T ) 2 /4 (F 2. + F. F, 1 +F 2 )/4 3 v x x +1-1 ' ' " 77 V F 2 - F 2 / 4 x(f 2 1 +F.F x l + F 2 )/l2 ' " \/377 F (F 2. + F. F,_ + F 2, 1 ) / "

4 238 ELEMENTARY PROBLEMS AND SOLUTIONS Oct. - V I, ( F F. 1 ) ( F ^ 1 + F _ l F F^+1 )/24 = ^ F + 1 " F -1>/ 2 4 ' We remark that the area A of the curved surface of the frustrum is A = F (F,. + F, )/2 = {/2)F L. +1-1" ' Also solved by Carole Baia, Gary C. McDoald, Keeth E. Newcomer, C. B. A. Peck, M. N. S. Swamy, Howard L. Walto, Joh Wesser, Charles Ziegefus, ad the proposer. McDoald also added the formula for the curved surface. B - o O Proposed by Verer E. Hoggatt, Jr., Sa Jose State College, Sa Jose, Califoria 2 Show that L^ L~ x o + 5F 0, 1 = 1, where F ad L are the l -th Fiboacci umber ad Lucas umber, respectively. Solutio by 2d Lt. Charles R. Wall, U. S. Army, A. P. 0., Sa Fracisco, Calif. Usig my secod aswer to B-22 (Vol. 2, No. 1, p. 78), L 2(+1) L 2 = 5F (+l)+ + L (+l)- = 5 F 2 + l + L l = 5 F Ll + 1 Thus L 2+2 L 2 " 5F 2+l = l ' 2 Also solved by J. L. Brow, Jr.; J. A. H. Huter; Douglas Lid, Kathlee Marafio, Gary C. McDoald, C. B. A. Peck, Bejami Sharpe, M. N. S. Swamy, Howard L. Walto, Joh Wesser, Kathlee M. Wickett, David Zeitli, Charles Ziegefus, ad the proposer. Also by David KLarer. MODULO THREE B - 6 l Proposed by J. A. H. Huter, Toroto, Otario Defie a sequece U,, U?,... by U, =3 ad U = U, l for > Prove that U = 0(mod ) if $ 0 (mod 3). Solutio by Joh Wesser, Melboure, Florida

5 1965 ELEMENTARY PROBLEMS AND SOLUTIONS 239 A alterative represetatio for U is 7 U = 1 (k^+k+1). k=l Upo expadig the idividual sums ivolved we obtai U = [(2+l)(+l)/6] + [(+l)/2] + = (/3) [(+2)(+l)+3]. Hece, U = 0(mod ) if ad oly if (+l)(+2) = 0(mod 3). This coditio obtais if ad oly if ^ 0(mod 3). Also solved by Robert J. Hursey, Jr., Douglas Lid, Gary C. McDoald, Robert McGee, C. B. A. Peck, Charles R. Wall, David Zeitli, ad the proposer. UNIQUE SUM OF SQUARES B Proposed by Brother U. Alfred, St. Mary's College, Califoria Prove that a Fiboacci umber with odd subscript caot be r e p - reseted as the sum of squares of two Fiboacci umbers i m o r e tha oe way. Solutio by J. L. Brow, jr., Pesylvaia State Uiversity, State College, Pa. F r o m the idetity 3 F 0,, = F + F,,,, (> 1) it follows that 2+l +1 F 2., < (F + F,,) = F 2 Therefore, ay represetatio x +2 2+l +1' «+?. F.., = F, + F 2 (k < m) m u s t have both k ad m < +1. The 2+l k m x ' k > (otherwise 1 Ff + F Z < F^ + F z ^, = F,,. for k > 2). k r +1 2+l ' Also solved by Douglas Lid, Joseph A. Orjechouski ad Robert McGee (joitly), C. B. A. Peck, ad the proposer. A N I S O S C E L E S T R I A N G L E A old problem whose source is ukow, suggested by Sidey Kravitz, Dover, New Jersey. I A ABC let sides AB ad AC be equal. Let there be a poit D o side AB such that AD = CD = BC. the golde mea. 2cos $ A = AB/BC = (1 + \/5)/2, Show that Solutio by Joh Wesser, Melboure, Florida By ispectio of the figure ad the law of cosies AD 2 = CD 2 + A C 2-2CD* AC cos f A.

6 240 ELEMENTARY PROBLEMS AND SOLUTIONS Oct. Sice AD = CD = BC ad AB = AC, it follows immediately that 2 cos $ A = AC/CD = AB/BC. The secod result comes from the fact that $ B = $ BDC = $ A + $ DCA = 2 $ A ad hece $ A = 36 ad 2 cos A = (1 + >/5)/2. (See N. N. Vorobyov: The Fiboacci Numbers (New York, (1961) p. 56.) Also solved by Herta Taussig Freitag, Cheryl Hedrix, Kathlee Marafio, ad Carol Barrigto (joitly), ]. A. H. Huter, Douglas Lid, James Leisser, C. B. A. Peck, Kathlee M. Wickett, ad the proposer. XXXXXXXXXXXXXXX Cotiued from page H. Withrop, "The Mathematics Of The Roud Robi, " M a t h e - matics Magazie (I P r e s s ). 6. H. Withrop, "A Mathematical Model For The Study Of The Propagatio Of Novel Social Behavior, " Idia Sociological Bulleti, July 1965, Vol. II. (I P r e s s ) 7. H. Withrop, "Some Geeralizatios Of The Dyig Rabbit P r o b - lem, " (I Preparatio). 8. N. N. Vorob'ev, Fiboacci Numbers, Blaisdell Publishig Compay, New York, XXXXXXXXXXXXX ASSOCIATION PUBLISHES BOOKLET Brother U. Alfred has just completed a ew booklet etitled: Itroductio to Fiboacci Discovery. This booklet for t e a c h e r s, r e s e a r c h e r s, ad bright studets ca be secured for $1.50 each or 4 copies for $5.00 from Brother U. Alfred, St. M a r y ' s College, Calif.