(i) ^ ( ^ ) 5 *, <±i) 21- ( ^ I J ^ - l,

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1 Edited by A. P. Hillma Please sed all material for ELEMENTARY PROBLEMS AND SOLUTIONS to Dr. A. P. HILLMAN; 709 SOLANO DR., S.E.; ALBUQUERQUE, NM Each solutio should be o a separate sheet (or sheets) ad must be received withi six moths of publicatio of the problem. Solutios typed i the format used below will be give preferece. Proposers of problems should iclude solutios. Ayoe desirig acowledgmet of cotributios should eclose a stamped, self-addressed card (or evelope). BASIC FORMULAS The Fiboacci umbers F ad the Lucas umbers L s a t i s f y F + 2 L + 2 F + 1 L F > L > F 0 h 0, 2 5 F l L l Also, a (1 + / 5 ) / 2, 3 (1 - / 5 ) / 2, F (a - 3 ) / / 5, ad L a + 3. i ; l. PROBLEMS PROPOSED IN THIS ISSUE B-658 Proposed by Joseph J. Kostal, U. of Illiois at Chicago Prove t h a t Q\ + Q\ + + Q,\ P 2 (mod 2 ), where the P ad Q a r e the P e l l umbers defied by P + 2 2P + l +P > P 0 > P l ^ l + Q > ^0 l >' «1 1 ' B-659 Proposed by Richard Adre-Jeai, Sfax, Tuisia For > 3, what i s the e a r e s t i t e g e r to F V51 B-86Q Proposed by Herta T. Freitag, Roaoe, VA Fid closed forms f o r : [/21. KH + D / 2 ] (i) ^ ( ^ ) 5 *, <±i) 21- ( ^ I J ^ - l, where [ ] is the greatest iteger i t. 1990] 85

2 B-661 Proposed by Herta T. Freitag, Roaoe, VA Let T() ( + l ) / 2. I B-646, i t was see t h a t T() i s a i t e g r a l d i v i s o r of T(2T()) for a l l i 2 + { 1, 2,... }. Fid the i Z + such t h a t T(ji) i s a i t e g r a l d i v i s o r of JT i l T(2T(i)). B-662 Proposed by H.-J. Seiffert, Berli, Germay Let H L P, where the L ad P a r e the Lucas ad P e l l umbers, r e s p e c - t i v e l y. Prove the followig cogrueces modulo 9: (1) H^ 3, (2) E h+l 3 + 1, (3) ^ + 2 E 3 + 6, (4) # ^ + 3 E B-663 Proposed by Clar. Kimberlig, U. of Evasville, Evasville, IN Let t\ 1, ti 2, ad t ( 3 / 2 ) t _ x - t _ 2 for?z 3, 4,.... D e t e r - mie lim sup t. SOLUTIONS Whe Is 2 (mod 5)? B-634 Proposed by P. L. Maa, Albuquerque, NM For how may i t e g e r s with 1 < < 10 6 i s 2 E (mod 5)? Solutio by Has Kappus, Rodersdorf, Switzerlad More g e e r a l l y, we show t h a t the umber of s o l u t i o s of 2 E (mod 5) (*) with 1 < < 1 0 r i s 2 1CF" 1. I f a c t, i t i s e a s i l y checed t h a t (2 - ) mod 5 i s p e r i o d i c with period p 20 s i c e p 20 i s the s m a l l e s t umber such t h a t 2 ( 2 p - 1) E p (mod 5) for a l l N. Now the oly s o l u t i o s of (*) with 1 < < 20 are 3, 14, 16, 17. Hece, the umber of s o l u t i o s of (*) i the i t e r v a l [ 1, 10 p ] i s 4-10 p /20 2 lo 2 " - 1. Also solved by R. Adre-Jeai, Charles Ashbacher, Paul S. Brucma, Joh Caell, Nicolas D. Diamatis, Alberto Facchii, Piero Filippoi, Russell Jay Hedel, H. Klauser & M. Wachtel, Joseph J. Kostal, L. Kuipers, Y. H. Harris Kwog, Carl Libis, Sahib Sigh, Lawrece Somer, Amitabha Tripathi, Gregory Wulczy, ad the proposer. Applicatio of t h e Iequality o t h e Meas B-635 Proposed by Mohammad K. Azaria, U. of Evasville, Evasville, IN For a l l p o s i t i v e i t e g e r s 9 prove t h a t (l)\ < (w + 2) +l. L -i J 86 [Feb.

3 Solutio by Bob Prielipp, U. of Wiscosi-Oshosh E(fc!&) E ((fe + D! - fc!) ( + D! - 1 l 1 Thus, the required iequality is equivalet to < +!),< ("-tip. This iequality follows immediately from the Arithmetic Mea-Geometric Mea Iequality, sice ( + 1) _ ( + 1) ( + 2) ~ 2( + 1 ) 2 Also solved by.r. Adre-Jeai, Charles Ashbacher, Paul S. Brucma, J. E. Chace, Nicholas D. Diamatis, Russell Euler, Piero Filippoi, Has Kappus, Y. H. Harris Kwog, Carl Libis, Alejadro Necochea, H.-J. Seiffert, Sahib Sigh, Amitabha Tripathi, Gregory Wulczy, ad the proposer. Differece Equatio B-636 Proposed by Mohammad K. Azaria, U. of Evasville, Evasville, IN Solve the d i f f e r e c e equatio x + l ( + l)x + X( + l ) 3 [! (! - 1)] for x i terms of X, x Q9 ad «Solutio by Y. H. Harris Kwog, SUNY College at Fredoia, Fredoia, NY Divide the r e c u r r e c e r e l a t i o by ( + 1 )! :, * w + * % + X[( + l ) ( + 1)! - ( + l ) 2 ]. ( + 1)! \ Let a x /l. We the have a +l a + X[( + 1)( + 1)! - ( + l) 2 ] for > 0, from which it follows immediately that a a + A E ( ^! ^ ~ ^2) 0 i Sice V \ ( + 1)! - 1, 2 ( + 1)(2 + l ) / 6, ad a x Q, we o b t a i x rz!{x 0 + X[( + 1)! ( + l ) ( 2 + l ) / 6 ] }. AZso solved by R. Adre-Jeai, Paul S. Brucma, Nicholas D. Diamatis, Guo-Gag Gao, Has Kappus, L. Kuipers, H.-J. Seiffert, Amitabha Tripathi, ad the proposer. 1990] 87

4 Golde Geometric Series B-637 Proposed by Joh Turer, U. of Waiato, Hamilto, New Zealad Show that t - - i. l F + af +l where a is the golde mea (1 + / 5 ) / 2. Solutio by Sahib Sigh, Clario U. of Pesylvaia, Clario, PA By iductio, F + af + i a +l. Thus, the give sum equals 1 E - ± - Sice 1/a < 1, the sum of this geometric series is 1/a _ 1 - (1/a) a(a - 1) 1 Also solved by R. Adre-Jeai, Paul S. Brucma, Joh Caell, J.. Chace, Nicolas D. Diamatis, Russell Euler, Piero Filippoi, Herta T. Freitag, Guo-Gag Gao, Russell Jay Hedel, Has Kappus, Joseph J. Kostal, L. Kuipers, Y. H. Harris Kwog, Carl Libis, Alejadro Necochea, Oxford Ruig Club (U. of Mississippi), Bob Prielipp, Elmer D. Robiso, H.-J. Seiffert, A. G. Shao, Amitabha Tripathi, Gregory Wulczy, ad the proposer. Summig Every Fourth Fiboacci Number B-638 Proposed by Herta T. Freitag, Roaoe, VA Fid s ad t as fuctio of ad such that ^ F -h + hi~2 F s F t' i 1 Solutio by Paul S. Brucma, Edmods, WA -1 \ ~ l L^F-h + hi-2 jl> F -hi-2 ~^L^^L-hi 1 ^0 i 0 L -hi-h' 5 (L " L -h) F 2 F -2' Hece, we may tae s 2, t - 2 (or s - 2, t 2). Also solved by R. Adre-Jeai, Piero Filippoi, Russell Jay Hedel, L. Kuipers, Y. H. Harris Kwog, Bob Prielipp, H.-J. Seiffert, Sahib Sigh, Amitabha Tripathi, Gregory Wulczy, ad the proposer. [Feb.

5 Lucas Aalogue B-639 Proposed by Herta T. Freitag, Roaoe, VA Fid s ad t as fuctios of ad such that JH L -i* + i*i-2 i l F s L f Solutio by Y. H. Harris Kwog, SUNY College at Fredoia, Fredoia, NY It is well ow that L a + b, where a ad b are the zeros of x x - 1; so we ca employ the same techique used i solvig B-638. Alterately, usig the result (from B-638) )-</-h + hi-2 F 2 F -2 9 ^ 1 ad the fact that L F^.-, - F 1 3 a solutio follows immediately: 2^ L->4 + L +i-2.2-* F + l- + L±i-2 + X«> F -\-h + hi-2 i \ ^ 1 ^ l F 2 F +\-2 + F 2 F -l-2 F 2 L -2 ' Also solved by Paul S. Brucma, R. Adre-Jeai, Piero Filippoi, L. Kuipers, Bob Prielipp, H.-J. Seiffert, Sahib Sigh, Amitabha Tripathi, Gregory Wulczy, ad the proposer. 1990] 89