1/(1 -x n ) = \YJ k=0

Transcription

1 ADVANCED PROBLEMS AND SOLUTIONS Edited by RAYMOND E.WHITNEY Lock Have State College, Lock Have, Pesylvaia Sed all commuicatios cocerig Advaced Problems ad Solutios to Eaymod E. Whitey, Mathematics Departmet, Lock Have State College, Lock Have, Pesylvaia This departmet especially welcomes problems believed to be ew or extedig old results. Proposers should submit solutios or other iformatio that will assist the editor. To facilitate their cosideratio, solutios should be submitted o separate siged sheets withi two moths after publicatio of the problems. H-25 Proposed by L Carlitz, Duke Uiversity, Durham, North Carolia. Evaluate the determiats of o r d e r z - qz z - z - q 2 z D = - ~ 2 i - q z - q z A = -2 - z q - z H-26 Proposed by P. Bruckma, Uiversity of Illiois, Urbaa, Illiois. Prove the idetity: /( -x ) = \YJ - k= /{ " X 2km/ x e ) H-27 Proposed by C. Bridger, Sprigfield, Illiois. Defie G, (x) by the relatio - (x 2 + )B 2 - xs 3 Z G k (x)sk = where x is idepedet of s.. Fid a recursio formula coectig the G, (x). 72

2 Feb. 973 ADVANCED PROBLEMS AND SOLUTIONS Put x = ad fid G, () i terms of Fiboacci umbers. 3. Also with x =, show that the sum of ay four cosecutive G-umbers is a Lucas umber. H-28 Proposed by P. Erdos, Budapest, Hugary. Assume a - ^! - a k! x (a, s> 2, < i ^ k) "l is a iteger. Show that the max > a. -o? i=l where the maximum is to be take with respect to all choices of the a. f s ad k. H-29 Proposed by L Carlitz, Duke Uiversity, Durham, North Carolia. Put + a - B +l u =, a - B $ where a = 3 = #/3 = z. Determie the coefficiets C(,k) such that "* = E C( ' k)u -k + l k=l ( * «//-<?/# Proposed by G. Wulczy, Buckell Uiversity, Lewisburg, Pesylvaia. is a square. Show that a positive iteger is a Lucas umber if ad oly if or H-2 Proposed by S. Krishma, Orissa, Idia. A. Show that f J is of the form 2 3 k + 2 whe is prime ad > 3. / 2-2 \ B. Show that f ~ f ) is of the form 3 k - 2 -, whe is prime. (?) ml represets the biomial coefficiet,. f (S _ A\ H-22 Proposed by V. Hoggatt, Jr., Sa Jose State Uiversity, Sa Jose, Califoria. Let be a positive iteger. Cosider edge-coected squares. How may cofiguratios are there if each row starts k squares to the right of the row above? (k deotes a o-egative iteger.)

3 74 ADVANCED PROBLEMS AND SOLUTIONS [Feb. H-23 Proposed by V. Hoggatt, Jr., Sa Jose State Uiversity, Sa Jose, Califoria. A. Let A be the left adjusted Pascal triagle, with rows ad colums ad T s above the mai diagoal. Thus A ' - ' 2 T T Fid A A where A represets the traspose of matrix, A B. Let C = 2 X where the i colum of C is the i row of Pascal's triagle adjusted to the,-p mai diagoal ad the other etries are T s. Fid C A. H-24 Proposed by Karst, Uiversity of Arizoa, Tucso, Arizoa. Let x = y 2 + z 2 be the first prime i a sequece of primes i A. P. ad x = (y ) 2 + (z ) 2 the first prime i aother sequece of primes i A.P. where both sequeces have the same commo differece. The secod member after the prime i the first sequece is divisible by 7 ad has a factor which is the square of a 3-digit prime; the secod member before the first prime i the secod sequece is also divisible by 7, ad its first three digits a r e a permutatio of the last three digits which form a perfect square. The commo differece cosists of prime factors, each of them smaller tha 7. Fid x, y, ad z. SOLUTIONS AN OLD FRIEND REVISITED H-8 Proposed by G. Ledi, Jr., Sa Fracisco, Califoria. Solve the differece equatio with Ci = a, C 2 = b, ad F, the Fiboacci umber.

4 9^3] ADVANCED PROBLEMS AND SOLUTIONS Solutio by Clyde A. Bridger, Sprigfield, Illiois. 75 Write the followig series of equatios, begiig with = l, C 3 = F 3 C 2 + a C 4 = F 4 C 3 + C 2 We see at oce that = F +2 C + +l b a C 3 = F a b + a = - F 3 C 4 = F 4 (F 3 b + a) + b a F 3 - F4 etc. So the solutio i determiat form is ""+2 b a F 3 - F 4 - F 5 "+ - *+2 as may be verified by expadig i terms of the miors of the last row. The ratio of two adjacet solutios of the differece equatio ca be developed ito a cotiued fractio. Write, usig the above sets of equatios, ' F3 + E c 4 c 3 F 4, C +2 C + TP -I +2 F _,, F 3 + f

5 76 ADVANCED PROBLEMS AND SOLUTIONS [Feb. Also solved by R. Whitey. ANOTHER O L D TIMER H-8 Proposed by H. E. Hutley, Hutto, Somerset, U.K. Fid the sides of a tetrahedro, the faces of which are all scalee triagles similar to each other, ad havig sides of itegral legths. Solutio by the Proposer. The iterestig article, "Mystery Puzzle ad Phi, by Marvi H. Holt (Fiboacci Quarterly, Vol. 3, No. 2, p. 35) cotais a solutio. See H. E. Hutley s The Divie Proportio, Dover, New York, N. Y., 97, pp. 8-9, Sectio etitled "The Tetrahedro Problem. " SHADES OF THE PAST H-86 Proposed by V. E. Hoggatt, Jr., Sa Jose State Uiversity, Sa Jose, Calif. (Corrected) P q Let p,q be itegers such that p + q ^ l, q ^ O ; show that if x (x - ) M - = has roots r l 9 r 2, r, ad (x - l ) p - x p = has roots si, l s p+q p+q, the s. i for i =, 2,, p + q. Solutio by L Carlitz, Duke Uiversity, Durham, North Carolia. Presumably the problem should read: Show that if x p (x - l ) q - = has roots r l9 r 2,, r ad (y - l ) p + q - y p = J J p+q has roots Si, s 2,, s,, the the roots ca be so umbered that p+q' rp + Q i s? (i =, 2, ', p + q)

6 973]' ADVANCED PROBLEMS AND SOLUTIONS 77 Proof. Cosider the trasformatio This implies x - l - y - ' x Hece S if x satisfies x^(x - l ) q =, we get y ^ = x Q x = P+q (x - l ) q x P (x - l ) q = XP + Q This evidetly yields the stated result, P A R T I A L SOLUTION H-25 Proposed by Staley Rabiowitz, Far Rockaway, New York. Defie a sequece of positive itegers to be left-ormal if give ay strig of digits, there exists a member of the give sequece begiig with this strig of digits, ad defie the sequece to be right-ormal if there exists a member of the sequece edig with this strig of digits. Show that the sequeces whose but ot right-ormal. terms are give by the followig are left-ormal a. P(), where P(x) is a polyomial fuctio with itegral coefficiets. b. P, where P is the prime. s F c. l d. F, where F is the Fiboacci umber. Partial Solutio by R. Whitey, Lock Have State College, Lock Have, Pesylvaia. b. The article "Iitial Digits for the Sequece of P r i m e s, " by R. E. Whitey (Amer. Math. Mothly, Vol. 79, No. 2, 972, pp. 5-52) established apositive relative logarithmic desity for the set of primes with iitial digit sequece ( a, a _-, ", a-} i the set of primes. Thus P is left-ormal. O the other had, o member of P eds i " 4, " so P is ot right-ormal. & I believe that the left-ormality of F Editorial Note ca also be established with a desity argumet. The followig list represets those problems for which o solutios have bee submitted. Let's fight problem pollutio! H-76, H-84, H-87, H-9, H-9, H-84, H-, H-, H-3, H-4, H-5, H-6* H-22, H-25 (partial), H-3, H-46, H-48, H-52, H-7, H-74, H-79, H-82. This list represets problems less tha or equal to H-85.