# ELEMENTARY PROBLEMS AND SOLUTIONS

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1 Edited by A. P. HILLMAN Assistant Editors GLORIA C. PADILLA CHARLES R. WALL Send all communications regarding ELEMENTARY PROBLEMS SOLUTIONS to PROFESSOR A. P. HILLMAN; 709 Solano Dr., S.E.,; Albuquerque, NM Each solution or problem should be on a separate sheet (or sheets). Preference will be given to those typed with double spacing in the format used below. Solutions should be received within four months of the publication date. Proposed problems should be accompanied by their solutions. DEFINITIONS The Fibonacci numbers F n the Lucas numbers L n satisfy F n+2 L n+2 = Fn f n» ^0 = > ^ 1 = 1 = L n+1 + L n> L 0 = 2 > L i = 1 ' Also, a 3 designate the roots (1 + v5)/2 (1 - v5)/2, respectively, of x 2 - x - 1 = 0. PROBLEMS PROPOSED IN THIS ISSUE B-520 Proposed by Charles R. Wall, Trident Technical College, Charleston, SC (a) Suppose that one has a table for multiplication (mod 10) in which a, b,,.., j have been substituted for 0, 1,..., 9 in some order. How many decodings of the substitution are possible? (B) Answer the analogous question for a table of multiplication (mod 12). B-521 Proposed by Charles R. Wall, Trident Technical College, Charleston, SC See the previous problem. Find all moduli m > 1 for which the multiplication (mod m) table can be decoded in only one way. B-522 Proposed by loan Tomescu, University of Bucharest, Romania Find the number A(n) of sequences (a l5 a 2,..., a k ) of integers ai satisfying 1 < a t < a i + 1 < n a i ai E 1 (mod 2) for i = 1, 2,..., k - 1. [Here k is variable, but of course 1 < k < n. For example, the three allowable sequences for n = 2 are (1), (2), (1,2).] B-523 Proposed by Laszlo Cseh Imre Merenyi, Cluj, Romania Let p, a Q, a-t,..., a n be integers with p a positive prime such that gcd(a 0, p) = 1 = gcd(a n, p). 1984] 183

2 Prove that in {0, 1,..., p - 1} there are as many solutions of the congruence as there are of the congruence a n x n + a n _ 1 x n ~ ciyz + a 0 = 0 (mod p) a 0 x n + a x x n ~ x + '" + (Zn-iX + a = 0 (mod p ). B-524 Proposed by Herta T. Freitag, Roanoke, VA Let S n = Ff n _ 1 + F n F n. 1 (F 2n _ 1 + F n 2 ) + 3F n F n + 1 (F 2n _ 1 + F F _i). Show that S n is the square of a Fibonacci number. B-525 Proposed by Walter Blumberg, Coral Springs, FL Let x,y, z be positive integers such that 2 X - 1 = / s # > 1. Prove that z = 1. SOLUTIONS Fibonacci-Lucas Centroid B-496 Proposed by Stanley Rabinowitz, Digital Equip. Corp., Merrimack, NH Show that the centroid of the triangle whose vertices have coordinates is (F n + h, L n+h ). (F n, L n ), (F n+1, L n+1 ), (F n + 6, n + 6 ) Solution, independently, by Walter Blumberg, Coral Springs, FL; Wray G. Brady, Slippery Rock, PA; Paul S. Bruckman, Sacramento, CA; Laszlo Cseh, Cluj, Romania; Leonard Dresel, Reading, Engl; Herta T. Freitag, Roanoke, VA; L. Kuipers, Switzerl; Stanley Rabinowitz, Merimack, NH; Imre Merenyi, Cluj, Romania; John W. Milsom, Butler, PA; Bob Prielipp, Oshkosh, WI; Sahib Singh, Clarion, PA; Lawrence Somer, Washington, CD; Gregory Wulczyn, Lewisburg, PA. The coordinates (x, y) of the centroid are given by 3x = F n + F n+1 + Fn+6 = F n F n + h + F n+5 ~ ^n + 2 "*" ^n+h + ^n+3 ^n + h similarly, Hence, the centroid is (F n+h, L n+h ). 3y = L n + L n+1 + L n+6 = 3L n+ h. Area of a Fibonacci-Lucas Triangle B-kS7 Proposed by Stanley Rabinowitz, Digital Equip. Corp., Merrimack, NH For d an odd positive integer, find the area of the triangle with vertices (F n9 L n ), (F n + di L n+d ), (F n + 2d> L n+2d ). 184 [May

3 Solution by Paul S. Bruckman, Sacramento, CA By means of a well-known determinant formula,the area of the given triangle is given by A =i i i 1 II (In the above expression, the inner bar is the determinant symbol, the outer bar represents absolute value.) Then A = -2\(F n + 2d L n + d " Fn + d^n + ld} ~ ( F n + 2d^n ~ F n L n + 2d^ + ( F n+d^n ~ F n L n + d ) \. Using the relation this becomes: F U L V - F V L U = 2(-l) v F u _ v, (2) A = (-l) n + d F d - {-l) n F 2d + (-l) n F d = (.-l) d F d - F 2d + F d> which equals F 2d when d is odd ( equals F 2d - 2F d when d is even). Also solved by Walter Blumberg, Wray G. Brady, Leonard Dresel, Herta T. Freltag, L 0 Kuipers, Graham Lord, John W. Milsom, Bob Prielipp, Sahib Singh, Gregory Wulczyn, the proposer. Fibonacci Recursions Modulo 10 B-498 Proposed by Herta T. Freitag, Roanoke, VA Characterize the positive integers k such that, for all positive integers n, F n + F n + k = F n + 2k (mod 10). When k is odd, we have the identity F m+k - F m k = F m L k. Applying this with m = n + k 9 we have F m = F m Lk (mod 10), this will be satisfied whenever L k = 1 (mod 10). On the other h, when k is even, we have F m + k - F m _ k = L m F k, it is not possible to satisfy the given recurrence for even k. Returning to the case of odd k, the condition L k = 1 (mod 10) is equivalent to L k = 1 (mod 2) L k = 1 (mod 5). The first condition implies that k is not divisible by 3; with the help of the Binet formula for L ki the second condition reduces to 2 " = 1 (mod 5), which gives that k - 1 in a multiple of 4. Combining these results, we have k = ov k = 12t + 5 (t = 0, 1, 2, 3,... ). Also solved by Paul S. Bruckman, Laszlo Cseh, L. Kuipers, Imre Merenyi, Bob Prielipp, Sahib Singh, Lawrence Somer, Gregory Wulczyn, the proposer. 1984] 185

4 Lucas Recursions Modulo 12 B-499 Proposed by Herta T. Freitag, Roanoke, VA Do the Lucas numbers analogue of B-498. We have the identit ies L m + k Lm- k L m L k when k is odd, L m + k Lm-k 5F m F k when k is even. Hence, putting m = n + k 9 the relation L n + L n + k E L n + 2k (mod 10) leads, for odd k, to L m ~ L m L k (mod 10), so that we require L k = 1 (mod 10), leading to the same values of k as in B-498 above. Also solved by Paul S. Bruckman, Laszlo Cseh, L. Kuipers, Imre Merenyi, Bob Prielipp, Lawrence Somer, Gregory Wulczyn, the proposer. Two Kinds of Divisibility B-500 Proposed by Philip L. Mana, Albuquerque, NM Let A(n) B(n) be polynomials of positive degree with integer coefficients such that B(k)\A(k) for all integers k. Must there exist a nonzero integer h a polynomial C(n) with integer coefficients such that ha(n) = 5(n)C(n)? Solution by the proposer. Using the division algorithm multiplying by an integer In so as to make all coefficients into integers, one has ha(n) = Q(n)B(n) + R(n), (*) where Q(n) R(n) are polynomials in n with integral coefficients R(n) is either the zero polynomial or has degree less than B(n). The hypothesis that B(n)\A(n) (*) imply that B(n)\R(n) for all integers n. If R(n) is not the zero polynomial, R(n) has lower degree than B(n) so lim [R(n)/B(n)] = 0; also R(n) is zero for only a finite number of integers n. Thus 0 < R(n)/B(n) < 1 for some large enough n 9 contradicting B(n)\R(n). HenceR(n) is the zero polynomial (?v) shows that the answer is "yes." Also solved by Paul S. Bruckman L. Kuipers. Doubling Back on a Sequence B-501 Proposed by J. O. Shallit & J. P. Yamron, U.C. Berkeley, CA Let a be the mapping that sends a sequence X = (x l9 x 2,..., x, ) of length 2k to the sequence of length k, (X\A ) \X -^X 2k 9 2 2k - 1 ' > k k + 1 ) * Let V = (1, 2, 3,..., 2 h ), a 2 (7) = a(a(f)), etc. Prove that a(7), a 2 (7),..., a ~ (7) are all strictly increasing sequences. 186 [May

5 Suppose the nubmers a ls a 2, a 3, a h form a strictly increasing sequence, subject to the condition a 1 + a h = a 2 + a 3 = S, then gives hence, (a h - a x ) 2 > (a 3 - a 2 ) 2 (a h - a x ) 2 = (a 3 + a 2 ) 2-4a r+ a 1 > -4a 3 a 2 a^^. < a 2 a 3. Now any two consecutive terms of a(7) are of the form a^^, a 2 a 3, with a x + a^ = a 3 + a 2 = h, so that it follows that a(7) is a strictly increasing sequence. Next, consider a (7). To avoid a notational forest, we shall apply our method to the specific case where h = 4, with 7 = (1, 2, 3,..., 16). Then, using a dot to denote multiplication, we have where a(7) = (1-16, 2 15, 3-14,..., 8-9) a 2 (T) = ( , ,..., ) = ( , ,..., ) = (b 1 c ls b 2 e 2,..., 2^ e 4 ) (b is i 2, Z> 3, \ ) = a(l, 2, 3,..., 8) (o 13 c 2, c 3, c h ) = a(9, 10, 11,..., 16). By our previous argument, b± is strictly increasing, similarly c^ is. Thus a 2 (7) = (b%ci) is a strictly increasing sequence. Similarly, we can show that a 3 (7) = (d 1 e 1 f 1 g 19 d 2 e 2 f 2 g 2 ), where (d 2, d 2 ) = a(l, 2, 3, 4), (e l5 e 2 ) = a(5, 6, 7, 8), etc., is strictly increasing. The above arguments can be generalized to apply to any value of h, Also solved by Paul S. Bruckman, L. Kuipers, the proposers. 1984] 187

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