PUTNAM PROBLEMS SEQUENCES, SERIES AND RECURRENCES. Notes
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1 PUTNAM PROBLEMS SEQUENCES, SERIES AND RECURRENCES Notes. x n+ = ax n has the general solution x n = x a n. 2. x n+ = x n + b has the general solution x n = x + (n )b. 3. x n+ = ax n + b (with a ) can be rewritten x n+ + k = a(x n + k) where (a )k = b and so reduces to the recurrence. 4. x n+ = ax n + bx n has different general solution depending on the discriminant of the characteristic polynomial t 2 at b. (a) If a 2 4b 0 and the distinct roots of the characteristic polynomial are r and r 2, then the general solution of the recurrence is x n = c r n + c 2 r n 2 where the constants c and c 2 are chosen so that x = c r + c 2 r 2 and x 2 = c r 2 + c 2 r 2 2. (b) If a 2 4b = 0 and r is the double root of the characteristic polynomial, then where c and c 0 are chosen so that x n = (c n + c 0 )r n x = (c + c 0 )r and x 2 = (2c + c 0 )r x n+ = ( s)x n + sx n + r can be rewritten x n+ x n = s(x n x n ) + r and solved by a previous method for x n+ x n. 6. x n+ = ax n + bx n + c where a + b can be rewritten (x n+ + k) = a(x n + k) + b(x n + k) where (a + b )k = c and solved for x n + k. 7. The general homogeneous linear recursion has the form Its characteristic polynomials is x n+k = a k x n+k + + a x n+ + a 0. t k a k t k a t a 0. Let r be a root of this polynomial of multiplicity m; then the nth term of the recurrence is a linear combination of terms of the type (c m r m + + c r + c 0 )r n. Putnam questions 206-B-. Let x 0, x, x 2,... be the sequence such that x 0 = and for n 0, x n+ = ln(e xn x n ) (as usual, the function ln is the natural logarithm). Show that the infinite series x 0 + x + x 2 +
2 converges and find its sum. 206-B-6. Evaluate ( ) k k= k n=0 k2 n A-2. Let a 0 =, a = 2, and a n = 4a n a n 2 for n 2. Find an odd prime factor of a B-5. Let P n be the number of permutations π of {, 2,..., n} such that i j = implies π(i) π(j) 2 for all i, j in {, 2,..., n}. Show that for n 2, the quantity P n+5 P n+4 P n+3 + P n does not depend on n, and find its value. 204-A-. Prove that every nonzero coefficient of the Taylor series of ( x + x 2 )e x about x = 0 is a rational number whose numerator (in lowest terms) is either or a prime number. 204-A-3. Let a 0 = 5/2 and a k = a 2 k 2 for k. Compute ) ( ak in closed form. k=0 203-B-. For positive integers n, let the number c(n) be determined by the rules c() =, c(2n) = c(n), and c(2n + ) = ( ) n c(n). Find the value of 203 c(n)c(n + 2). n= 200-B-. Is there an infinite sequence of real numbers a, a 2, a 3,... such that a m + a m 2 + a m 3 + = m for every positive integer m? 2009-B-6. Prove that for every positive integer n, there is a sequence a 0, a,..., a 2009 with a 0 = 0 and a 2009 = n such that each term after a 0 is either an earlier term plus 2 k for some nonegative integer k or of the form b mod c for some earlier positive terms b and c. (Here b mod c denotes the remainder when b is divided by c, so 0 (bmodc) < c.) 2007-B-3. Let x 0 = and for n 0, let x n+ = 3x n + x n 5. In particular, x = 5, x 2 = 26, x 3 = 36, x 4 = 72. Find a closed-form expression for x ( a means the largest integer a,) 2006-A-3. Let, 2, 3,, 2005, 2006, 2007, 2009, 202, 206, be a sequence defined by x k = k for k =, 2,, 2006 and x k+ = x k +x k 2005 for k Show that the sequence has 2005 consecutive terms each divisible by
3 2006-B-6. Let k be an integer greater than. Suppose a k > 0, and define a n+ = a n + k an for n 0. Evaluate lim n a k+ n n k A-3. Define a sequence {u n } n=0 by u 0 = u = u 2 =, and thereafter by the condition that ( ) un u det n+ = n! u n+2 u n+3 for all n 0. Show that u n is an integer for all n. (By convention, 0! =.) 2002-A-5. Define a sequence by a 0 =, together with the rules a 2n+ = a n and a 2n+2 = a n + a n+ for each integer n 0. Prove that every positive rational number appears in the set { } { an : n = a n, 2, 2, 3, 3 } 2,. 200-B-3. For any positive integer n let n denote the closest integer to n. Evaluate n= 2 n + 2 n 2 n. 200-B-6. Assume that {a n } n is an increasing sequence of positive real numbers such that lim a n /n = 0. Must there exist infinitely many positive integers n such that for i =, 2,, n? a n + a n+i < 2a n 2000-A-. Let A be a positive real number. What are the possible values of j=0 x2 j, given that x 0, x, x 2, are positive numbers for which j=0 x j = A? 2000-A-6. Let f(x) be a polynomial with integer coefficients. Define a sequence a 0, a, of integers such that a 0 = 0 and a n+ = f(a n ) for all n 0. Prove that if there exists a positive integer m for which a m = 0, then either a = 0 or a 2 = A-3. Consider the power series expansion 2x x 2 = a n x n. n=0 Prove that, for each integer n 0, there is an integer m such that a 2 n + a 2 n+ = a m. 999-A-4. Sum the series m= n= m 2 n 3 m (n3 m + m3 n ). 3
4 999-A-6. The sequence {a n } n is defined by a =, a 2 = 2, a 3 = 24, and for n 4, Show that, for all n, a n is an integer multiple of n. a n = 6a2 n a n 3 8a n a 2 n 2 a n 2 a n B-3. Let A = {(x, y) : 0 x, y }. For (x, y) A, let S(x, y) = x m y n, 2 m n 2 where the sum ranges over all pairs (m, n) of positive integers satisfying the indicated inequalities. Evaluate lim{( xy 2 )( x 2 y)s(x, y) : (x, y) (, ), (x, y) A}. 998-A-4. Let A = 0 and A 2 =. For n > 2, the number A n is defined by concatenating the decimal expansions of A n and A n 2 from left to right. For example, A 3 = A 2 A = 0, A 4 = A 3 A 2 = 0, A 5 = A 4 A 3 = 00, and so forth. Determine all n such that divides A n. 998-B-4. Find necessaary and sufficient conditions on positive integers m and n so that ( ) i/m + i/n = 0. mn i=0 997-A-6. For a positive integer n and any real number c, define x k recursively by x 0 = 0, x =, and for k 0, x k+2 = cx k+ (n k)x k k + Fix n and then take c to be the largest value for which x n+ = 0. Find x k in terms of n and k, k n. 994-A-. Suppose that a sequence a, a 2, a 3, satisfies 0 < a n a 2n + a 2n+ for all n. Prove that the series n= a n diverges. 994-A-5. Let (r n ) n 0 be a sequence of positive real numbers such that lim n r n = 0. Let S be the set of numbers representable as a sum r i + r i2 + + r i994 with i < i 2 < < i 994. Show that every nonempty interval (a, b) contains a nonempty subinterval (c, d) that does not intersect S. 993-A-2. Let (x n ) n 0 be a sequence of nonzero real numbers such that x 2 n x n x n+ = for n =, 2, 3,. Prove that there exists a real number a such that x n+ = ax n x n for all n. 993-A-6. The infinite sequence of 2 s and 3 s 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, has the property that, if one forms a second sequence that records the number of 3 s between successive 2 s, the result is identical to the given sequence. Show that there exists a real number r such that, for any n, 4.
5 the nth term of the sequence is 2 if and only if n = + rm for some nonnegative integer m. (Note: x denotes the largest integer less than or equal to x. 992-A-. Prove that f(n) = n is the only integer-valued function defined on the integers that satisfies the following conditions (i) f(f(n)) = n, for all integers n; (ii) f(f(n + 2) + 2) = n for all integers n; (iii) f(0) =. 992-A-5. For each positive integer n, let a n = { 0, if the number of s in the binary representation of n is even,, if the number of s in the binary representation of n is odd. Show that there do not exist integers k and m such that for 0 j m. a k+j = a k+m+j = a k+2m+j 99-B-. For each integer n 0, let S(n) = n m 2, where m is the greatest integer with m 2 n. Define a sequence (a k ) k=0 by a 0 = A and a k+ = a k + S(a k ) for k 0. For what positive integers A is this sequence eventually constant? 990-A-. Let and for n 3, The first few terms are T 0 = 2, T = 3, T 2 = 6, T n = (n + 4)T n 4nT n 2 + (4n 8)T n 3. 2, 3, 6, 4, 40, 52, 784, 568, 40576, Find, with proof, a formula for T n of the form T n = A n +B n, where (A n ) and (B n ) are well-known sequences. 988-B-4. Prove that if n= a n is a convergent series of positive real numbers, then so is (a n ) n/(n+). n= 985-A-4. Define a sequence {a i } by a = 3 and a i+ = 3 ai for i. Which integers between 00 and 99 inclusive occur as the last two digits in the decimal expansion of infinitely many a i? 979-A-3. Let x, x 2, x 3, be a sequence of nonzero real numbers satisfying x n = x n 2x n 2x n 2 x n 2 for n = 3, 4, 5,. Establish necessary and sufficient conditions on x and x 2 for x n to be an integer for infinitely many values of n. 975-B-6. Show that, if s n = n, then (a) n(n + ) /n < n + s n for n >, and (b) (n )n /(n ) < n s n for n > 2. 5
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