Download Set 6

PUTNAM PRACTICE—SEQUENCES AND COMBINATORICS
Problem 1
P∞
(1986) Evaluate n=0 Arccot(n2 + n + 1), where Arccot t for t ≥ 0 denotes the number θ in the interval
0 < θ ≤ π/2 with cot θ = t.
Problem 2
(1987) The sequence of digits
123456789101112131415161718192021 . . .
is obtained by writing the positive integers in order. If the 10n -th digit in this sequence occurs in the part
of the sequence in which the m-digit numbers are placed, define f (n) to be m. For example, f (2) = 2
because the 100th digit enters the sequence in the placement of the two-digit integer 55. Find, with proof,
f (1987).
Problem 3
(1990) Let
T0 = 2, T1 = 3, T2 = 6, and for n ≥ 3,
Tn = (n + 4)Tn−1 − 4nTn−2 + (4n − 8)Tn−3 .
The first few terms are
2, 3, 6, 14, 40, 152, 784, 5168, 40576.
Find, with proof, a formula for Tn of the form Tn = An + Bn , where {An } and {Bn } are well-known
sequences.
Problem 4
(1991) For each integer n ≥ 0, let S(n) = n − m2 , where m is the greatest integer with m2 ≤ n. Define a
sequence (ak )∞
k=0 by a0 = A and ak+1 = ak + S(ak ) for k ≥ 0. For what positive integers A is this sequence
eventually constant?
Problem 5
(1992) For any pair (x, y) of real numbers, a sequence (an (x, y))n≥0 is defined as follows:
a0 (x, y) = x,
(an (x, y))2 + y 2
, for n ≥ 0.
2
converges}.
an+1 (x, y) =
Find the area of the region {(x, y)|(an (x, y))n≥0
Problem 6
(1993) Let (xn )n≥0 be a sequence of nonzero real numbers such that x2n − xn xn+1 = 1 for n = 1, 2, 3, · · · .
Prove there is a real number a such that xn+1 = axn − xn−1 for all n ≥ 1.
Date: 11/22, 2013.
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PUTNAM PRACTICE—SEQUENCES AND COMBINATORICS
Problem 7
(1994) For n ≥ 1, let dn be the greatest common divisor of the entries of An − I, where
3 2
1 0
A=
and
I=
.
4 3
0 1
Show that lim dn = ∞.
n→∞
Problem 8
Pn
(1997) For each positive integers n, write the sum m=1 1/m in the form pn /qn , where pn and qn are
relative prime positive integers. Determine all n such that 5 des note divide qn .
Problem 9
(1998) Let A1 = 0 and A2 = 1. For n > 2, the number An is defined by concatenating the decimal
expansions of An−1 and An−2 from the left to right. For example A3 = A2 A1 = 10, A4 = A3 A2 = 101,
A5 = A4 A3 = 10110, and so forth. Determine all n such that 11 divides An .
Problem 10
1
(1999) Consider the power series expansion 1−2x−x
2 =
2
2
there is an integer m such that an + an+1 = am .
P∞
n=1
an xn . Prove that, for each integer n ≥ 0,
Problem 11
(1999) Sum the series
P∞
m=1
m2 n
n=1 em (n3m +m3n ) .
P∞
Problem 12
(2001) For each integer m, consider the polynomial Pm (x) = x4 − (2m + 4)x2 + (m − 2)2 . For what values
of m is Pm (x) the product of two nonconstant polynomials with integer coefficients?
Problem 13
(2001) For any positive integer n, let < n > denote the closest integer to
√
n. Evaluate
P∞
n=1
2<n> +2−<n>
.
2n
Problem 14
(2002) Let n ≥ 2 be integer and Tn be the number of non-empty subsets S of {1, 2, 3, · · · , n} with the
property that the average of the elements of S is an integer. Prove that Tn − n is always even.
Problem 15
(2003) Let n be a positive integer. Starting with the sequence 1, 12 , 13 , . . . , n1 , form a new sequence of n − 1
5
2n−1
entries 34 , 12
, . . . , 2n(n−1)
by taking the averages of two consecutive entries in the first sequence. Repeat
the averaging of neighbors on the second sequence to obtain a third sequence of n − 2 entries, and continue
until the final sequence produced consists of a single number xn . Show that xn < 2/n.