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NUMBER THEORY
1999A6. The sequence is defined by and for n ≥ 4,
Show that, for all n, a_{n} is an integer multiple of n .
1999B6. Let S be a finite set of integers, each greater than 1. Suppose that
for each integer n there
is some s ∈ S such that gcd (s, n) = 1 or gcd (s, n) = s. Show that there exists
s, t ∈ S such that gcd (s, t)
is prime. [Here gcd (a, b) denotes the greatest common divisor of a and b.]
1998A4. Let A_{1} = 0 and A_{2} = 1. For n > 2, the number A _{n} is defined by
concatenating the
decimal expansions of
and
from left to right. For example,
and so forth. De termine all n such that 11 divides A _{n}.
1998B5. Let N be the positive integer with 1998 decimal digits, all of them 1;
that is, N = 1111 · · · 11
(1998 digits). Find the thousandth digit after the decomal point of
1998B6. Prove that, for any integers a, b, c, there exists a positive integer
n such that
is not an integer.
1997A5. Let N_{n} denote the number of ordered n −tuples of positive integers (a_{1},
a_{2}, · · · , a_{n}) such
that 1/a_{1} + 1/a_{2} + · · · + 1/a_{n} = 1. Determine whether
is even or odd.
1997B3. For each positive integer n write the sum
in the form
where p_{n} and q_{n} are
relatively prime positive integers. Determine all n such that 5 does not divide
q_{n}.
1997B5. Prove that for n ≥ 2,
1996A5. If p is a prime number greater than 3, and prove that the sum
of binomial coefficients is divisible by p^{2}.
(For example,)
1995A3. The number d_{1}d_{2} · · · d_{9} has nine (not necessarily distinct) decimal
digits. The number
e_{1}e_{2} · · · e_{9} is such that each of the nine 9digit numbers formed by replacing
just one of the digits d_{i} in
d_{1}d_{2} · · · d_{9} by the corresponding digit e_{i} (1 ≤ i ≤ 9) is divisible by 7. The
number f_{1}f_{2} · · · f_{9} is related
to e_{1}e_{2} · · · e_{9} in the same way; that is, each of the nine numbers formed by
replacing one of the e_{i} by
the corresponding f_{i} is divisible by 7. Show that, for each i, d_{i} − f_{i} is
divisible by 7. [For example, if
d_{1}d_{2} · · · d_{9} = 199501996, then e_{6} may be 2 or 9, since 199502996 and 199509996
are multiples of 7.]
1995A4. Suppose we have a necklace of n beads. Each bead is labelled with an
integer and the sum
of all these labels is n − 1. Prove that we can cut the necklace to form a
string whose consecutive labels x_{1},
x_{2}, · · ·, x_{n} satisfy
for k = 1, 2, · · · n .
1994B1. Find all positive integers that are within 250 of exactly 15 perfect
squares . (Note: A perfect
square is the square of an integer; that is, a member of the set {0, 1, 4, 9,
16, · · · , }. a is within n of b if
b − n ≤ a ≤ b + n.)
1994B6. For any integer a, set Show that for 0 ≤ a, b, c, d ≤ 99,
(mod10100)
implies {a, b} = {c, d}.
1993A4. Let
be positive integers each of which is less
than or equal to 93. Let
be positive integers each of which is less than or equal to
19. Prove that there exists a
(nonempty) sum of some x_{i}’s equal to a sum of some y_{j} ’s.
1993B1. Find the smallest positive integer n such that for every integer m,
with 0 < m < 1993, there
exists an integer k for which
1993B5. Show there do not exist four points in the Euclidean plane such that
the pairwise distances
between the points are all odd integers.
1993B6. Let S be a set of three, not necessarily distinct, positive integers.
Show that one can
transform S into a set containing 0 by a finite number of applications of the
following rule : Select two of the
three integers, say x and y, where x ≤ y, and replace them with 2x and y − x.
1992A3. For a given positive integer m, find all triples (n, x, y) of positive
integers, with n relatively
prime to m, which satisfy
1992A5. For each positive integer n, let
Show that there do not exist positive integers k and m such that
for 0 ≤ j ≤ m − 1 .
1989A1. How many primes among the positive integers, written as usual in base
10, are such that
their digits are alternating 1’s and 0’s, beginning and ending with 1?
1988B1. A composite (positive integer) is a product ab with a and b not
necessarily distinct integers
in {2, 3, 4, · · ·}. Show that every composite is ex pressible as xy + xz + yz +
1, with x, y, and z positive
integers.
1988B6. Prove that there exist an infinite number of ordered pairs (a, b) of
integers such that for
every positive integer t the number at+b is a triangular number if and only if t
is a triangular number. (The
triangular numbers are the t_{n} = n(n + 1)/2 with n in {0, 1, 2, · · ·}.
1987A2. The sequence of digits
1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 · · ·
is obtained by writing the positive integers in order. If the 10^{n}th digit in
this sequence occurs in the part of
the sequence in hich 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).
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