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NUMBER THEORY

1999-A-6. The sequence is defined by and for n ≥ 4,

Show that, for all n, an is an integer multiple of n .

1999-B-6. 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.]

1998-A-4. Let A1 = 0 and A2 = 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.

1998-B-5. 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

1998-B-6. Prove that, for any integers a, b, c, there exists a positive integer n such that
is not an integer.

1997-A-5. Let Nn denote the number of ordered n −tuples of positive integers (a1, a2, · · · , an) such
that 1/a1 + 1/a2 + · · · + 1/an = 1. Determine whether is even or odd.

1997-B-3. For each positive integer n write the sum in the form where pn and qn are
relatively prime positive integers. Determine all n such that 5 does not divide qn.

1997-B-5. Prove that for n ≥ 2,

1996-A-5. If p is a prime number greater than 3, and prove that the sum

of binomial coefficients is divisible by p2.
(For example,)

1995-A-3. The number d1d2 · · · d9 has nine (not necessarily distinct) decimal digits. The number
e1e2 · · · e9 is such that each of the nine 9-digit numbers formed by replacing just one of the digits di in
d1d2 · · · d9 by the corresponding digit ei (1 ≤ i ≤ 9) is divisible by 7. The number f1f2 · · · f9 is related
to e1e2 · · · e9 in the same way; that is, each of the nine numbers formed by replacing one of the ei by
the corresponding fi is divisible by 7. Show that, for each i, di − fi is divisible by 7. [For example, if
d1d2 · · · d9 = 199501996, then e6 may be 2 or 9, since 199502996 and 199509996 are multiples of 7.]

1995-A-4. 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 x1,
x2, · · ·, xn satisfy

for k = 1, 2, · · · n .

1994-B-1. 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.)

1994-B-6. For any integer a, set Show that for 0 ≤ a, b, c, d ≤ 99,

(mod10100)

implies {a, b} = {c, d}.

1993-A-4. 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 xi’s equal to a sum of some yj ’s.

1993-B-1. Find the smallest positive integer n such that for every integer m, with 0 < m < 1993, there
exists an integer k for which

1993-B-5. Show there do not exist four points in the Euclidean plane such that the pairwise distances
between the points are all odd integers.

1993-B-6. 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.

1992-A-3. For a given positive integer m, find all triples (n, x, y) of positive integers, with n relatively
prime to m, which satisfy

1992-A-5. For each positive integer n, let

Show that there do not exist positive integers k and m such that

for 0 ≤ j ≤ m − 1 .

1989-A-1. 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?

1988-B-1. 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.

1988-B-6. 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 tn = n(n + 1)/2 with n in {0, 1, 2, · · ·}.

1987-A-2. 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 10n-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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