# NUMBER THEORY

**2006-A-3. **Let 1, 2, 3, · · · , 2005, 2006, 2007,
2009, 2012, 2016, · · · be a sequence defined by x_{k} = k for

k = 1, 2, · · · , 2006 and
for k
≥ 2006. Show that the sequence
has 2005 consecutive terms

each divisible by 2006.

**2005-A-1. **Show that every positive integer is a sum of one or more numbers of
the form
, where

r and s are nonnegative integers and no summand divides another. (For example,
23 = 9 + 8 + 6.)

**2005-B-2. **Find all positive integers n,
such that
and

**2005-B-4.** For positive integers m and n, let f(m, n) denote the number of n−tuples

of integers such that
Show that f(m, n) = f(n,m).

**2004-A-1. **Basketball star Shanille O’Keal’s team statistician keeps track of the
number, S(N), of

successful free throws she has made in her first N attempts of the season. Early
in the season, S(N) as

less than 80% of N, but by the end of the season, S(N) was more than 80% of N.
Was there necessarily a

moment in between when S(N) was exactly 80% of N?

**2004-A-3.** Define a sequence
and thereafter by the condition that

for all n ≥ 0. Show that u_{n} is an integer for all n. (By convention, 0! = 1.)

**2004-B-2.** Let m and n be positive integers. Show that

**2004-B-6.** Let A be a non-empty set of positive integers, and let N(x) denote the
number of elements

of A not exceeding x. Let B denote the set of positive integers b that can be
written in the form b = a − a'

with and . Let
be the members of B, listed in
increasing order. Show that if the

sequence
is unbounded, then

**2003-A-1.** Let n be a fixed positive integer. How many ways are there to write n
as a sum of positive

integers,

with k an arbitrary positive integer, and
For example,
with n = 4, there are

four ways: 4, 2+2, 1+1+2, 1+1+1+1.

**2003-A-6.** For a set S of nonnegative integers, let
denote the number of
ordered pairs (s_{1}, s_{2})

such that s_{1} ∈ S, s_{2} ∈ S, s_{1}
≠ s_{2}, and s_{1} + s_{2} = n. Is it possible to partition
the nonnegative integers into

two sets A and B in such a way that
for all n?

**2003-B-2. **Let n be a positive integer. Starting
with the sequence
for a new sequence

of n − 1 entries
by taking the averages of two consecutive entries in the first
sequence.

Repeat the averaging of neighbours on the second sequence to obtain a sequence
of n−2 entries and continue

until the final sequence consists of a single number x_{n}. Show that x_{n} < 2/n.

**2003-B-3. **Show that for each positive integer n,

(Here lcm denotes the least common multiple, and [x] denotes the greatest
integer ≤ x.)

**2003-B-4.** Let where a, b, c, d, e are

integers, a ≠ 0. Show that if r_{1} + r_{2} is a rational number, and if
then r_{1}r_{2} is a rational

number.

**2002-A-3.** Let n ≥ 2 be an integer and T_{n} 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 T_{n} − n is always even.

**2002-A-5. **Define a sequence by
, together with the rules
for

each integer n ≥ 0. Prove that every positive rational number appears in the set

**2002-A-6.** Fix an integer b ≥ 2. Let f(1) = 1, and f(2) = 2, and for each n
≥ 3,
define f(n) = nf(d),

where d is the number of base−b digits of n. For which values of b does

converge?

**2002-B-5.** A palindrome in base b is a positive integer whose base−b digits read
the same forwards

and backward; for example, 2002 is a 4-digits palindrome in base 10. Note that
200 is not a palindrome in

base 10, but it is the 3-digit palindrome 242 in base 9, and 404 in base 7.
Prove that there is an integer

which is a 3-digit palindrome in base b for at least 2002 different values of b.

**2001-A-5. **Prove that there are unique positive integers a, n such that

**2001-B-1.** Let n be an even positive integer. Write the numbers 1, 2, · · · , n^{2}
in the squares of an n×n

grid so that the kth row, from right to left is

(k − 1)n + 1, (k − 1)n + 2, · · · , (k − 1)n + n .

Colour the squares of the grid so that half the squares in each row and in each
column are red and the other

half are black (a checkerboard colouring is one psosibility). Prove that for
each such colouring, the sum of

the numbers on the red squares is equal to the sum of the numbers in the black
square.

**2001-B-3.** For any positive integer n let
denote the closest integer to
Evaluate

**2001-B-4. **Let S denote the set of rational numbers different from −1, 0 and 1.
Define f : S → S by

f(x) = x − (1/x). Prove or disprove that

where (n times). (Note: f(S) denotes the set of all values f(s) for s ∈ S,)

**2000-A-2. **Prove that there exist infinitely many integers n such that n, n + 1,
and n + 2 are each the

sum of two squares of integers. (Example: 0 = 0^{2} + 0^{2}, 1 = 0^{2} + 1^{2}, and 2 = 1^{2} +
1^{2}.)

**2000-B-1.** Let a_{j} , b_{j} , and c_{j} be integers for 1
≤ j ≤ N. Assume, for each j, that
at least one of a_{j} , b_{j} ,

c_{j} is odd. Show that there exist integers r, s, t such that ra_{j} + sb_{j} + tc_{j} is
odd for at least 4N/7 values of j,

1 ≤ j ≤ N.

**2000-B-2.** Prove that the expression

is an integer for all pairs of integers n ≥ m ≥ 1. [Here
and gcd (m, n) is the greatest common

divisor of m and n.]

**2000-B-5. **Let
be a finite set of positive integers. We define sets
of positive integers as

follows:

Integer a is in
if and only if exactly one of a − 1 or a is in S_{n}.

Show that there exists infinitely many integers N for which