Download Problem List 3

Putnam Practice: Number theory
The first few problems here are exercises in using the pigenhole principle: if you have n+1 pigeons and n holes, you have to put at least two
pigeons in the same hole. More generally, if you have mn + 1 pigeons
and n holes, you have to put at least m + 1 pigeons in the same hole.
(1)
Let A be any set of 20 distinct integers chosen from the
arithmetic progression 1, 4, 7, . . . , 100. Prove that there must
be two distinct integers in A whose sum is 104.
(2)
A lattice point in the plane is a point (x, y) such that both
x and y are integers. Find the smallest number n such that
given n lattice points in the plane, there exist two whose
midpoint is also a lattice point.
(3)
Show that some multiple of 2017 is a number all of whose
digits are zeroes and ones.
(4)
Let α be an irrational number. Prove that there are infinitely many integer pairs (h, k) with k > 0 such that
α − h < 1 .
k k2
(5)
Prove that, for every set X = {x1 , x2 , . . . , xn } of n real
numbers, there exists a non-empty subset S of X and an
integer m such that
X
1
s ≤
.
m +
n+1
s∈S
Next are three problems involving congruences and Fermat’s little theorem. Let n be a positive integer. The Euler function φ(n) is equal to
the number of positive integers k = 1, 2, . . . , n−1 relatively prime to n.
The Euler function can be easily calculated using that φ(pk ) = pk −pk−1
for p prime, and φ(mn) = φ(m)φ(n) when m and n are relatively prime.
Fermat’s theorem states that if a is relatively prime to n, then
aφ(n) ≡ 1 (mod n).
There is also the Chinese remainder theorem: if m and n are two
relatively prime integers, then the system of congruences
x ≡ a (mod m) ,
x ≡ b (mod n)
has a unique solution mod mn.
(6)
(7)
17
Deterrmine the last digit of 1717 .
Find the last three digits of 79999 .
1
2
2
22
Show that for every n the sequence 2, 22 , 22 , 22 ,. . . (mod
n) is eventually constant.
Here are two more “themed” problems, involving Pell’s equation: if d
is a positive non-square integer, then solutions of
(8)
x2 − dy 2 = 1
with positive x and y are (xk , yk ), k = 1, 2, . . . where (x1 , y1 ) is the
solution with smallest x and y and
√
√
xk + yk d = (x1 + y1 d)k .
(9)
(10)
Show that there are infinitely many integers such that n2 +
(n + 1)2 is a perfect square.
Let m and n be two integers (1 ≤ m ≤ 1981, 1 ≤ n ≤ 1981)
that satisfy the equation
(n2 − nm − m2 )2 = 1.
Find the maximal value for n2 + m2 .
And here are three more (with no particular theme), the last one very
hard.
(11)
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 expressible as xy + xz + yz + 1, with x, y,
and z positive integers.
(12)
Let S be the set of all ordered triples (p, q, r) of prime
numbers for which at least one rational number x satisfies px2 + qx + r = 0. Which primes appear in seven or
more elements of S?
(13)
Let p be an odd prime number such that p ≡ 2 (mod 3).
Define a permutation π of the residue classes modulo p by
π(x) ≡ x3 (mod p). Show that π is an even permutation if
and only if p ≡ 3 (mod 4).