1986 USAMO Problems
Problems from the 1986 USAMO.
Problem 1
Does there exist 14 consecutive positive integers each of which is divisible by one or more primes
from the interval
?
Does there exist 21 consecutive positive integers each of which is divisible by one or more primes
from the interval
?
Problem 2
During a certain lecture, each of five mathematicians fell asleep exactly twice. For each pair of mathematicians, there was some moment when both were asleep simultaneously. Prove that, at some moment, three of them were sleeping simultaneously.
Problem 3
What is the smallest integer , greater than one, for which the root-mean-square of the first
positive integers is an integer?
The root-mean-square of
numbers
is defined to be
Problem 4
Two distinct circles and
are drawn in the plane. They intersect at points
and
, where
is the diameter of
. A point
on
and inside
is also given.
Using only a "T-square" (i.e. an instrument which can produce a straight line joining two points and the perpendicular to a line through a point on or off the line), find a construction for two points and
on
such that
is perpendicular to
and
is a right angle.
Problem 5
By a partition of an integer
, we mean here a representation of
as a sum of one or more positive integers where the summands must be put in nondecreasing order. (E.g., if
, then the partitions
are
,
,
, and
).
For any partition , define
to be the number of
's which appear in
, and define
to be the number of distinct integers which appear in
. (E.g., if
and
is the partition
, then
and
).
Prove that, for any fixed , the sum of
over all partitions of
of
is equal to the sum of
over all partitions of
of
.
See Also
1986 USAMO (Problems • Resources) | ||
Preceded by 1985 USAMO |
Followed by 1987 USAMO | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.