Difference between revisions of "2018 USAMO Problems"
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Note: For any geometry problem whose statement begins with an asterisk (<math>*</math>), the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction. | Note: For any geometry problem whose statement begins with an asterisk (<math>*</math>), the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction. | ||
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==Problem 1== | ==Problem 1== |
Revision as of 11:40, 21 April 2018
Contents
Day 1
Note: For any geometry problem whose statement begins with an asterisk (), the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction.
Problem 1
For each positive integer , find the number of
-digit positive integers that satisfy both of the following conditions:
no two consecutive digits are equal, and
the last digit is a prime.
Solution
Problem 1
Let be positive real numbers such that
. Prove that
Solution
Problem 2
Find all functions such that
for all
with
Solution
Problem 3
For a given integer let
be the set of positive integers less than
that are relatively prime to
Prove that if every prime that divides
also divides
then
is divisible by
for every positive integer
Solution
Day 2
Note: For any geometry problem whose statement begins with an asterisk (), the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction.
Problem 4
Let be a prime, and let
be integers. Show that there exists an integer
such that the numbers
produce at least
distinct remainders upon division by
.
Solution
Problem 5
In convex cyclic quadrilateral we know that lines
and
intersect at
lines
and
intersect at
and lines
and
intersect at
Suppose that the circumcircle of
intersects line
at
and
, and the circumcircle of
intersects line
at
and
, where
and
are collinear in that order. Prove that if lines
and
intersect at
, then
Solution
Problem 6
Let be the number of permutations
of the numbers
such that the
ratios
for
are all distinct. Prove that
is odd for all
Solution