2018 USAMO Problems
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
Let be positive real numbers such that . Prove that
Problem 2
Find all functions such that
for all with
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
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 .
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
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
2018 USAMO (Problems • Resources) | ||
Preceded by 2017 USAMO Problems |
Followed by 2019 USAMO Problems | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |