2019 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 the set of positive integers. A function satisfies the equation for all positive integers . Given this information, determine all possible values of .
Problem 2
Let be a cyclic quadrilateral satisfying . The diagonals of intersect at . Let be a point on side satisfying . Show that line bisects .
Problem 3
Let be the set of all positive integers that do not contain the digit in their base- representation. Find all polynomials with nonnegative integer coefficients such that whenever .
Day 2
Problem 4
Let be a nonnegative integer. Determine the number of ways that one can choose sets , for integers with , such that: for all , the set has elements; and whenever and .
Problem 5
Two rational numbers and are written on a blackboard, where and are relatively prime positive integers. At any point, Evan may pick two of the numbers and written on the board and write either their arithmetic mean or their harmonic mean on the board as well. Find all pairs such that Evan can write on the board in finitely many steps.
Problem 6
Find all polynomials with real coefficients such that holds for all nonzero real numbers satisfying .
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
2019 USAJMO (Problems • Resources) | ||
Preceded by 2018 USAMO |
Followed by 2020 USAMO | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |