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 ![\[\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(n)\ldots))=\frac{n^2}{f(f(n))}\]](http://latex.artofproblemsolving.com/2/e/7/2e73ff2f75b103e9d6a4004a42e4ed7f4ee64a67.png)
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 ![\[\frac{P(x)}{yz}+\frac{P(y)}{zx}+\frac{P(z)}{xy}=P(x-y)+P(y-z)+P(z-x)\]](http://latex.artofproblemsolving.com/2/1/b/21b97a1d7f99072ab6e0a4d493fbfa30f7e7d32a.png)
holds for all nonzero real numbers 
satisfying 
.
