Difference between revisions of "2019 USAJMO Problems"
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− | {{USAJMO box|year=2019|before=[[2018 USAJMO Problems]]|after=[[2020 | + | {{USAJMO box|year=2019|before=[[2018 USAJMO Problems]]|after=[[2020 USAJMO Problems]]}} |
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 01:38, 10 March 2024
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
There are bowls arranged in a row, numbered
through
, where
and
are given positive integers. Initially, each of the first
bowls contains an apple, and each of the last
bowls contains a pear.
A legal move consists of moving an apple from bowl to bowl
and a pear from bowl
to bowl
, provided that the difference
is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first
bowls each containing a pear and the last
bowls each containing an apple. Show that this is possible if and only if the product
is even.
Problem 2
Let be the set of all integers. Find all pairs of integers
for which there exist functions
and
satisfying
for all integers
.
Problem 3
Let
be a cyclic quadrilateral satisfying
. The diagonals of
intersect at
. Let
be a point on side
satisfying
. Show that line
bisects
.
Day 2
Problem 4
Let
be a triangle with
obtuse. The
-excircle is a circle in the exterior of
that is tangent to side
of the triangle and tangent to the extensions of the other two sides. Let
be the feet of the altitudes from
and
to lines
and
, respectively. Can line
be tangent to the
-excircle?
Problem 5
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 6
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.
2019 USAJMO (Problems • Resources) | ||
Preceded by 2018 USAJMO Problems |
Followed by 2020 USAJMO Problems | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.