2019 USAJMO Problems

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 $a+b$ bowls arranged in a row, numbered $1$ through $a+b$ , where $a$ and $b$ are given positive integers. Initially, each of the first $a$ bowls contains an apple, and each of the last $b$ bowls contains a pear.

A legal move consists of moving an apple from bowl $i$ to bowl $i+1$ and a pear from bowl $j$ to bowl $j-1$ , provided that the difference $i-j$ is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first $b$ bowls each containing a pear and the last $a$ bowls each containing an apple. Show that this is possible if and only if the product $ab$ is even.

Solution

Problem 2

Let $\mathbb Z$ be the set of all integers. Find all pairs of integers $(a,b)$ for which there exist functions $f:\mathbb Z\rightarrow\mathbb Z$ and $g:\mathbb Z\rightarrow\mathbb Z$ satisfying $f(g(x))=x+a\quad\text{and}\quad g(f(x))=x+b$ for all integers $x$ .

Solution

Problem 3

$(*)$ Let $ABCD$ be a cyclic quadrilateral satisfying $AD^2+BC^2=AB^2$ . The diagonals of $ABCD$ intersect at $E$ . Let $P$ be a point on side $\overline{AB}$ satisfying $\angle APD=\angle BPC$ . Show that line $PE$ bisects $\overline{CD}$ .

Solution

Day 2

Problem 4

Problem 5

Problem 6

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

2019 USAJMO (Problems • Resources)
Preceded by 2018 USAJMO	Followed by 2020 USAJMO
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2019 USAJMO Problems

Contents

Day 1

Problem 1

Problem 2

Problem 3

Day 2

Problem 4

Problem 5

Problem 6