# 2018 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

For each positive integer $n$, find the number of $n$-digit positive integers that satisfy both of the following conditions: $\bullet$ no two consecutive digits are equal, and $\bullet$ the last digit is a prime.

### Problem 2

Let $a,b,c$ be positive real numbers such that $a+b+c=4\sqrt{abc}$. Prove that $$2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.$$ Solution

### Problem 3

( $*$) Let $ABCD$ be a quadrilateral inscribed in circle $\omega$ with $\overline{AC} \perp \overline{BD}$. Let $E$ and $F$ be the reflections of $D$ over lines $BA$ and $BC$, respectively, and let $P$ be the intersection of lines $BD$ and $EF$. Suppose that the circumcircle of $\triangle EPD$ meets $\omega$ at $D$ and $Q$, and the circumcircle of $\triangle FPD$ meets $\omega$ at $D$ and $R$. Show that $EQ = FR$.

## 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

Triangle $ABC$ is inscribed in a circle of radius 2 with $\angle ABC \geq 90^\circ$, and $x$ is a real number satisfying the equation $x^4 + ax^3 + bx^2 + cx + 1 = 0$, where $a=BC,b=CA,c=AB$. Find all possible values of $x$.

### Problem 5

Let $p$ be a prime, and let $a_1, \dots, a_p$ be integers. Show that there exists an integer $k$ such that the numbers $$a_1 + k, a_2 + 2k, \dots, a_p + pk$$produce at least $\tfrac{1}{2} p$ distinct remainders upon division by $p$.

### Problem 6

Karl starts with $n$ cards labeled $1,2,3,\dots,n$ lined up in a random order on his desk. He calls a pair $(a,b)$ of these cards swapped if $a>b$ and the card labeled $a$ is to the left of the card labeled $b$. For instance, in the sequence of cards $3,1,4,2$, there are three swapped pairs of cards, $(3,1)$, $(3,2)$, and $(4,2)$.

He picks up the card labeled 1 and inserts it back into the sequence in the opposite position: if the card labeled 1 had $i$ card to its left, then it now has $i$ cards to its right. He then picks up the card labeled $2$ and reinserts it in the same manner, and so on until he has picked up and put back each of the cards $1,2,\dots,n$ exactly once in that order. (For example, the process starting at $3,1,4,2$ would be $3,1,4,2\to 3,4,1,2\to 2,3,4,1\to 2,4,3,1\to 2,3,4,1$.)

Show that, no matter what lineup of cards Karl started with, his final lineup has the same number of swapped pairs as the starting lineup.

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 2018 USAJMO (Problems • Resources) Preceded by2017 USAJMO Followed by2019 USAJMO 1 • 2 • 3 • 4 • 5 • 6 All USAJMO Problems and Solutions