USAJMO 2018

For each positive integer $n$ , find the number of $n$ -digit positive integers that satisfy both of the following conditions:

• no two consecutive digits are equal, and
• the last digit is a prime.

Let $a,b,c$ be positive real numbers such that $a+b+c=4\sqrt[3]{abc}$. Prove that $$2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.$$

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 $EQ = FR$ .

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 $p$ .

Proposed by Ankan Bhattacharya

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.