USAJMO 2018

For each positive integer $n$, find the number of $n$-digit positive integers for which no two consecutive digits are equal and the last digit is prime.

Proposed by Zachary Franco, Zuming Feng.

Let $a,b,c\gt 0$ satisfy $a+b+c=4\sqrt[3]{abc}$. Prove that

$$2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.$$

Proposed by Titu Andreescu.

Let $ABCD$ be a quadrilateral inscribed in circle $\omega$ with $AC\perp BD$. Let $E,F$ be the reflections of $D$ across $BA,BC$, and let $P=BD\cap EF$. The circumcircles of $EPD,FPD$ meet $\omega$ again at $Q,R$, respectively, different from $D$. Prove that $EQ=FR$.

Proposed by Ray Li.

Find all real $x$ for which there is a triangle $ABC$ of circumradius $2$, with $\angle ABC\ge90^\circ$, such that $x^4+ax^3+bx^2+cx+1=0$, where $a=BC$, $b=CA$, $c=AB$.

Proposed by Titu Andreescu.

Let $p$ be prime and $a_1,\ldots,a_p$ be integers. Prove that some integer $k$ makes $a_1+k,a_2+2k,\ldots,a_p+pk$ have at least $p/2$ distinct remainders modulo $p$.

Proposed by Ankan Bhattacharya.

Karl starts with cards $1,2,\ldots,n$ in any order. A pair $(a,b)$ is swapped if $a\gt b$ and card $a$ is left of card $b$. He moves card 1 to its opposite position: if it had $i$ cards to its left, it now has $i$ cards to its right. He does the same for cards $2,3,\ldots,n$, each exactly once in that order.

For $n=4$, the source example is

$$3142\longrightarrow3412\longrightarrow2341\longrightarrow2431\longrightarrow2341,$$

with three swapped pairs initially and finally. Prove that the final order always has the same number of swapped pairs as the initial order.

Proposed by Maria Monks Gillespie.