Question bank USAMO 2018

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. Source: Evan Chen’s third-party compilation, not an official publication. One editorial point; manual grading only.

Find all $f:(0,\infty)\to(0,\infty)$ such that

$$f(x+1/y)+f(y+1/z)+f(z+1/x)=1$$

for all $x,y,z\gt 0$ with $xyz=1$.

Proposed by Titu Andreescu, Nikolai Nikolov. Source: Evan Chen’s third-party compilation, not an official publication. One editorial point; manual grading only.

Let $n\ge2$, and let $a_1,\ldots,a_m$ be the $m=\varphi(n)$ positive integers less than $n$ and relatively prime to $n$. Suppose every prime divisor of $m$ divides $n$. Prove $m\mid a_1^k+\cdots+a_m^k$ for every integer $k\gt 0$.

Proposed by Ivan Borsenco. Source: Evan Chen’s third-party compilation, not an official publication. One editorial point; manual grading only.

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. Source: Evan Chen’s third-party compilation, not an official publication. One editorial point; manual grading only.

Let $ABCD$ be a convex cyclic quadrilateral. Put $E=AC\cap BD$, $F=AB\cap CD$, $G=DA\cap BC$. Circle $ABE$ meets line $CB$ again at $P$, and circle $ADE$ meets line $CD$ again at $Q$. Suppose the orders are $C,B,P,G$ and $C,Q,D,F$. Put $M=FP\cap GQ$. Prove $\angle MAC=90^\circ$.

Proposed by Kada Williams. Source: Evan Chen’s third-party compilation, not an official publication. One editorial point; manual grading only.

Let $a_n$ count permutations $(x_1,\ldots,x_n)$ of $(1,\ldots,n)$ for which all ratios $x_k/k$ are distinct. Prove that $a_n$ is odd for every $n\ge1$.

Proposed by Richard Stong. Source: Evan Chen’s third-party compilation, not an official publication. One editorial point; manual grading only.