USAMO 1997 — Days 1 and 2

Let $p_1,p_2,p_3,\ldots$ be the primes in increasing order, and let $0\lt x_0\lt 1$. For every positive integer $k$, define $$x_k=\begin{cases}0,&x_{k-1}=0,\\ \left\{p_k/x_{k-1}\right\},&x_{k-1}\ne0.\end{cases}$$ Here $\{t\}=t-\lfloor t\rfloor$ is the fractional part. Find, with proof, all starting values $x_0$ for which the sequence eventually becomes $0$.

Let $ABC$ be a triangle. Choose noncollinear points $D,E,F$ on the perpendicular bisectors of $BC,CA,AB$, respectively. Prove that the lines through $A,B,C$ perpendicular to $EF,FD,DE$, respectively, are concurrent.

Prove that for every integer $n$ there is a unique polynomial $Q$ whose coefficients belong to $\{0,1,\ldots,9\}$ and such that $Q(-2)=Q(-5)=n$.

To clip a convex $n$-gon, choose consecutive sides $AB,BC$, let $M,N$ be their respective midpoints, and replace $AB,BC$ by $AM,MN,NC$. Thus triangle $MBN$ is cut off, giving a convex $(n+1)$-gon. Start with a regular hexagon $\mathcal P_6$ of area $1$ and repeatedly clip, obtaining $\mathcal P_7,\mathcal P_8,\ldots$. Prove that every possible sequence of clippings satisfies $\operatorname{area}(\mathcal P_n)\gt 1/3$ for all $n\ge6$.

For positive real numbers $a,b,c$, prove $$\frac1{a^3+b^3+abc}+\frac1{b^3+c^3+abc}+\frac1{c^3+a^3+abc}\le\frac1{abc}.$$

Nonnegative integers $a_1,a_2,\ldots,a_{1997}$ satisfy $$a_i+a_j\le a_{i+j}\le a_i+a_j+1$$ for all $i,j\ge1$ with $i+j\le1997$. Prove that there exists a real number $x$ such that $a_n=\lfloor nx\rfloor$ for every $1\le n\le1997$.