Let $ABCD$ be a quadrilateral whose sides are all equal and $\angle ABC=60^\circ$. Let $\ell$ pass through $D$ without meeting the quadrilateral elsewhere. Let $E,F$ be the intersections of $\ell$ with the lines $AB,BC$, respectively, and let $M=CE\cap AF$. Prove that $CA^2=CM\cdot CE$.
Find the total number of different integer values taken by
$$f(x)=\lfloor x\rfloor+\lfloor2x\rfloor+\left\lfloor\frac{5x}{3}\right\rfloor+\lfloor3x\rfloor+\lfloor4x\rfloor$$
for real $x$ with $0\le x\le100$. Here $\lfloor t\rfloor$ is the greatest integer not exceeding $t$.
Let $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$ and $g(x)=c_{n+1}x^{n+1}+c_nx^n+\cdots+c_0$ be nonzero polynomials with real coefficients such that $g(x)=(x+r)f(x)$ for some real $r$. If $a=\max(|a_n|,\ldots,|a_0|)$ and $c=\max(|c_{n+1}|,\ldots,|c_0|)$, prove that $a/c\le n+1$.
Let $P_1,P_2,\ldots,P_{1993}=P_0$ be distinct points in the $xy$-plane such that (i) both coordinates of each $P_i$ are integers; (ii) for $i=0,1,\ldots,1992$, the segment $P_iP_{i+1}$ contains no lattice point other than its endpoints. Prove that for some $i$ there is a point $Q=(q_x,q_y)$ on $P_iP_{i+1}$ such that both $2q_x$ and $2q_y$ are odd integers.
Schematic of an odd closed chain with lattice-point vertices; the problem has $1993$ vertices.
