Subcontests
(5)Points on xy-plane
Let P1, P2, …, P1993=P0 be distinct points in the xy-plane
with the following properties:
(i) both coordinates of Pi are integers, for i=1,2,…,1993;
(ii) there is no point other than Pi and Pi+1 on the line segment joining Pi with Pi+1 whose coordinates are both integers, for i=0,1,…,1992.
Prove that for some i, 0≤i≤1992, there exists a point Q with coordinates (qx,qy) on the line segment joining Pi with Pi+1 such that both 2qx and 2qy are odd integers. Polynomial
Let
\begin{eqnarray*} f(x) & = & a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 \ \ \mbox{and} \\ g(x) & = & c_{n+1} x^{n+1} + c_n x^n + \cdots + c_0 \end{eqnarray*}
be non-zero polynomials with real coefficients such that g(x)=(x+r)f(x) for some real number r. If a=max(∣an∣,…,∣a0∣) and c=max(∣cn+1∣,…,∣c0∣), prove that ca≤n+1.