CN-LEAGUE 2003

From point $P$ outside a circle draw two tangents to the circle touching at $A$ and $B$ . Draw a secant line intersecting the circle at points $C$ and $D$ , with $C$ between $P$ and $D$ . Choose point $Q$ on the chord $CD$ such that $\angle DAQ=\angle PBC$ . Prove that $\angle DBQ=\angle PAC$ .

Let the three sides of a triangle be $\ell, m, n$ , respectively, satisfying $\ell>m>n$ and $\left\{\frac{3^\ell}{10^4}\right\}=\left\{\frac{3^m}{10^4}\right\}=\left\{\frac{3^n}{10^4}\right\}$ , where $\{x\}=x-\lfloor{x}\rfloor$ and $\lfloor{x}\rfloor$ denotes the integral part of the number $x$ . Find the minimum perimeter of such a triangle.

Let a space figure consist of $n$ vertices and $l$ lines connecting these vertices, with $n=q^2+q+1$ , $l\ge q^2(q+1)^2+1$ , $q\ge2$ , $q\in\mathbb{N}$ . Suppose the figure satisfies the following conditions: every four vertices are non-coplaner, every vertex is connected by at least one line, and there is a vertex connected by at least $p+2$ lines. Prove that there exists a space quadrilateral in the figure, i.e. a quadrilateral with four vertices $A, B, C, D$ and four lines $AB, BC, CD, DA$ in the figure.