CMO 1990

Given a convex quadrilateral $ABCD$ , side $AB$ is not parallel to side $CD$ . The circle $O_1$ passing through $A$ and $B$ is tangent to side $CD$ at $P$ . The circle $O_2$ passing through $C$ and $D$ is tangent to side $AB$ at $Q$ . Circle $O_1$ and circle $O_2$ meet at $E$ and $F$ . Prove that $EF$ bisects segment $PQ$ if and only if $BC\parallel AD$ .

Let $x$ be a natural number. We call $\{x_0,x_1,\dots ,x_l\}$ a factor linkof $x$ if the sequence $\{x_0,x_1,\dots ,x_l\}$ satisfies the following conditions: (1) $x_0=1, x_l=x$ ; (2) $x_{i-1}<x_i, x_{i-1}|x_i, i=1,2,\dots,l$ . Meanwhile, we define $l$ as the length of the factor link $\{x_0,x_1,\dots ,x_l\}$ . Denote by $L(x)$ and $R(x)$ the length and the number of the longest factor link of $x$ respectively. For $x=5^k\times 31^m\times 1990^n$ , where $k,m,n$ are natural numbers, find the value of $L(x)$ and $R(x)$ .

A function $f(x)$ defined for $x\ge 0$ satisfies the following conditions: i. for $x,y\ge 0$ , $f(x)f(y)\le x^2f(y/2)+y^2f(x/2)$ ; ii. there exists a constant $M$ ( $M>0$ ), such that $|f(x)|\le M$ when $0\le x\le 1$ . Prove that $f(x)\le x^2$ .

Given a positive integer number $a$ and two real numbers $A$ and $B$ , find a necessary and sufficient condition on $A$ and $B$ for the following system of equations to have integer solution: $$\left\{\begin{array}{cc} x^2+y^2+z^2=(Ba)^2 x^2(Ax^2+By^2)+y^2(Ay^2+Bz^2)+z^2(Az^2+Bx^2)=\dfrac{1}{4}(2A+B)(Ba)^4\end{array}\right.$$

Given a finite set $X$ , let $f$ be a rule such that $f$ maps every even-element-subset $E$ of $X$ (i.e. $E \subseteq X$ , $|E|$ is even) into a real number $f(E)$ . Suppose that $f$ satisfies the following conditions: (I) there exists an even-element-subset $D$ of $X$ such that $f(D)>1990$ ; (II) for any two disjoint even-element-subsets $A,B$ of $X$ , equation $f(A\cup B)=f(A)+f(B)-1990$ holds. Prove that there exist two subsets $P,Q$ of $X$ satisfying: (1) $P\cap Q=\emptyset$ , $P\cup Q=X$ ; (2) for any non-even-element-subset $S$ of $P$ (i.e. $S\subseteq P$ , $|S|$ is odd), we have $f(S)>1990$ ; (3) for any even-element-subset $T$ of $Q$ , we have $f(T)\le 1990$ .

A convex $n$ -gon and its $n-3$ diagonals which have no common point inside the polygon form a subdivision graph. Show that if and only if $3|n$ , there exists a subdivision graphthat can be drawn in one closed stroke. (i.e. start from a certain vertex, get through every edges and diagonals exactly one time, finally back to the starting vertex.)