JMO 1991

Let $P,\ Q,\ R$ be the points such that $BP: PC=CQ: QA=AR: RB=t: 1-t\ (0<t<1)$ for a triangle $ABC$. Denote $K$ by the area of the triangle with segments $AP,\ BQ,\ CR$ as side lengths and $L$ by triangle $ABC$, find $\frac{K}{L}$ in terms of $t$.

Let $N$ be the set of the whole of positive integers. The mapping from $N$ to $N$ is defined as follows: $p(1)=2,\ p(2)=3,\ p(3)=4,\ p(4)=1,\ \ \ p(n)=n\ (n\geq 5)$, $q(1)=3,\ q(2)=4,\ q(3)=2,\ q(4)=1,\ \ \ p(n)=n\ (n\geq 5)$. Answer the following questions.

(1) If you make a mapping $f: N\rightarrow N$ sucessfully, we have $f$ such that $f(f(n))=p(n)+2$. Give an example.

(2) Prove that it is impossible that $f(f(n))=q(n)+2$ holds in regardless of any definition for $f: N\rightarrow N$.

Let $A$ be a positive 16 digit integer.If you take out some consecutive digits integers among $A$, prove that we can make the product of the numbers be square number. For example if some digit of $A$ is 4, you may take out only the digit.

A rectangular of a $10*14$ is divided into small 140 unit squares and painted in red and white like chess board as below. We put $0$ or $1$ in the square such that each row and column has an odd numbers of $1$. Prove that the number of $1$ contained in red-painted square is even.

The pattern arranged by a red and a white square alternatively.

RWRWRWRW..... WRWRWRWR..... RWRWRWRW..... WRWRWRWR.....

Let $A$ be a set of $n\geq 2$ points on a plane. Prove that there exists a circle which contains at least $\left[\frac{n}{3}\right]$ points of $A$ among circles (involving perimeter) with some end points taken from $A$ as the diameter, where $[x]$ is the greatest integer which is less than or equal to $x$.