JMO 1995

Let $n\geq 2$ be integers and $r$ be positive integers such that $r$ is not the multiple of $n$, and let $g$ be the greatest common measure of $n$ and $r$. Prove that $$\sum_{i=1}^{n-1} \left\{\frac{ri}{n} \right\}\ =\frac{1}{2}(n-g).$$ where $\left\{x\right\}$ is the fractional part, that is to say, which means the value by subtracting $x$ from the maximum integer value which is equal or less than $x$.

Find all the non-constant rational function of $f(x)$ and real numbers $a$ satisfying $\{f(x)\}^2-a=f(x^2).$ Here a rational function of $x$ is the equation expressed by the ratio of two polynominals of $x.$

Given a convex pentagon $ABCDE.$ Let $S,R$ be the intersection points of $AC$ or $AD$ and $BE$ respectively, and let the intersection points $T,P$ of $CA$ or $CE$ and $BD$ respectively.Let $Q$ be the intersection point of $CE$ and $AD.$ If all of $\triangle{ASR},\ \triangle{BTS},\ \triangle{CPT},\ \triangle{DQP},$ and $\triangle{ERQ}$ have the area of $1,$ then find the area of the following pentagons.

(1) The pentagon $PQRST.$ (2) The pentagon $ABCDE.$

The sequence $\{a_1,\ a_2,\ \cdots\}$ is defined by $a_{2n}=a_n,\ a_{2n+1}=(-1)^n.$ A point $P$ moves on the coordinate plane as follows.

(1) Let $P_0$ be the origin, $P$ moves in a distance of $1$ from $P_0$ toward in the positive direction of $x$ - axis, Denote this point by $P_i.$

(2) After $P$ has moved to $P_i,$ it turns $90^\circ$ to the left and moves in a distance of $1$ when $a_i=1,$ and turns $90^\circ$ to the right and moves in a distance of $1$ when $a_i=-1.$ Denote this point by $P_{i+1},$ w here $i=1,2,\cdots.$ Prove that $P$ can't pass on the same segment more than two times.

Let $k,n$ be integers such that $1\leq k\leq n,$ a nd let $a_1,\ a_2,\ \cdots,\ a_k$ be numbers satisfying the following equations.

$$\begin{cases} a_1+\cdots\cdots +a_k=n \\ a_1^2+\cdots\cdots +a_k^2=n\\ \ \ \ \ \ \ \ \ \ \ \ \ \vdots \\ a_1^k+\cdots\cdots +a_k^k=n \end{cases}$$

Prove that $$(x+a_1)(x+a_2)\cdots(x+a_k)=x^k+\ _n C_1\ x^{k-1}+\ _nC_2\ x^{k-2}+\cdots+\ _nC_k.$$

where $_iC_j$ is a binomial coefficient which means $\frac{i\cdot (i-1)\cdots(i-j+1)}{j\cdot (j-1)\cdots 2\cdot 1}.$