OIMU 2006

Let $m,n$ be positive integers greater than $1$ . We define the sets $P_m=\left\{\frac{1}{m},\frac{2}{m},\cdots,\frac{m-1}{m}\right\}$ and $P_n=\left\{\frac{1}{n},\frac{2}{n},\cdots,\frac{n-1}{n}\right\}$ .

Find the distance between $P_m$ and $P_n$ , that is defined as $$\min\{|a-b|:a\in P_m,b\in P_n\}$$

Prove that for any positive integer $n$ and any real numbers $a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n$ we have that the equation $$a_1 \sin(x) + a_2 \sin(2x) +\cdots+a_n\sin(nx)=b_1 \cos(x)+b_2\cos(2x)+\cdots +b_n \cos(nx)$$ has at least one real root.

Let $p_1(x)=p(x)=4x^3-3x$ and $p_{n+1}(x)=p(p_n(x))$ for each positive integer $n$ . Also, let $A(n)$ be the set of all the real roots of the equation $p_n(x)=x$ .

Prove that $A(n)\subseteq A(2n)$ and that the product of the elements of $A(n)$ is the average of the elements of $A(2n)$ .

Prove that for any interval $[a,b]$ of real numbers and any positive integer $n$ there exists a positive integer $k$ and a partition of the given interval $$a = x (0) < x (1) < x (2) < \cdots < x (k-1) < x (k) = b$$ such that $$\int_{x(0)}^{x(1)}f(x)dx+\int_{x(2)}^{x(3)}f(x)dx+\cdots=\int_{x(1)}^{x(2)}f(x)dx+\int_{x(3)}^{x(4)}f(x)dx+\cdots$$ for all polynomials $f$ with real coefficients and degree less than $n$ .

A regular $n$ -gon is inscribed in a circle of radius $1$ . Let $a_1,\cdots,a_{n-1}$ be the distances of one of the vertices of the polygon to all the other vertices. Prove that $$(5-a_1^2)\cdots(5-a_{n-1}^2)=F_n^2$$ where $F_n$ is the $n^{th}$ term of the Fibonacci sequence $1,1,2,\cdots$

Let $x_0(t)=1$ , $x_{k+1}(t)=(1+t^{k+1})x_k(t)$ for all $k\geq 0$ ; $y_{n,0}(t)=1$ , $y_{n,k}(t)=\frac{t^{n-k+1}-1}{t^k-1}y_{n,k-1}(t)$ for all $n\geq 0$ , $1\leq k \leq n$ .

Prove that $\sum_{j=0}^{n-1}(-1)^j x_{n-j-1}(t)y_{n,j}(t)=\frac{1-(-1)^n}{2}$ for all $n\geq 1$ .

Consider the multiplicative group $A=\{z\in\mathbb{C}|z^{2006^k}=1, 0<k\in\mathbb{Z}\}$ of all the roots of unity of degree $2006^k$ for all positive integers $k$ .

Find the number of homomorphisms $f:A\to A$ that satisfy $f(f(x))=f(x)$ for all elements $x\in A$ .