OIMU 2007

For each pair of integers $(i,k)$ such that $1\le i\le k$ , the linear transformation $P_{i,k}:\mathbb{R}^k\to\mathbb{R}^k$ is defined as: $P_{i,k}(a_1,\cdots,a_{i-1},a_i,a_{i+1},\cdots,a_k)=(a_1,\cdots,a_{i-1},0,a_{i+1},\cdots,a_k)$ Prove that for all $n\ge2$ and for every set of $n-1$ linearly independent vectors $v_1,\cdots,v_{n-1}$ in $\mathbb{R}^n$ , there is an integer $k$ such that $1\le k\le n$ and such that the vectors $P_{k,n}(v_1),\cdots,P_{k,n}(v_{n-1})$ are linearly independent.

Prove that for all positive integers $n$ and for all real numbers $x$ such that $0\le x\le1$ , the following inequality holds: $\left(1-x+\frac{x^2}{2}\right)^n-(1-x)^n\le\frac{x}{2}$ .

Let $f:\mathbb{R}\to\mathbb{R}^+$ be a continuous and periodic function. Prove that for all $\alpha\in\mathbb{R}$ the following inequality holds: $\int_0^T\frac{f(x)}{f(x+\alpha)}dx\ge T$ ,

where $T$ is the period of $f(x)$ .

Consider an infinite sequence $a_1,a_2,\cdots$ whose terms all belong to $\left\{1,2\right\}$ . A positive integer with $n$ digits is said to be good if its decimal representation has the form $a_ra_{r+1}\cdots a_{r+(n-1)}$ , for some positive integer $r$ . Suppose that there are at least $2008$ good numbers with a million digits. Prove that there are at least $2008$ good numbers with $2007$ digits.

Determine all pairs of polynomials $f,g\in\mathbb{C}[x]$ with complex coefficients such that the following equalities hold for all $x\in\mathbb{C}$ : $f(f(x))-g(g(x))=1+i$ $f(g(x))-g(f(x))=1-i$

Let $F$ be a field whose characteristic is not $2$ , let $F^*=F\setminus\left\{0\right\}$ be its multiplicative group and let $T$ be the subgroup of $F^*$ constituted by its finite order elements. Prove that if $T$ is finite, then $T$ is cyclic and its order is even.

The height of a positive integer is defined as being the fraction $\frac{s(a)}{a}$ , where $s(a)$ is the sum of all the positive divisors of $a$ . Show that for every pair of positive integers $N,k$ there is a positive integer $b$ such that the height of each of $b,b+1,\cdots,b+k$ is greater than $N$ .