OIMU 2008

Let $n$ be a positive integer that is not divisible by either $2$ or $5$ . In the decimal expansion of $\frac{1}{n}= 0.a_1a_2a_3\cdots$ a finite number of digits after the decimal point are chosen arbitrarily to be deleted. Clearly the decimal number obtained by this procedure is also rational, so it's equal to $\frac{a}{b}$ for some integers $a,b$ . Prove that $b$ is divisible by $n$ .

Prove that for each natural number $n$ there is a polynomial $f$ with real coefficients and degree $n$ such that $p(x)=f(x^2-1)$ is divisible by $f(x)$ over the ring $\mathbb{R}[x]$ .

Prove that $x+\frac{1}{x^x}<2$ for $0<x<1$ .

Two vertices $A,B$ of a triangle $ABC$ are located on a parabola $y=ax^2 + bx + c$ with $a>0$ in such a way that the sides $AC,BC$ are tangent to the parabola. Let $m_c$ be the length of the median $CC_1$ of triangle $ABC$ and $S$ be the area of triangle $ABC$ . Find $$\frac{S^2}{m_c^3}$$

Find all positive integers $n$ such that there are positive integers $a_1,\cdots,a_n, b_1,\cdots,b_n$ that satisfy

$$(a_1^2+\cdots+a_n^2)(b_1^2+\cdots+b_n^2)-(a_1b_1+\cdots+a_nb_n)^2=n$$

a) Determine if there are matrices $A,B,C\in\mathrm{SL}_{2}(\mathbb{Z})$ such that $A^2+B^2=C^2$ .b) Determine if there are matrices $A,B,C\in\mathrm{SL}_{2}(\mathbb{Z})$ such that $A^4+B^4=C^4$ .Note: The notation $A\in \mathrm{SL}_{2}(\mathbb{Z})$ means that $A$ is a $2\times 2$ matrix with integer entries and $\det A=1$ .

Let $A$ be an abelian additive group such that all nonzero elements have infinite order and for each prime number $p$ we have the inequality $|A/pA|\leq p$ , where $pA = \{pa |a \in A\}$ , $pa = a+a+\cdots+a$ (where the sum has $p$ summands) and $|A/pA|$ is the order of the quotient group $A/pA$ (the index of the subgroup $pA$ ).

Prove that each subgroup of $A$ of finite index is isomorphic to $A$ .