CMO 1998

Determine the number of real solutions $a$ to the equation: $$\left[\,\frac{1}{2} a\,\right]+\left[\,\frac{1}{3} a\,\right]+\left[\,\frac{1}{5} a\,\right] = a.$$ Here, if $x$ is a real number, then $[ x\,]$ denotes the greatest integer that is less than or equal to $x$ .

Find all real numbers $x$ such that: $$x = \sqrt{ x - \frac{1}{x} } + \sqrt{ 1 - \frac{1}{x} }$$

Let $n$ be a natural number such that $n \geq 2$ . Show that $$\frac {1}{n + 1} \left( 1 + \frac {1}{3} + \cdot \cdot \cdot + \frac {1}{2n - 1} \right) > \frac {1}{n} \left( \frac {1}{2} + \frac {1}{4} + \cdot \cdot \cdot + \frac {1}{2n} \right).$$

Let $ABC$ be a triangle with $\angle{BAC} = 40^{\circ}$ and $\angle{ABC}=60^{\circ}$ . Let $D$ and $E$ be the points lying on the sides $AC$ and $AB$ , respectively, such that $\angle{CBD} = 40^{\circ}$ and $\angle{BCE} = 70^{\circ}$ . Let $F$ be the point of intersection of the lines $BD$ and $CE$ . Show that the line $AF$ is perpendicular to the line $BC$ .

Let $m$ be a positive integer. Define the sequence $a_0, a_1, a_2, \cdots$ by $a_0 = 0, a_1 = m,$ and $a_{n+1} = m^2a_n - a_{n-1}$ for $n = 1,2,3,\cdots$ . Prove that an ordered pair $(a,b)$ of non-negative integers, with $a \leq b$ , gives a solution to the equation $${ \frac{a^2 + b^2}{ab + 1} = m^2}$$ if and only if $(a,b)$ is of the form $(a_n,a_{n+1})$ for some $n \geq 0$ .