Spain Mathematical Olympiad 1986

1. Define the distance between real numbers $x$ and $y$ by $$d(x,y) =\sqrt{([x]-[y])^2+(\{x\}-\{y\})^2}.$$ Determine (as a union of intervals) the set of real numbers whose distance from $3/2$ is less than $202/100$ .
2. A segment $d$ is said to divide a segment $s$ if there is a natural number $n$ such that $s = nd = d+d+ ...+d$ ($n$ times).
   a) Prove that if a segment $d$ divides segments $s$ and $s'$ with $s < s'$, then it also divides their difference $s'-s$.
   b) Prove that no segment divides the side $s$ and the diagonal $s'$ of a regular pentagon (consider the pentagon formed by the diagonals of the given pentagon without explicitly computing the ratios).
3. Find all natural numbers $n$ such that $5^n+3$ is a power of $2$
4. Denote by $m(a,b)$ the arithmetic mean of positive real numbers $a,b$. Given a positive real function $g$ having positive derivatives of the first and second order, define $\mu (a,b)$ the mean value of $a$ and $b$ with respect to $g$ by $$2g(\mu (a,b)) = g(a)+g(b).$$ Decide which of the two mean values $m$ and $\mu$ is larger.
5. Consider the curve $\Gamma$ defined by the equation $y^2 = x^3 +bx+b^2$, where $b$ is a nonzero rational constant. Inscribe in the curve $\Gamma$ a triangle whose vertices have rational coordinates.
6. Evaluate $$\prod_{k=1}^{14} \cos \big(\frac{k\pi}{15}\big)$$