CMO 1999

Find all real solutions to the equation $4x^2 - 40 \lfloor x \rfloor + 51 = 0$ .

Let $ABC$ be an equilateral triangle of altitude 1. A circle with radius 1 and center on the same side of $AB$ as $C$ rolls along the segment $AB$ . Prove that the arc of the circle that is inside the triangle always has the same length.

Determine all positive integers $n$ with the property that $n = (d(n))^2$ . Here $d(n)$ denotes the number of positive divisors of $n$ .

Suppose $a_1,a_2,\cdots,a_8$ are eight distinct integers from $\{1,2,\cdots,16,17\}$ . Show that there is an integer $k > 0$ such that the equation $a_i - a_j = k$ has at least three different solutions. Also, find a specific set of 7 distinct integers from $\{1,2,\ldots,16,17\}$ such that the equation $a_i - a_j = k$ does not have three distinct solutions for any $k > 0$ .

Let $x$ , $y$ , and $z$ be non-negative real numbers satisfying $x + y + z = 1$ . Show that $$x^2 y + y^2 z + z^2 x \leq \frac {4}{27}$$ and find when equality occurs.