CANADA_MO/md/en-exam1999.md

THE 1999 CANADIAN MATHEMATICAL OLYMPIAD

1. Find all real solutions to the equation $4 x^{2}-40[x]+51=0$.

Here, if $x$ is a real number, then $[x]$ denotes the greatest integer that is less than or equal to $x$. 2. Let $A B C$ be an equilateral triangle of altitude 1 . A circle with radius 1 and center on the same side of $A B$ as $C$ rolls along the segment $A B$. Prove that the arc of the circle that is inside the triangle always has the same length. 3. Determine all positive integers $n$ with the property that $n=(d(n))^{2}$. Here $d(n)$ denotes the number of positive divisors of $n$. 4. Suppose $a_{1}, a_{2}, \ldots, a_{8}$ are eight distinct integers from ${1,2, \ldots, 16,17}$. 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 $k>0$. 5. Let $x, y$, and $z$ be non-negative real numbers satisfying $x+y+z=1$. Show that

$$x^{2}y+y^{2}z+z^{2}x\le \frac{4}{27}x^\{2\}\; y+y^\{2\}\; z+z^\{2\}\; x\; \backslash leq\; \backslash frac\{4\}\{27\}$$

and find when equality occurs.