Professor E.J. Barbeau

85.

Find all pairs
$(a,b)$ of positive integers
with
$a\ne b$ for which the system
$\mathrm{cos}\mathrm{ax}+\mathrm{cos}\mathrm{bx}=0$
$a\mathrm{sin}\mathrm{ax}+b\mathrm{sin}\mathrm{bx}=0$
has a solution. If so, determine its solutions.

86.

Let
$\mathrm{ABCD}$ be a convex quadrilateral with
$\mathrm{AB}=\mathrm{AD}$ and
$\mathrm{CB}=\mathrm{CD}$. Prove that


(a) it is possible to inscribe a circle in it;


(b) it is possible to circumscribe a circle about
it if and only if
$\mathrm{AB}\perp \mathrm{BC}$;


(c) if
$\mathrm{AB}\perp \mathrm{AC}$ and
$R$ and
$r$ are the respective
radii of the circumscribed and inscribed circles, then the
distance between the centres of the two circles is equal to
the square root of
${R}^{2}+{r}^{2}r\sqrt{{r}^{2}+4{R}^{2}}$.

87.

Prove that, if the real numbers
$a$,
$b$,
$c$, satisfy the equation
$\lfloor \mathrm{na}\rfloor +\lfloor \mathrm{nb}\rfloor =\lfloor \mathrm{nc}\rfloor $
for each positive integer
$n$, then at least one of
$a$ and
$b$ is an integer.

88.

Let
$I$ be a real interval of length
$1/n$. Prove
that
$I$ contains no more than
$\frac{1}{2}(n+1)$ irreducible
fractions of the form
$p/q$ with
$p$ and
$q$ positive integers,
$1\le q\le n$ and the greatest common divisor of
$p$ and
$q$ equal to 1.

89.

Prove that there is only one triple of positive
integers, each exceeding 1, for which the product of any two of
the numbers plus one is divisible by the third.

90.

Let
$m$ be a positive integer, and
let
$f(m)$ be the smallest value of
$n$ for which
the following statement is true:


given any set of
$n$ integers, it
is always possible to find a subset of
$m$ integers
whose sum is divisible by
$m$
Determine
$f(m)$.