PROBLEMS FOR FEBRUARY
Solutions should be submitted to
Prof. E.J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than March 31, 2001

61.

Let
$S=1!2!3!\dots 99!100!$
(the product of the first 100 factorials).
Prove that there exists an integer
$k$ for
which
$1\le k\le 100$ and
$S/k!$ is a
perfect square. Is
$k$ unique? (Optional:
Is it possible to find such a number
$k$ that
exceeds 100?)

62.

Let
$n$ be a positive integer.
Show that, with three exceptions,
$n!+1$
has at least one prime divisor that exceeds
$n+1$.

63.

Let
$n$ be a positive integer
and
$k$ a nonnegative integer.
Prove that
$n!=(n+k){}^{n}\left(\genfrac{}{}{0ex}{}{n}{1}\right)(n+k1){}^{n}+\left(\genfrac{}{}{0ex}{}{n}{2}\right)(n+k2){}^{n}\dots \pm \left(\genfrac{}{}{0ex}{}{n}{n}\right){k}^{n}\hspace{1em}.$

64.

Let
$M$ be a point in the interior of
triangle
$\mathrm{ABC}$, and suppose that
$D$,
$E$,
$F$
are points on the respective side
$\mathrm{BC}$,
$\mathrm{CA}$,
$\mathrm{AB}$. Suppose
$\mathrm{AD}$,
$\mathrm{BE}$ and
$\mathrm{CF}$
all pass through
$M$.
(In technical terms, they are cevians.)
Suppose that the areas and the perimeters of
the triangles
$\mathrm{BMD}$,
$\mathrm{CME}$,
$\mathrm{AMF}$ are equal.
Prove that triangle
$\mathrm{ABC}$ must be equilateral.

65.

Suppose that
$\mathrm{XTY}$ is a straight line
and that
$\mathrm{TU}$ and
$\mathrm{TV}$ are two rays emanating from
$T$ for which
$\angle \mathrm{XTU}=\angle \mathrm{UTV}=\angle \mathrm{VTY}={60}^{\u02c6}$. Suppose that
$P$,
$Q$ and
$R$ are respective points on the rays
$\mathrm{TY}$,
$\mathrm{TU}$ and
$\mathrm{TV}$ for which
$\mathrm{PQ}=\mathrm{PR}$.
Prove that
$\angle \mathrm{QPR}={60}^{\u02c6}$.

66.

(a) Let
$\mathrm{ABCD}$ be a square and let
$E$
be an arbitrary point on the side
$\mathrm{CD}$. Suppose that
$P$ is a point on the diagonal
$\mathrm{AC}$ for which
$\mathrm{EP}\perp \mathrm{AC}$ and that
$Q$ is a point on
$\mathrm{AE}$ produced for which
$\mathrm{CQ}\perp \mathrm{AE}$. Prove that
$B,P,Q$ are collinear.


(b) Does the result hold if the hypothesis is
weakened to require only that
$\mathrm{ABCD}$ is a rectangle?