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# 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

$n!=\left(n+k\right){}^{n}-\left(\genfrac{}{}{0}{}{n}{1}\right)\left(n+k-1\right){}^{n}+\left(\genfrac{}{}{0}{}{n}{2}\right)\left(n+k-2\right){}^{n}-\dots ±\left(\genfrac{}{}{0}{}{n}{n}\right){k}^{n} .$

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 $\angle \mathrm{XTU}=\angle \mathrm{UTV}=\angle \mathrm{VTY}={60}^{ˆ}$. 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 $\angle \mathrm{QPR}={60}^{ˆ}$.

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?

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