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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 $1 \leq k \leq 100$ 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} - (\binom{n}{1}) (n + k - 1)^{n} + (\binom{n}{2}) (n + k - 2)^{n} - \dots \pm (\binom{n}{n}) k^{n} .

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

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

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

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