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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!¼99!100! (the product of the first 100 factorials). Prove that there exists an integer k for which 1 £ k £ 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 - æç è n 1 ö÷ ø (n + k - 1)n + æç è n 2 ö÷ ø (n + k - 2)n- ¼± æç è n n ö÷ ø kn .

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 ÐXTU = ÐUTV = ÐVTY = 60°. Suppose that P, Q and R are respective points on the rays TY, TU and TV for which PQ = PR. Prove that ÐQPR = 60°.

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?
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