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PROBLEMS FOR MAY

Professor E.J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than June 30, 2001
Notes. A set in any space is convex if and only if, given any two points in the set, the line segment joining them is also contained in the set. A closed set is one that contains its boundary. A real sequence { xn } converges if and only if there is a number c, called its limit, such that, as n increases, the number xn gets closer and closer to c. If the sequences is increasing (i.e., xn+1 ³ xn for each index n) and bounded above (i.e., there is a number M for which xn £ M for each n, then it must converge. [Do you see why this is so?] Similarly, a decreasing sequence that is bounded below converges. [Supply the definitions and justify the statement.] An infinite series is an expression of the form åk=a¥ xk = xa + xa+1 + xa+2 + ¼+ xk + ¼, where a is an integer, usually 0 or 1. The nth partial sum of the series is sn º åk=an xk. The series has sum s if and only if its sequence { sn } of partial sums converges and has limit s; when this happens, the series converges. If the sequence of partial sums fails to converge, the series diverges. If every term in the series is nonnegative and the sequence of partial sums is bounded above, then the series converges. If a series of nonnegative terms converges, then it is possible to rearrange the order of the terms without changing the value of the sum.

79.
Let x0, x1, x2 be three positive real numbers. A sequence { xn } is defined, for n ³ 0 by

 xn+3 = xn+2 + xn+1 + 1xn .
Determine all such sequences whose entries consist solely of positive integers.

80.
Prove that, for each positive integer n, the series

 ¥å k=1 kn2k
converges to twice an odd integer not less than (n+1)!.

81.
Suppose that x ³ 1 and that x = ëx û+ { x }, where ëx û is the greatest integer not exceeding x and the fractional part { x } satisfies 0 £ x < 1. Define

f(x) =
 Ö ëx û + Ö {x }

Öx
.
(a) Determine the small number z such that f(x) £ z for each x ³ 1.
(b) Let x0 ³ 1 be given, and for n ³ 1, define xn = f(xn-1). Prove that limn ® ¥ xn exists.

82.
(a) A regular pentagon has side length a and diagonal length b. Prove that

 b2a2 + a2b2 = 3 .
(b) A regular heptagon (polygon with seven equal sides and seven equal angles) has diagonals of two different lengths. Let a be the length of a side, b be the length of a shorter diagonal and c be the length of a longer diagonal of a regular heptagon (so that a < b < c). Prove that:

 a2b2 + b2c2 + c2a2 = 6
and

 b2a2 + c2b2 + a2c2 = 5 .

83.
Let C be a circle with centre O and radius 1, and let F be a closed convex region inside C. Suppose from each point C, we can draw two rays tangent to F meeting at an angle of 60°. Describe F.

84.
Let ABC be an acute-angled triangle, with a point H inside. Let U, V, W be respectively the reflected image of H with respect to axes BC, AC, AB. Prove that H is the orthocentre of DABC if and only if U, V, W lie on the circumcircle of DABC,

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