PROBLEMS FOR APRIL

The first five problems appeared on the annual University of Toronto Undergraduate Competition. Please send your solutions to

Professor E.J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3

no later than May 15, 2002.

139.

Let A, B, C be three pairwise orthogonal faces of a tetrahedran meeting at one of its vertices and having respective areas a, b, c. Let the face D opposite this vertex have area d. Prove that

d2 = a2 + b2 + c2 .

140.

Angus likes to go to the movies. On Monday, standing in line, he noted that the fraction x of the line was in front of him, while 1/n of the line was behind him. On Tuesday, the same fraction x of the line was in front of him, while 1/(n+1) of the line was behind him. On Wednesday, the same fraction x of the line was in front of him, while 1/(n+2) of the line was behind him. Determine a value of n for which this is possible.

141.

In how many ways can the rational 2002/2001 be written as the product of two rationals of the form (n+1)/n, where n is a positive integer?

142.

Let x, y > 0 be such that x³ + y³ £ x - y. Prove that x² + y² £ 1.

143.

A sequence whose entries are 0 and 1 has the property that, if each 0 is replaced by 01 and each 1 by 001, then the sequence remains unchanged. Thus, it starts out as 010010101001 ¼. What is the 2002th term of the sequence?

144.

Let a, b, c, d be rational numbers for which bc ¹ ad. Prove that there are infinitely many rational values of x for which Ö[((a + bx)(c + dx))] is rational. Explain the situation when bc = ad.