PROBLEMS FOR AUGUST

Please send your solution to

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

227.

[Since the original statement of this problem in May was incorrect and not everyone picked up the correction, it is reposed.] Let n be an integer exceeding 2 and let a₀, a₁, a₂, ¼, a_n, a_n+1 be positive real numbers for which a₀ = a_n, a₁ = a_n+1 and

ai-1 + ai+1 = ki ai

for some positive integers k_i, where 1 £ i £ n.

Prove that

2n £ k1 + k2 + ¼+ kn £ 3n .

241.

[Corrected.] Determine

sec40° + sec80° + sec160° .

248.

Find all real solutions to the equation

æ Ö x + 3 - 4 Ö x - 1   +   æ Ö x + 8 - 6 Ö x - 1   = 1 .

249.

The non-isosceles right triangle ABC has ÐCAB = 90^°. Its inscribed circle with centre T touches the sides AB and AC at U and V respectively. The tangent through A of the circumscribed circle of triangle ABC meets UV in S. Prove that:

(a) ST || BC;

(b) |d₁ - d₂ | = r , where r is the radius of the inscribed circle, and d₁ and d₂ are the respective distances from S to AC and AB.

250.

In a convex polygon \frakP, some diagonals have been drawn so that no two have an intersection in the interior of \frakP. Show that there exists at least two vertices of \frak P, neither of which is an enpoint of any of these diagonals.

251.

Prove that there are infinitely many positive integers n for which the numbers { 1, 2, 3, ¼, 3n } can be arranged in a rectangular array with three rows and n columns for which (a) each row has the same sum, a multiple of 6, and (b) each column has the same sum, a multiple of 6.

252.

Suppose that a and b are the roots of the quadratic x² + px + 1 and that c and d are the roots of the quadratic x² + qx + 1. Determine (a - c)(b - c)(a + d)(b + d) as a function of p and q.

253.

Let n be a positive integer and let q = p/(2n+1). Prove that cot² q, cot² 2q, ¼, cot² nq are the solutions of the equation

æ è 2n + 1 1 ö ø xn - æ è 2n + 1 3 ö ø xn-1 + æ è 2n+1 5 ö ø xn-2 -¼ = 0 .

254.

Determine the set of all triples (x, y, z) of integers with 1 £ x, y, z £ 1000 for which x² + y² + z² is a multiple of xyz.