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## PROBLEMS FOR AUGUST

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, an, an+1 be positive real numbers for which a0 = an, a1 = an+1 and
 ai-1 + ai+1 = ki ai
for some positive integers ki, 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
 æè 2n + 1 1 öø xn - æè 2n + 1 3 öø xn-1 + æè 2n+1 5 öø 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.
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