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PROBLEMS FOR SEPTEMBER

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

255.
Prove that there is no positive integer that, when written to base 10, is equal to its kth multiple when its initial digit (on the left) is transferred to the right (units end), where 2 £ k £ 9 and k ¹ 3.

256.
Find the condition that must be satisfied by y1, y2, y3, y4 in order that the following set of six simultaneous equations in x1, x2, x3, x4 is solvable. Where possible, find the solution.

 x1 + x2 = y1 y2      x1 + x3 = y1 y3      x1 + x4 = y1 y4

 x2 + x3 = y2 y3      x2 + x4 = y2 y4      x3 + x4 = y3 y4 .

257.
Let n be a positive integer exceeding 1. Discuss the solution of the system of equations:

 ax1 + x2 + ¼+ xn = 1

 x1 + ax2 + ¼+ xn = a

 ¼

 x1 + x2 + ¼+ axi + ¼+ xn = ai-1

 ¼

 x1 + x2 + ¼+ xi + ¼+ axn = an-1 .

258.
The infinite sequence { an ; n = 0, 1, 2, ¼} satisfies the recursion
 an+1 = an2 + (an - 1)2
for n ³ 0. Find all rational numbers a0 such that there are four distinct indices p, q, r, s for which ap - aq = ar - as.

259.
Let ABC be a given triangle and let A¢BC, AB¢C, ABC¢ be equilateral triangles erected outwards on the sides of triangle ABC. Let W be the circumcircle of A¢B¢C¢ and let A", B", C" be the respective intersections of W with the lines AA¢, BB¢, CC¢.
Prove that AA¢, BB¢, CC¢ are concurrent and that
 AA" + BB" + CC" = AA¢ = BB¢ = CC¢ .

260.
TABC is a tetrahedron with volume 1, G is the centroid of triangle ABC and O is the midpoint of TG. Reflect TABC in O to get T¢A¢B¢C¢. Find the volume of the intersection of TABC and T¢A¢B¢C¢.

261.
Let x, y, z > 0. Prove that
x

 x + Ö (x+y)(x+z)
+  y

 y + Ö (x+y)(y+z)
+  z

 z + Ö (x+z)(y+z)
£ 1 .
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