PROBLEMS FOR SEPTEMBER
Please send your solution to
Edward J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than October 15, 2003.
It is important that your complete mailing address
and your email address appear on the front page.

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
y_{1}, y_{2}, y_{3}, y_{4} in order that the following set
of six simultaneous equations in x_{1}, x_{2}, x_{3}, x_{4} is solvable.
Where possible, find the solution.
x_{1} + x_{2} = y_{1} y_{2} x_{1} + x_{3} = y_{1} y_{3} x_{1} + x_{4} = y_{1} y_{4} 

x_{2} + x_{3} = y_{2} y_{3} x_{2} + x_{4} = y_{2} y_{4} x_{3} + x_{4} = y_{3} y_{4} . 


257.

Let n be a positive integer exceeding 1. Discuss the
solution of the system of equations:
ax_{1} + x_{2} + ¼+ x_{n} = 1 

x_{1} + ax_{2} + ¼+ x_{n} = a 

x_{1} + x_{2} + ¼+ ax_{i} + ¼+ x_{n} = a^{i1} 

x_{1} + x_{2} + ¼+ x_{i} + ¼+ ax_{n} = a^{n1} . 


258.

The infinite sequence { a_{n} ; n = 0, 1, 2, ¼} satisfies the recursion
a_{n+1} = a_{n}^{2} + (a_{n}  1)^{2} 

for n ³ 0. Find all rational numbers a_{0} such that there are
four distinct indices p, q, r, s for which a_{p}  a_{q} = a_{r}  a_{s}.

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