PROBLEMS FOR OCTOBER
Please send your solution to
Ed Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than November 30, 2003.
It is important that your complete mailing address
and your email address appear on the front page.

262.

Let ABC be an acute triangle. Suppose that
P and U are points on the side BC so that P lies
between B and U, that Q and V are points on the
side CA so that Q lies between C and V, and that
R and W are points on the side AB so that R lies
between A and W. Suppose also that
ÐAPU = ÐAUP = ÐBQV = ÐBVQ = ÐCRW = ÐCWR . 

The lines AP, BQ and CR bound a triangle T_{1}
and the lines AU, BV and CW bound a triangle T_{2}.
Prove that all six vertices of the triangles T_{1} and
T_{2} lie on a common circle.

263.

The ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
are each used exactly once altogether to form three
positive integers for which the largest is the sum of the
other two. What are the largest and the smallest possible
values of the sum?

264.

For the real parameter a, solve for real x
the equation
A complete answer will discuss the circumstances under which
a solution is feasible.

265.

Note that 959^{2} = 919681, 919 + 681 = 40^{2};
960^{2} = 921600, 921 + 600 = 39^{2}; and 961^{2} = 923521,
923 + 521 = 38^{2}. Establish a general result of which these
are special instances.

266.

Prove that, for any positive integer n,
(2n  n) divides the least common multiple of the
numbers 1, 2, 3, ¼, 2n1, 2n.

267.

A nonorthogonal reflection in an axis a takes
each point on a to itself, and each point P not on a
to a point P¢ on the other side of a in such a way that
a intersects PP¢ at its midpoint and PP¢ always makes
a fixed angle q with a. Does this transformation
preserves lines? preserve angles? Discuss the image of a
circle under such a transformation.

268.

Determine all continuous real functions f of
a real variable for which
f(x + 2f(y)) = f(x) + y + f(y) 

for all real x and y.