PROBLEMS FOR MARCH
Please send your solution to
Edward J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than April 30, 2003.
It is important that your complete mailing address
and your email address appear on the front page.
Suppose that each side and each diagonal of a
regular hexagon A1A2A3A4A5A6 is coloured either red or
blue, and that no triangle AiAjAk has all of its sides
coloured blue. For each k = 1, 2, ¼, 6, let
rk be the number of segments AkAj (j ¹ k) coloured
red. Prove that
Let S be a circle with centre O and
radius 1, and let
Pi (1 £ i £ n) be points chosen on the
(circumference of the) circle for which åi=1n[( ®) || ( OPi)] = 0. Prove that, for each
point X in the plane, å|XPi | ³ n.
Find all values of the parameter a for which the
equation 16x4 - ax3 + (2a + 17)x2 - ax + 16 = 0 has exactly
four real solutions which are in geometric progression.
Let x be positive and let 0 < a £ 1.
(1 - xa)(1 - x)-1 £ (1 + x )a-1.|
Let the three side lengths of a scalene triangle
be given. There are two possible ways of orienting the triangle
with these side lengths, one obtainable from the other by
turning the triangle over, or by reflecting in a mirror.
Prove that it is possible to slice the triangle in one of
its orientations into finitely many pieces that can be
rearranged using rotations and translations in the plane (but not
reflections and rotations out of the plane) to form the other.
Let ABC be a triangle. Suppose that D is a point
on BA produced and E a point on the side BC, and that
DE intersects the side AC at F. Let BE + EF = BA + AF.
Prove that BC + CF = BD + DF.
There are two definitions of an ellipse.
(1) An ellipse is the locus of points P such that the sum of
its distances from two fixed points F1
) is constant.
(2) An ellipse is the locus of points P such that, for some
real number e (called the eccentricity) with 0 < e < 1,
the distance from P to a fixed point F (called a focus)
is equal to e times its perpendicular distance to a fixed
straight line (called the directrix).
Prove that the two definitions are compatible.