PROBLEMS FOR NOVEMBER
Please send your solution to
Ms. Valeria Pandelieva
641 Kirkwood Avenue
Ottawa, ON K1Z 5X5
It is very important that the front page contain your
complete mailing address and your email address. The deadline
for this set is December 21, 2002.
Notes: The sides of a rightangled triangle that are adjacent
to the right angle are called legs. The centre of
gravity or centroid of a collection of
$n$ mass particles is
the point where the cumulative mass can be regarded as concentrated so
that the motion of this point, when exposed to outside forces such as
gravity, is identical to that of the whole collection. To illustrate
this point, imagine that the mass particles are connected to a point
by rigid nonmaterial sticks (with mass 0) to form a structure.
The point where the tip of a needle could be put so that this
structure is in a state of balance is its centroid. In addition, there
is an intuitive definition of a centroid of a lamina, and of a solid:
The centroid of a lamina is the point, which would cause equilibrium
(balance) when the tip of a needle is placed underneath to support it.
Likewise, the centroid of a solid is the point, at which the
solid ``balances'', i.e., it will not revolve if force is
applied. The centroid,
$G$ of a set of points is defined
vectorially by
$\mathrm{OG}=\frac{\sum _{i=1}^{n}{m}_{i}\xb7\mathrm{OM}{}_{i}}{\sum _{i=1}^{n}{m}_{i}}$
where
${m}_{i}$ is the mass of the particle at a position
${M}_{i}$
(the summation extending over the whole collection). Problem 181
is related to the centroid of an assembly of three particles placed
at the vertices of a given triangle. The circumcentre of
a triangle is the centre of its circumscribed circle. The
orthocentre of a triangle is the intersection point of
its altitudes. An unbounded region in the plane is one
not contained in the interior of any circle.

185.

Find all triples of natural numbers
$a$,
$b$,
$c$, such
that all of the following conditions hold: (1)
$a<1974$;
(2)
$b$ is less than
$c$ by 1575; (3)
${a}^{2}+{b}^{2}={c}^{2}$.

186.

Find all natural numbers
$n$ such that there exists
a convex
$n$sided polygon whose diagonals are all of the same
length.

187.

Suppose that
$p$ is a real parameter and that
$f(x)={x}^{3}(p+5){x}^{2}2(p3)(p1)x+4{p}^{2}24p+36\hspace{1em}.$


(a) Check that
$f(3p)=0$.


(b) Find all values of
$p$ for which two of the roots
of the equation
$f(x)=0$ (expressed in terms of
$p$) can be
the lengths of the two legs in a rightangled triangle with a
hypotenuse of
$4\sqrt{2}$.

188.

(a) The measure of the angles of an acute triangle
are
$\alpha $,
$\beta $ and
$\gamma $ degrees. Determine (as an
expression of
$\alpha $,
$\beta $,
$\gamma $) what masses must be
placed at each of the triangle's vertices for the centroid
(centre of gravity) to coincide with (i) the orthocentre of
the triangle; (ii) the circumcentre
of the triangle.


(b) The sides of an arbitrary triangle are
$a$,
$b$,
$c$
units in length. Determine (as an expression of
$a$,
$b$,
$c$) what
masses must be placed at each of the triangle's vertices for the
centroid (centre of gravity) to coincide with (i) the centre of
the inscribed circle of the triangle; (ii) the intersection
point of the three segments joining the vertices of the triangle
to the points on the opposite sides where the inscribed circle
is tangent (be sure to prove that, indeed, the three segments
intersect in a common point).

189.

There are
$n$ lines in the plane, where
$n$ is an
integer exceeding 2. No three of them are concurrent and no two
of them are parallel. The lines divide the plane into regions;
some of them are closed (they have the form of a convex polygon);
others are unbounded (their borders are broken lines consisting of
segments and rays).


(a) Determine as a function of
$n$ the number of unbounded
regions.


(b) Suppose that some of the regions are coloured, so that
no two coloured regions have a common side (a segment or ray). Prove
that the number of regions coloured in this way does not exceed
$\frac{1}{3}({n}^{2}+n)$.

190.

Find all integer values of the parameter
$a$ for which
the equation
$\Vert 2x+1\Vert +\Vert x2\Vert =a$
has exactly one integer among its solutions.

191.

In Olymonland the distances between every two
cities is different. When the transportation program of the country
was being developed, for each city, the closest of the other cities
was chosen and a highway was built to connect them. All highways
are line segments. Prove that


(a) no two highways intersect;


(b) every city is connected by a highway to no more than
5 other cities;


(c) there is no closed broken line composed of highways only.