PROBLEMS FOR JULY AND AUGUST 
Because of the variability of summer plans, the usual ration
of problems has been doubled and the deadline set later so that
students can have a chance to organize their work conveniently.
Send your solutions to Prof. E.J. Barbeau, Department of Mathematics,
University of Toronto, Toronto, ON M5S 3G3 no later than
September 10, 2002. Please make sure that the front page of
your solution contains your complete mailing address and your email
address.
Notes. A composite integer is one that has positive
divisors other than 1 and itself; it is not prime. A set of
point in the plane is concyclic (or cyclic, inscribable)
if and only if there is a circle that passes through all of them.

157.

Prove that if the quadratic equation
${x}^{2}+\mathrm{ax}+b+1=0$ has nonzero integer solutions, then
${a}^{2}+{b}^{2}$ is a composite
integer.

158.

Let
$f(x)$ be a polynomial with real coefficients for
which the equation
$f(x)=x$ has no real solution. Prove that the
equation
$f(f(x))=x$ has no real solution either.

159.

Let
$0\le a\le 4$. Prove that the area of the
bounded region enclosed by the curves with equations
$y=1\Vert x1\Vert $
and
$y=\Vert 2xa\Vert $
cannot exceed
$\frac{1}{3}$.

160.

Let
$I$ be the incentre of the triangle
$\mathrm{ABC}$ and
$D$ be the point of contact of the inscribed circle with the side
$\mathrm{AB}$. Suppose that
$\mathrm{ID}$ is produced outside of the triangle
$\mathrm{ABC}$ to
$H$ so that the length
$\mathrm{DH}$ is equal to the semiperimeter
of
$\Delta \mathrm{ABC}$. Prove that the quadrilateral
$\mathrm{AHBI}$
is concyclic if and only if angle
$C$ is equal to
${90}^{\u02c6}$.

161.

Let
$a,b,c$ be positive real numbers for which
$a+b+c=1$. Prove that
$\frac{{a}^{3}}{{a}^{2}+{b}^{2}}+\frac{{b}^{3}}{{b}^{2}+{c}^{2}}+\frac{{c}^{3}}{{c}^{2}+{a}^{2}}\ge \frac{1}{2}\hspace{1em}.$

162.

Let
$A$ and
$B$ be fixed points in the plane.
Find all positive integers
$k$ for which the following assertion
holds:


among all triangles
$\mathrm{ABC}$ with
$\Vert \mathrm{AC}\Vert =k\Vert \mathrm{BC}\Vert $, the one with the largest area is isosceles.

163.

Let
${R}_{i}$ and
${r}_{i}$ re the respective circumradius and
inradius of triangle
${A}_{i}{B}_{i}{C}_{i}$ (
$i=1,2$). Prove that, if
$\angle {C}_{1}=\angle {C}_{2}$ and
${R}_{1}{r}_{2}={r}_{1}{R}_{2}$, then the two
triangles are similar.

164.

Let
$n$ be a positive integer and
$X$ a set with
$n$ distinct elements. Suppose that there are
$k$ distinct subsets
of
$X$ for which the union of any four contains no more that
$n2$ elements. Prove that
$k\le {2}^{n2}$.

165.

Let
$n$ be a positive integer. Determine all
$n$tples
$\{{a}_{1},{a}_{2},\dots ,{a}_{n}\}$ of positive integers for which
${a}_{1}+{a}_{2}+\dots +{a}_{n}=2n$ and there is no subset of them
whose sum is equal to
$n$.

166.

Suppose that
$f$ is a realvalued function defined on
the reals for which
$f(\mathrm{xy})+f(yx)\ge f(y+x)$
for all real
$x$ and
$y$. Prove that
$f(x)\ge 0$ for all real
$x$.

167.

Let
$u=(\sqrt{5}2){}^{1/3}(\sqrt{5}+2){}^{1/3}$
and
$v=(\sqrt{189}8){}^{1/3}(\sqrt{189}+8){}^{1/3}$. Prove
that, for each positive integer
$n$,
${u}^{n}+{v}^{n+1}=0$.

168.

Determine the value of
$\mathrm{cos}{5}^{\u02c6}+\mathrm{cos}{77}^{\u02c6}+\mathrm{cos}{149}^{\u02c6}+\mathrm{cos}{221}^{\u02c6}+\mathrm{cos}{293}^{\u02c6}\hspace{1em}.$

169.

Prove that, for each positive integer
$n$ exceeding 1,
$\frac{1}{{2}^{n}}+\frac{1}{{2}^{1/n}}<1\hspace{1em}.$

170.

Solve, for real
$x$,
$x\xb7{2}^{1/x}+\frac{1}{x}\xb7{2}^{x}=4\hspace{1em}.$