location:
 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 $f\left(x\right)$ be a polynomial with real coefficients for which the equation $f\left(x\right)=x$ has no real solution. Prove that the equation $f\left(f\left(x\right)\right)=x$ 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

$y=1-‖x-1‖$

and

$y=‖2x-a‖$

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 semi-perimeter of $\Delta \mathrm{ABC}$. Prove that the quadrilateral $\mathrm{AHBI}$ is concyclic if and only if angle $C$ is equal to ${90}^{ˆ}$.

161.
Let $a,b,c$ be positive real numbers for which $a+b+c=1$. Prove that

$\frac{{a}^{3}}{{a}^{2}+{b}^{2}}+\frac{{b}^{3}}{{b}^{2}+{c}^{2}}+\frac{{c}^{3}}{{c}^{2}+{a}^{2}}\ge \frac{1}{2} .$

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 $‖\mathrm{AC}‖=k‖\mathrm{BC}‖$, 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 $n-2$ elements. Prove that $k\le {2}^{n-2}$.

165.
Let $n$ be a positive integer. Determine all $n-$tples $\left\{{a}_{1},{a}_{2},\dots ,{a}_{n}\right\}$ 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 real-valued function defined on the reals for which

$f\left(\mathrm{xy}\right)+f\left(y-x\right)\ge f\left(y+x\right)$

for all real $x$ and $y$. Prove that $f\left(x\right)\ge 0$ for all real $x$.

167.
Let $u=\left(\sqrt{5}-2\right){}^{1/3}-\left(\sqrt{5}+2\right){}^{1/3}$ and $v=\left(\sqrt{189}-8\right){}^{1/3}-\left(\sqrt{189}+8\right){}^{1/3}$. Prove that, for each positive integer $n$, ${u}^{n}+{v}^{n+1}=0$.

168.
Determine the value of

$\mathrm{cos}{5}^{ˆ}+\mathrm{cos}{77}^{ˆ}+\mathrm{cos}{149}^{ˆ}+\mathrm{cos}{221}^{ˆ}+\mathrm{cos}{293}^{ˆ} .$

169.
Prove that, for each positive integer $n$ exceeding 1,

$\frac{1}{{2}^{n}}+\frac{1}{{2}^{1/n}}<1 .$

170.
Solve, for real $x$,

$x·{2}^{1/x}+\frac{1}{x}·{2}^{x}=4 .$

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