location:
 PROBLEMS FOR JULY AND AUGUST

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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