## PROBLEMS FOR SEPTEMBER

171.
Let $n$ be a positive integer. In a round-robin match, $n$ teams compete and each pair of teams plays exactly one game. At the end of the match, the $i$th team has ${x}_{i}$ wins and ${y}_{i}$ losses. There are no ties. Prove that

${x}_{1}^{2}+{x}_{2}^{2}+\dots +{x}_{n}^{2}={y}_{1}^{2}+{y}_{2}^{2}+\dots +{y}_{n}^{2} .$

172.
Let $a$, $b$, $c$, $d$. $e$, $f$ be different integers. Prove that

$\left(a-b\right){}^{2}+\left(b-c\right){}^{2}+\left(c-d\right){}^{2}+\left(d-e\right){}^{2}+\left(e-f\right){}^{2}+\left(f-a\right){}^{2}\ge 18 .$

173.
Suppose that $a$ and $b$ are positive real numbers for which $a+b=1$. Prove that

$\left(a+\frac{1}{a}{\right)}^{2}+\left(b+\frac{1}{b}{\right)}^{2}\ge \frac{25}{2} .$

Determine when equality holds.

174.
For which real value of $x$ is the function

$\left(1-x\right){}^{5}\left(1+x\right)\left(1+2x\right){}^{2}$

maximum? Determine its maximum value.

175.
$\mathrm{ABC}$ is a triangle such that $\mathrm{AB}<\mathrm{AC}$. The point $D$ is the midpoint of the arc with endpoints $B$ and $C$ of that arc of the circumcircle of $\Delta \mathrm{ABC}$ that contains $A$. The foot of the perpendicular from $D$ to $\mathrm{AC}$ is $E$. Prove that $\mathrm{AB}+\mathrm{AE}=\mathrm{EC}$.

176.
Three noncollinear points $A$, $M$ and $N$ are given in the plane. Construct the square such that one of its vertices is the point $A$, and the two sides which do not contain this vertex are on the lines through $M$ and $N$ respectively. [Note: In such a problem, your solution should consist of a description of the construction (with straightedge and compasses) and a proof in correct logical order proceeding from what is given to what is desired that the construction is valid. You should deal with the feasibility of the construction.]

177.
Let ${a}_{1}$, ${a}_{2}$, $\dots$, ${a}_{n}$ be nonnegative integers such that, whenever $1\le i$, $1\le j$, $i+j\le n$, then

${a}_{i}+{a}_{j}\le {a}_{i+j}\le {a}_{i}+{a}_{j}+1 .$

(a) Give an example of such a sequence which is not an arithmetic progression.
(b) Prove that there exists a real number $x$ such that ${a}_{k}=⌊\mathrm{kx}⌋$ for $1\le k\le n$.

Since solutions are still being marked for the June set of problems, their solutions will not be published until October.