CMO 1974

i) If $x = \left(1+\frac{1}{n}\right)^{n}$ and $y=\left(1+\frac{1}{n}\right)^{n+1}$ , show that $y^{x}= x^{y}$ .

ii) Show that, for all positive integers $n$ , $$1^{2}-2^{2}+3^{2}-4^{2}+\cdots+(-1)^{n}(n-1)^{2}+(-1)^{n+1}n^{2}= (-1)^{n+1}(1+2+\cdots+n).$$

Let $ABCD$ be a rectangle with $BC=3AB$ . Show that if $P,Q$ are the points on side $BC$ with $BP = PQ = QC$ , then $$\angle DBC+\angle DPC = \angle DQC.$$

Let $$f(x) = a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n}x^{n}$$ be a polynomial with coefficients satisfying the conditions: $$0\le a_{i}\le a_{0}, i=1,2,\ldots,n.$$ Let $b_{0},b_{1},\ldots,b_{2n}$ be the coefficients of the polynomial

\begin{align*}\left(f(x)\right)^{2}&= \left(a_{0}+a_{1}x+a_{2}x^{2}+\cdots a_{n}x^{n}\right) &= b_{0}+b_{1}x+b_{2}x^{2}+\cdots+b_{2n}x^{2n}. \end{align*}

Prove that $b_{n+1}\le \frac{1}{2}\left(f(1)\right)^{2}$ .

Let $n$ be a fixed positive integer. To any choice of real numbers satisfying $$0\le x_{i}\le 1, i=1,2,\ldots, n,$$ there corresponds the sum $$\sum_{1\le i<j\le n}|x_{i}-x_{j}|.$$ Let $S(n)$ denote the largest possible value of this sum. Find $S(n)$ .

Given a circle with diameter $AB$ and a point $X$ on the circle different from $A$ and $B$ , let $t_{a}$ , $t_{b}$ and $t_{x}$ be the tangents to the circle at $A$ , $B$ and $X$ respectively. Let $Z$ be the point where line $AX$ meets $t_{b}$ and $Y$ the point where line $BX$ meets $t_{a}$ . Show that the three lines $YZ$ , $t_{x}$ and $AB$ are either concurrent (i.e., all pass through the same point) or parallel.