IrMO 1989

A quadrilateral $ABCD$ is inscribed, as shown, in a square of area one unit. Prove that $$2\le |AB|^2+|BC|^2+|CD|^2+|DA|^2\le 4$$

$D$ $A$ $B$ $C$

Let $x = a_1a_2 \dots a_n$ be an n-digit number, where $a_1, a_2, \dots , an (a_1 \neq 0)$ are the digits. The $n$ numbers $x_1 = x = a_1 a_2 ... a_n,$ $x_2 = a_n a_1 ... a_{n-1},$ $x_3 = a_{n-1} a_n a _1 ... a_{n-2}$ , $x_4 = a_{n-2} a_{n-1} a_n a_1 , ... a_{n-3} ,$ $... , x_n = a_2 a_3 ... a_n a_1$ are said to be obtained from $x$ by the cyclic permutation of digits. [For example, if $n = 5$ and $x = 37001$ , then the numbers are $x_1 = 37001, x_2 = 13700,$ $x_3 = 01370(= 1370), x_4 = 00137(= 137),$ $x_5 = 70013.]$ Find, with proof, (i) the smallest natural number n for which there exists an n-digit number x such that the n numbers obtained from x by the cyclic permutation of digits are all divisible by 1989; and (ii) the smallest natural number x with this property.

Suppose $L$ is a fixed line, and $A$ is a fixed point not on $L$ . Let $k$ be a fixed nonzero real number. For $P$ a point on $L$ , let $Q$ be a point on the line $AP$ with $|AP|\cdot |AQ|=k^2$ . Determine the locus of $Q$ as $P$ varies along the line $L$ .

(i): Prove that if $n$ is a positive integer, then $$\binom{2n}{n}=\frac{(2n)!}{(n!)^2}$$ is a positive integer that is divisible by all prime numbers $p$ with $n<p\le 2n$ , and that $$\binom{2n}{n}<2^{2n}.$$ (ii): For $x$ a positive real number, let $\pi(x)$ denote the number of prime numbers $p \le x$ . [Thus, $\pi(10) = 4$ since there are $4$ primes, viz., $2$ , $3$ , $5$ , and $7$ , not exceeding $10$ .]Prove that if $n \ge 3$ is an integer, then (a) $$\pi(2n) < \pi(n) + {{2n}\over{\log_2(n)}};$$ (b) $$\pi(2^n) < {{2^{n+1}\log_2(n-1)}\over{n}};$$ (c) Deduce that, for all real numbers $x \ge 8$ , $$\pi(x) < {{4x \log_2(\log_2(x))}\over{\log_2(x)}}.$$