ZA-SAMO 2005

Let $F$ be the set of all fractions $m/n$ where $m$ and $n$ are positive integers with $m+n\le 2005$ . Find the largest number $a$ in $F$ such that $a < 16/23$ .

A warehouse contains $175$ boots of size $8$ , $175$ boots of size $9$ and $200$ boots of size $10$ . Of these $550$ boots, $250$ are for the left foot and $300$ for the right foot. Let $n$ denote the total number of usable pairs of boots in the warehouse. (A usable pair consists of a left and a right boot of the same size.)

(a) Is $n=50$ possible?

(b) Is $n=51$ possible?

The inscribed circle of triangle $ABC$ touches the sides $BC$ , $CA$ and $AB$ at $D$ , $E$ and $F$ respectively. Let $Q$ denote the other point of intersection of $AD$ and the inscribed circle. Prove that $EQ$ extended passes through the midpoint of $AF$ if and only if $AC = BC$ .

Let $x_1,x_2,\dots,x_n$ be positive numbers with product equal to 1. Prove that there exists a $k\in\{1,2,\dots,n\}$ such that $$\frac{x_k}{k+x_1+x_2+\cdots+x_k}\ge 1-\frac{1}{\sqrt[n]{2}}.$$

Consider the increasing sequence $1,2,4,5,7,9,10,12,14,16,17,19,\dots$ of positive integers, obtained by concatenating alternating blocks $\{1\},\{2,4\},\{5,7,9\},\{10,12,14,16\},\dots$ of odd and even numbers. Each block contains one more element than the previous one and the first element in each block is one more than the last element of the previous one. Prove that the $n$ -th element of the sequence is given by $$2n-\Big\lfloor\frac{1+\sqrt{8n-7}}{2}\Big\rfloor.$$ (Here $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$ .)