INMO 1996

a) Given any positive integer $n$ , show that there exist distint positive integers $x$ and $y$ such that $x + j$ divides $y + j$ for $j = 1 , 2, 3, \ldots, n$ ;

b) If for some positive integers $x$ and $y$ , $x+j$ divides $y+j$ for all positive integers $j$ , prove that $x = y$ .

Let $C_1$ and $C_2$ be two concentric circles in the plane with radii $R$ and $3R$ respectively. Show that the orthocenter of any triangle inscribed in circle $C_1$ lies in the interior of circle $C_2$ . Conversely, show that every point in the interior of $C_2$ is the orthocenter of some triangle inscribed in $C_1$ .

Solve the following system for real $a , b, c, d, e$ : $$\left\{ \begin{array}{ccc} 3a & = & ( b + c+ d)^3 3b & = & ( c + d +e ) ^3 3c & = & ( d + e +a )^3 3d & = & ( e + a +b )^3 3e &=& ( a + b +c)^3. \end{array}\right.$$

Let $X$ be a set containing $n$ elements. Find the number of ordered triples $(A,B, C)$ of subsets of $X$ such that $A$ is a subset of $B$ and $B$ is a proper subset of $C$ .

Define a sequence $(a_n)_{n \geq 1}$ by $a_1 =1$ and $a_2 =2$ and $a_{n+2} = 2 a_{n+1} - a_n + 2$ for $n \geq 1$ . prove that for any $m$ , $a_m a_{m+1}$ is also a term in this sequence.

There is a $2n \times 2n$ array (matrix) consisting of $0's$ and $1's$ and there are exactly $3n$ zeroes. Show that it is possible to remove all the zeroes by deleting some $n$ rows and some $n$ columns.