CONO-SUR 1996

In the following figure, the largest square is divided into two squares and three rectangles, as shown:

The area of each smaller square is equal to $a$ and the area of each small rectangle is equal to $b$ . If $a+b=24$ and the root square of $a$ is a natural number, find all possible values for the area of the largest square.

Consider a sequence of real numbers defined by: $a_{n + 1} = a_n + \frac{1}{a_n}$ for $n = 0, 1, 2, ...$ Prove that, for any positive real number $a_0$ , is true that $a_{1996}$ is greater than $63$ .

A shop sells bottles with this capacity: $1L, 2L, 3L,..., 1996L$ , the prices of bottles satifies this $2$ conditions: $1$ . Two bottles have the same price, if and only if, your capacities satifies $m - n = 1000$ $2$ . The price of bottle $m$ ( $1001>m>0$ ) is $1996 - m$ dollars. Find all pair(s) $m$ and $n$ such that: a) $m + n = 1000$ b) the cost is smallest possible!!! c) with the pair, the shop can measure $k$ liters, with $0<k<1996$ (for all $k$ integer) Note: The operations to measure are: i) To fill or empty any one of two bottles ii)Pass water of a bottle for other bottle We can measure $k$ liters when the capacity of one bottle plus the capacity of other bottle is equal to $k$

The sequence $0, 1, 1, 1, 1, 1,....,1$ where have $1$ number zero and $1995$ numbers one. If we choose two or more numbers in this sequence(but not the all $1996$ numbers) and substitute one number by arithmetic mean of the numbers selected, we obtain a new sequence with $1996$ numbers!!! Show that, we can repeat this operation until we have all $1996$ numbers are equal Note: It's not necessary to choose the same quantity of numbers in each operation!!!

We want to cover totally a square(side is equal to $k$ integer and $k>1$ ) with this rectangles: $1$ rectangle ( $1\times 1$ ), $2$ rectangles ( $2\times 1$ ), $4$ rectangles ( $3\times 1$ ),...., $2^n$ rectangles ( $n + 1 \times 1$ ), such that the rectangles can't overlap and don't exceed the limits of square. Find all $k$ , such that this is possible and for each $k$ found you have to draw a solution

Find all integers $n \leq 3$ such that there is a set $S_n$ formed by $n$ points of the plane that satisfy the following two conditions: Any three points are not collinear. No point is found inside the circle whose diameter has ends at any two points of $S_n$ .NOTE: The points on the circumference are not considered to be inside the circle.