CONO-SUR 1994

The positive integrer number $n$ has $1994$ digits. $14$ of its digits are $0$ 's and the number of times that the other digits: $1, 2, 3, 4, 5, 6, 7, 8, 9$ appear are in proportion $1: 2: 3: 4: 5: 6: 7: 8: 9$ , respectively. Prove that $n$ is not a perfect square.

Consider a circle $C$ with diameter $AB=1$ . A point $P_0$ is chosen on $C$ , $P_0 \ne A$ , and starting in $P_0$ a sequence of points $P_1, P_2, \dots, P_n, \dots$ is constructed on $C$ , in the following way: $Q_n$ is the symmetrical point of $A$ with respect of $P_n$ and the straight line that joins $B$ and $Q_n$ cuts $C$ at $B$ and $P_{n+1}$ (not necessary different). Prove that it is possible to choose $P_0$ such that:

i $\angle {P_0AB} < 1$ .

ii In the sequence that starts with $P_0$ there are $2$ points, $P_k$ and $P_j$ , such that $\triangle {AP_kP_j}$ is equilateral.

Let $p$ be a positive real number given. Find the minimun vale of $x^3+y^3$ , knowing that $x$ and $y$ are positive real numbers such that $xy(x+y)=p$ .

Pedro and Cecilia play the following game: Pedro chooses a positive integer number $a$ and Cecilia wins if she finds a positive integrer number $b$ , prime with $a$ , such that, in the factorization of $a^3+b^3$ will appear three different prime numbers. Prove that Cecilia can always win.

Consider a $\triangle {ABC}$ , with $AC \perp BC$ . Consider a point $D$ on $AB$ such that $CD=k$ , and the radius of the inscribe circles on $\triangle {ADC}$ and $\triangle {CDB}$ are equals. Prove that the area of $\triangle {ABC}$ is equal to $k^2$ .