CONO-SUR 1993

On a table there is a pile with $T$ tokens which incrementally shall be converted into piles with three tokens each. Each step is constituted of selecting one pile removing one of its tokens. And then the remaining pile is separated into two piles. Is there a sequence of steps that can accomplish this process?

a.) $T = 1000$ (Cono Sur) b.) $T = 2001$ (BWM)

Consider a circle with centre $O$ , and $3$ points on it, $A,B$ and $C$ , such that $\angle {AOB}< \angle {BOC}$ . Let $D$ be the midpoint on the arc $AC$ that contains the point $B$ . Consider a point $K$ on $BC$ such that $DK \perp BC$ . Prove that $AB+BK=KC$ .

Find the number of elements that a set $B$ can have, contained in $(1, 2, ... , n)$ , according to the following property: For any elements $a$ and $b$ on $B$ ( $a \ne b$ ), $(a-b) \not| (a+b)$ .

Prove that there exists a succession $a_1, a_2, ... , a_k, ...$ , where each $a_i$ is a digit ( $a_i \in (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)$ ) and $a_0=6$ , such that, for each positive integrer $n$ , the number $x_n=a_0+10a_1+100a_2+...+10^{n-1}a_{n-1}$ verify that $x_n^2-x_n$ is divisible by $10^n$ .

Prove that, given a positive integrer $n$ , there exists a positive integrer $k_n$ with the following property: Given any $k_n$ points in the space, $4$ by $4$ non-coplanar, and associated integrer numbers between $1$ and $n$ to each sharp edge that meets $2$ of this points, there's necessairly a triangle determined by $3$ of them, whose sharp edges have associated the same number.