TR-MO 1993

Prove that there is a number such that its decimal represantation ends with 1994 and it can be written as $1994\cdot 1993^{n}$ ( $n\in{Z^{+}}$ )

I centered incircle of triangle $ABC$ $(m(\hat{B})=90^\circ)$ touches $\left[AB\right], \left[BC\right], \left[AC\right]$ respectively at $F, D, E$ . $\left[CI\right]\cap\left[EF\right]={L}$ and $\left[DL\right]\cap\left[AB\right]=N$ . Prove that $\left[AI\right]=\left[ND\right]$ .

$n\in{Z^{+}}$ and $A={1,\ldots ,n}$ . $f: N\rightarrow N$ and $\sigma: N\rightarrow N$ are two permutations, if there is one $k\in A$ such that $(f\circ\sigma)(1),\ldots ,(f\circ\sigma)(k)$ is increasing and $(f\circ\sigma)(k),\ldots ,(f\circ\sigma)(n)$ is decreasing sequences we say that $f$ is good for $\sigma$ . $S_\sigma$ shows the set of good functions for $\sigma$ . a) Prove that, $S_\sigma$ has got $2^{n-1}$ elements for every $\sigma$ permutation. b) $n\geq 4$ , prove that there are permutations $\sigma$ and $\tau$ such that, $S_{\sigma}\cap S_{\tau}=\phi$ .

$a_{n}$ is a sequence of positive integers such that, for every $n\geq 1$ , $0<a_{n+1}-a_{n}<\sqrt{a_{n}}$ . Prove that for every $x,y\in{R}$ such that $0<x<y<1$ $x< \frac{a_{k}}{a_{m}}<y$ we can find such $k,m\in{Z^{+}}$ .

Prove that we can draw a line (by a ruler and a compass) from a vertice of a convex quadrilateral such that, the line divides the quadrilateral to two equal areas.

$n_{1},\ldots ,n_{k}, a$ are integers that satisfies the above conditions A)For every $i\neq j$ , $(n_{i}, n_{j})=1$ B)For every $i, a^{n_{i}}\equiv 1 (mod n_{i})$ C)For every $i, X^{a-1}\equiv 0(mod n_{i})$ . Prove that $a^{x}\equiv 1(mod x)$ congruence has at least $2^{k+1}-2$ solutions. ( $x>1$ )