1993 Turkey MO (2nd round)

Part of Turkey MO (2nd round)

Subcontests

(6)

1

turkey 1993 q6

n_{1},\ldots ,n_{k}, a

are integers that satisfies the above conditions A)For every

i\neq j

(n_{i}, n_{j})=1

i, a^{n_{i}}\equiv 1 (mod n_{i})

i, X^{a-1}\equiv 0(mod n_{i})

a^{x}\equiv 1(mod x)

congruence has at least

2^{k+1}-2

x>1

1

turkey 1993 q5

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.

1

turkey 1993 q4

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}

0<x<y<1

x< \frac{a_{k}}{a_{m}}<y

we can find such

k,m\in{Z^{+}}

1

turkey 1993 q3

n\in{Z^{+}}

A={1,\ldots ,n}

f: N\rightarrow N

\sigma: N\rightarrow N

are two permutations, if there is one

k\in A

(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

\sigma

S_\sigma

shows the set of good functions for

\sigma

. a) Prove that,

S_\sigma

2^{n-1}

elements for every

\sigma

permutation. b)

n\geq 4

, prove that there are permutations

\sigma

\tau

S_{\sigma}\cap S_{\tau}=\phi

1

turkey 1993 q2

I centered incircle of triangle

ABC

(m(\hat{B})=90^\circ)

\left[AB\right], \left[BC\right], \left[AC\right]

respectively at

F, D, E

\left[CI\right]\cap\left[EF\right]={L}

\left[DL\right]\cap\left[AB\right]=N

\left[AI\right]=\left[ND\right]

1

turkey 1993 q1

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^{+}}