1980 Austrian-Polish Competition

Part of Austrian-Polish

Subcontests

(3)

1

every positive integer k can be written as x_j -x_i for some i, j if x_{n+1}<=2n

A sequence of integers

1 = x_1 < x_2 < x_3 <...

x_{n+1} \le 2n

n

. Show that every positive integer

k

can be written as

x_j -x_i

i, j

1

equal segments wanted, tangents form endpoints of a diameter related

Through the endpoints

A

B

AB

of a given circle, the tangents

\ell

m

have been drawn. Let

C\ne A

\ell

q_1,q_2

be two rays from

C

q_i

cuts the circle in

D_i

E_i

D_i

C

E_i, i = 1,2

AD_1,AD_2,AE_1,AE_2

m

in the respective points

M_1,M_2,N_1,N_2

M_1M_2 = N_1N_2

.

1

Absolute Valued Sequence Inequality

a_1,a_2,a_3,\dots

be a sequence of real numbers satisfying the inequality |a_{k+m}-a_k-a_m| \leq 1 \text{for all} \ k,m \in \mathbb{Z}_{>0}. Show that the following inequality holds for all positive integers

k,m

\left| \frac{a_k}{k}-\frac{a_m}{m} \right| < \frac{1}{k}+\frac{1}{m}.