KURSCHAK 1984

Writing down the first $4$ rows of the Pascal triangle in the usual way and then adding up the numbers in vertical columns, we obtain $7$ numbers as shown above. If we repeat this procedure with the first $1024$ rows of the Pascal triangle, how many of the $2047$ numbers thus obtained will be odd?

$A_1B_1A_2$ , $B_1A_2B_2$ , $A_2B_2A_3$ ,..., $B_{13}A_{14}B_{14}$ , $A_{14}B_{14}A_1$ and $B_{14}A_1B_1$ are equilateral rigid plates that can be folded along the edges $A_1B_1$ , $B_1A_2$ , ..., $A_{14}B_{14}$ and $B_{14}A_1$ respectively. Can they be folded so that all $28$ plates lie in the same plane?

Given are $n$ integers, not necessarily distinct, and two positive integers $p$ and $q$ . If the $n$ numbers are not all distinct, choose two equal ones. Add $p$ to one of them and subtract $q$ from the other. If there are still equal ones among the $n$ numbers, repeat this procedure. Prove that after a finite number of steps, all $n$ numbers are distinct.