MathDB

7.1 / 6.3 All integers from 1 to 1966 are written on the board. Allowed is to erase any two numbers by writing their difference instead. Prove that repeating such an operation many times cannot ensure that There are only zeros left on the board.
7.2 Prove that the radius of a circle is equal to the difference between the lengths of two chords, one of which subtends an arc of

1/10

of a circle, and the other subtends an arc in

3/10

of a circle.
7.3 Prove that for any natural number

n

the number

n(2n+1)(3n+1)...(1966n + 1)

is divisible by every prime number less than

1966

.
7.4 What number needs to be put in place * so that the next the problem had a unique solution: “There are n straight lines on the plane, intersecting at * points. Find n.” ?
7.5 / 6.4 Black paint was sprayed onto a white surface. Prove that there are three points of the same color lying on the same line, and so, that one of the points lies in the middle between the other two.
7.6 There are

n

points on the plane so that any triangle with vertices at these points has an area less than

1

. Prove that all these points can be enclosed in a triangle of area

4

.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988082_1966_leningrad_math_olympiad]here.

Problem Statement