MathDB

8.1 / 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.” ?
8.2 / 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

.
8.3 / 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

.
8.4 Prove that the sum of all divisors of the number

n^2

is odd.
8.5 A quadrilateral has three obtuse angles. Prove that the larger of its two diagonals emerges from the vertex of an acute angle.
8.6 Numbers

x_1, x_2, . . .

are constructed according to the following rule:

x_1 = 2, x_2 = (x^5_1 + 1)/5x_1, x_3 = (x^5_2 + 1)/5x_2, ...

Prove that no matter how much we continued this construction, all the resulting numbers will be no less

1/5

and no more than

2

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

Problem Statement