MathDB

7.1 . The area of the quadrilateral is

3

^2

, and the lengths of its diagonals are

6

cm and

2

cm. Find the angle between the diagonals.
7.2 Prove that the number

1 + 2^{3456789}

is composite.
7.3

20

people took part in the chess tournament. The participant who took clear (undivided)

19

th place scored

9.5

points. How could they distribute points among other participants?
7.4 The sum of the distances between the midpoints of opposite sides of a quadrilateral is equal to its semi-perimeter. Prove that this quadrilateral is a parallelogram.
7.5

40

people travel on a bus without a conductor passengers carrying only coins in denominations of

10

15

and

20

kopecks. Total passengers have

49

coins. Prove that passengers will not be able to pay the required amount of money to the ticket office and pay each other correctly. (Cost of a bus ticket in 1963 was 5 kopecks.)
7.6 Some natural number

a

is divided with a remainder by all natural numbers less than

a

. The sum of all the different (!) remainders turned out to be equal to

a

. Find

a

.
7.7 Two squares were cut out of a chessboard. In what case is it possible and in what case not to cover the remaining squares of the board with dominoes (i.e., figures of the form

2\times 1

) without overlapping?
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here.

Problem Statement