MathDB

7.1. Prove that from the sides of an arbitrary quadrilateral you can fold a trapezoid.
7.2 / 6.2 The numbers

A

and

B

are relatively prime. What common divisors can have the numbers

A+B

and

A-B

?
7.3. / 6.4

15

magazines lie on the table, completely covering it. Prove that it is possible to remove eight of them so that the remaining magz cover at least

7/15

of the table area.
7.4 In a six-digit number that is divisible by

7

, the last digit has been moved to the beginning. Prove that the resulting number is also divisible at

7

.
[url=https://artofproblemsolving.com/community/c6h3391057p32066818]7.5* (asterisk problems in separate posts)
7.6 On sides

AB

and

BC

of triangle

ABC

, are constructed squares

ABDE

and

BCKL

with centers

O_1

and

O_2

M_1

and

M_2

are midpoints of segments

DL

and

AC

. Prove that

O_1M_1O_2M_2

is a square. https://cdn.artofproblemsolving.com/attachments/8/1/8aa816a84c5ac9de78b396096cf718063de390.png
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983459_1962_leningrad_math_olympiad]here.

Problem Statement