MathDB

7.1. / 6.5 Prove that out of any six people there will always be three pairs of acquaintances or three pairs of strangers.
7.2 Given a circle

O

and a square

K

, as well as a line

L

. Construct a segment of given length parallel to

L

and such that its ends lie on

O

and

K

respectively
7.3 The three-digit number

\overline{abc}

is divisible by

37

. Prove that the sum of the numbers

\overline{bca}

and

\overline{cab}

is also divisible by

37

. (typo corrected)
7.4. Point

C

is the midpoint of segment

AB

. On an arbitrary ray drawn from point

C

and not lying on line

AB

, three consecutive points

P

M

and

Q

so that

PM=MQ

. Prove that

AP+BQ>2CM

. https://cdn.artofproblemsolving.com/attachments/f/3/a8031007f5afc31a8b5cef98dd025474ac0351.png
7.5. Given

2n+1

different objects. Prove that you can choose an odd number of objects from them in as many ways as an even number.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c3983442_1961_leningrad_math_olympiad]here.

Problem Statement