MathDB

7.1 A rectangle that is not a square is inscribed in a square. Prove that its semi-perimeter is equal to the diagonal of the square.
7.2 Find five numbers whose pairwise sums are 0, 2, 4,5, 7, 9, 10, 12, 14, 17.
7.3 In a

1000

-digit number, all but one digit is a five. Prove that this number is not a perfect square.
7.4 / 6.5 Several teams took part in the volleyball tournament. Team A is considered stronger than team B if either A beat B or there is a team C such that A beat C, and C beat B. Prove that if team T is the winner of the tournament, then it is the strongest the rest of the teams.
7.5 In a pentagon

ABCDE

K

is the midpoint of

AB

L

is the midpoint of

BC

M

is the midpoint of

CD

N

is the midpoint of

DE

P

is the midpoint of

KM

Q

is the midpoint of

LN

. Prove that the segment

PQ

is parallel to side

AE

and is equal to its quarter. https://cdn.artofproblemsolving.com/attachments/2/5/be8e9b0692d98115dbad04f960e8a856dc593f.png
7.6 / 8.4 Several circles are arbitrarily placed in a circle of radius

3

, the sum of their radii is

25

. Prove that there is a straight line that intersects at least

9

of these circles.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988084_1968_leningrad_math_olympiad]here.

Problem Statement