MathDB

7.1 Given a convex

n

-gon all of whose angles are obtuse. Prove that the sum of the lengths of the diagonals in it is greater than the sum of the lengths of the sides.
7.2 Find all integer values for

x

and

y

such that

x^4 + 4y^4

is a prime number. (typo corrected)
7.3. Given a triangle

ABC

. Parallelograms

ABKL

BCMN

and

ACFG

are constructed on the sides, Prove that the segments

KN

MF

and

GL

can form a triangle. https://cdn.artofproblemsolving.com/attachments/a/f/7a0264b62754fafe4d559dea85c67c842011fc.png
7.4 / 6.2 Prove that a

10 \times 10

chessboard cannot be covered with

25

figures like https://cdn.artofproblemsolving.com/attachments/0/4/89aafe1194628332ec13ad1c713bb35cbefff7.png.
7.5 Find the greatest number of different natural numbers, each of which is less than

50

, and every two of which are coprime.
7.6. Given a triangle

ABC

D

and

E

are the midpoints of the sides

AB

and

BC

. Point

M

lies on

AC

ME > EC

. Prove that

MD < AD

. https://cdn.artofproblemsolving.com/attachments/e/c/1dd901e0121e5c75a4039d21b954beb43dc547.png
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983461_1964_leningrad_math_olympiad]here.

Problem Statement