MathDB

8.1 In the parallelogram

ABCD

, the diagonal

AC

is greater than the diagonal

BD

. The point

M

on the diagonal

AC

is such that around the quadrilateral

BCDM

one can circumscribe a circle. Prove that

BD

is the common tangent of the circles circumscribed around the triangles

ABM

and

ADM

. https://cdn.artofproblemsolving.com/attachments/b/3/9f77ff1f2198c201e5c270ec5b091a9da4d0bc.png
8.2

A

is an odd integer,

x

and

y

are roots of equation

t^2+At-1=0

. Prove that

x^4 + y^4

and

x^5+ y^5

are coprime integer numbers.
8.3 A regular triangle is reflected symmetrically relative to one of its sides. The new triangle is again reflected symmetrically about one of its sides. This is repeated several times. It turned out that the resulting triangle coincides with the original one. Prove that an even number of reflections were made.
8.4 /7.6 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.
8.5 All two-digit numbers that do not end in zero are written one after another so that each subsequent number begins with that the same digit with which the previous number ends. Prove that you can do this and find the sum of the largest and smallest of all multi-digit numbers that can be obtained in this way.
[url=https://artofproblemsolving.com/community/c6h3390996p32049528]8,6* (asterisk problems in separate posts)
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988084_1968_leningrad_math_olympiad]here.

Problem Statement