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.png8.2 A is an odd integer, x and y are roots of equation t2+Atā1=0. Prove that x4+y4 and x5+y5 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. leningrad math olympiadalgebrageometrynumber theorycombinatorics