[url=https://artofproblemsolving.com/community/c893771h1861957p12597232]8.1 The point E lies on the base [AD] of the trapezoid ABCD. The perimeters of the triangles ABE,BCE and CDE are equal. Prove that ∣BC∣=∣AD∣/2
8.2 In a convex pentagon, the lengths of all sides are equal. Find the point on the longest diagonal from which all sides are visible underneath angles not exceeding a right angle.
[url=https://artofproblemsolving.com/community/c893771h1862007p12597620]8.3 Every city in the certain state is connected by airlines with no more than with three other ones, but one can get from every city to every other city changing a plane once only or directly. What is the maximal possible number of the cities?
[url=https://artofproblemsolving.com/community/c893771h1861966p12597273]8.4*/7.4* (asterisk problems in separate posts)
[url=https://artofproblemsolving.com/community/c893771h1862002p12597605]8.5 Four different three-digit numbers starting with the same digit have the property that their sum is divisible by three of them without a remainder. Find these numbers.
[url=https://artofproblemsolving.com/community/c893771h1861967p12597280]8.6 Given a finite sequence of zeros and ones, which has two properties:
a) if in some arbitrary place in the sequence we select five digits in a row and also select five digits in any other place in a row, then these fives will be different (they may overlap);
b) if you add any digit to the right of the sequence, then property (a) will no longer hold true.
Prove that the first four digits of our sequence coincide with the last four.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988085_1969_leningrad_math_olympiad]here. leningrad math olympiadalgebrageometrycombinatoricsnumber theory