Problem 1. Submitted by Evguenia Kaganova. This problem was given to her in 1973.
Calculate: tan (pi/7) * tan (3pi/7) * tan (5pi/7). Here I used the notation pi instead of the corresponding Greek letter.
Problem 2. Submitted by Maxim Umansky. This problem was given to his friend in the early 70s.
Let p, q be two irrational numbers. Can pq be rational? Can pq be irrational?
You can find my solution, a solution submitted by Alexey Radul, and a solution submitted by Frederick Lewis here.
Problem 3. Submitted by Michael Shtilman. This problem was given to an acquaintance of his in 1969.
The graph of a monotonically increasing function is cut off with two horizontal lines. Find a point on the graph between intersections such that the sum of the two areas bounded by the lines, the graph and the vertical line through this point is minimum.
Problem 4. Submitted by Michael Shtilman. This problem was given to another acquaintance of his in 1982.
Given finite sequence of zeros and ones, a new sequence is created so that zeros are substituted with ones and vice versa. This new sequence is added at the end of the first one: If the first sequence was 0110, the new one will be 1001, and the final one will be 01101001. Starting with a one figure sequence the process is repeated infinitely. Now we consider this infinite sequence as decimals: 0.01101001... . Prove that this number is irrational.
Problem 5. Submitted by Michael Shtilman. This problem was given to yet a third acquaintance of his in 1974.
The radii of the circumscribed and inscribed circles for a triangle are R and r respectively. What are the maximum and minimum possible distances between the centers of these circles?
Comment: This problem is similar to problem 29 of the main list. In this wording, the problem becomes confusing, as the distance between the centers is uniquely specified by the radii.
Problem 6. Submitted by Dima Barboy. This problem was given to him in 1982.
Given an irregular octahedron, a sphere is built on each edge in such a way that the edge is a diameter of the corresponding sphere. Prove that the spheres cover the inside of the octahedron completely.
Problem 7. This problem is from a discussion on LiveJournal, submitted by the user "arbat". The solution is there too.
Take a cyclic quadrilateral. Build a perpendicular from the center of each side to the opposite side. Prove that all these perpendiculars intersect in one point.
Last revised August 2008