Řešitelský seminář, 4. 4. 2017
Problem 1. Prove that JS unbounded. (S(n) is the sum of the digits of positive integer n in its decimal
represen tation.
Problem 2. Suppose that a graph G is the union of three trees. Is it true that G can be covered by two planar graphs?
Problem 3. Show that if k < and T is a family ofkxk submatrices of and n x n matrix such that any two intersect then
T-l (n ~  ^ 2
-    \k ~ 1
Problem 4. Let n be a positive integer, and S\,... Sn be a collection of unite non-empty sets such that
\Sj n sj\ < 1
IS^ \ \S j\
l<z<j<n
Prove that there exits pairwaise distinct elements x\,... xn such that Xi is a member of Si for each index i.
Domácí úloha
Problem 5. Determine the integers k > 2 for which the sequence (2n), n = 1, 2, 3,... is eventually periodic.