a) What is the minimum number of pieces of wire needed in order to weld a cube’s frame?
b) What is the maximum length of a piece of wire that can be cut from this frame? (The length of the edge of the cube is 1 cm).
Prove that in a bipartite planar graph \(E \geq 2F\), if \(E \geq 2\) (\(E\) is the number of edges, \(F\) is the number of regions).
12 teams played a volleyball tournament in one round. Two teams scored exactly 7 wins.
Prove that there are teams \(A\), \(B\), \(C\) where \(A\) won against \(B\), \(B\) won against \(C\), and \(C\) won against \(A\).
A scone contains raisins and sultanas. Prove that inside the scone there will always be two points 1cm apart such that either both lie inside raisins, both inside sultanas, or both lie outside of either raisins or sultanas.
Does the number of 1999 occur in the Pascal triangle?
Find a natural number greater than one that occurs in the Pascal triangle a) more than three times; b) more than four times.
How many times greater is the sum of the numbers in the hundred and first line of the Pascal triangle than the sum of the numbers in the hundredth line?
Let’s put plus and minus signs in the 99th line of Pascal’s triangle. Between the first and second number there is a minus sign, between the second and the third there is a plus sign, between the third and the fourth there is a minus sign, then again a plus sign, and so on. Find the value of the resulting expression.
a) Can 4 points be placed on a plane so that each of them is connected by segments with three points (without intersections)?
b) Can 6 points be placed on a plane and connected by non-intersecting segments so that exactly 4 segments emerge from each point?
For which \(n > 3\), can a set of weights with masses of \(1, 2, 3, ..., n\) grams be divided into three groups of equal mass?