6

Part of 2011 Tournament of Towns

Problems(4)

Source: Tournament of towns

Two ants crawl along the sides of the

49

7 * 7

board. Each ant passes through all

64

vertices exactly once and returns to its starting point. What is the smallest possible number of sides covered by both ants?

2011 ToT Fall Junior A p6 sum of angles < 1100^o, fence along highway

Source:

A car goes along a straight highway at the speed of

60

100

meter long fence is standing parallel to the highway. Every second, the passenger of the car measures the angle of vision of the fence. Prove that the sum of all angles measured by him is less than

1100

2011 ToT Fall Senior A p6 1^1 + 3^3 + 5^5 + .. + (2^n - 1)^{2^n-1}

Source:

Prove that the integer

1^1 + 3^3 + 5^5 + .. + (2^n - 1)^{2^n-1}

is a multiple of

2^n

but not a multiple of

2^{n+1}

dividesdivisornumber theorypower of 2

Tournament of Towns, oral round, p6

Source:

On the plane there are centrally symmetric convex polygon with area 1 and two his copies (each obtained from a polygon by some parallel transfer). It is known that no point of the plane is not covered by the three polygons at once. Prove that the total area covered by polygons, at least 2.

geometrygeometry unsolved