MathDB

1

Part of 2006 Tournament of Towns

Problems(7)

Proving CB = MN

Source: Spring 2006 Tournament of Towns Junior O-Level #1

4/15/2015
Let ∠A\angle A in a triangle ABCABC be 60∘60^\circ. Let point NN be the intersection of ACAC and perpendicular bisector to the side ABAB while point MM be the intersection of ABAB and perpendicular bisector to the side ACAC. Prove that CB=MNCB = MN.
(3 points)
geometryperpendicular bisector
Billiard Balls

Source: Spring 2006 Tournament of Towns Junior A-Level #1

4/15/2015
There is a billiard table in shape of rectangle 2×12 \times 1, with pockets at its corners and at midpoints of its two largest sizes. Find the minimal number of balls one has to place on the table interior so that any pocket is on a straight line with some two balls. (Assume that pockets and balls are points).
(4 points)
Convex Polyhedron with 100 Edges

Source: Spring 2006 Tournament of Towns Senior O-Level #1

9/9/2015
All vertices of a convex polyhedron with 100 edges are cut off by some planes. The planes do not intersect either inside or on the surface of the polyhedron. For this new polyhedron find a) the number of vertices; (1 point) b) the number of edges. (2 points)
3D geometrygeometry
TOT 2006 Spring - Senior A-Level p1 50 points inside of any convex 100-gon

Source:

2/25/2020
Prove that one can always mark 5050 points inside of any convex 100100-gon, so that each its vertix is on a straight line connecting some two marked points. (4)
combinatoricscombinatorial geometry
TOT 2006 Fall - Junior O-Level p1 numbers iin blackboard

Source:

2/25/2020
Two positive integers are written on the blackboard. Mary records in her notebook the square of the smaller number and replaces the larger number on the blackboard by the difference of the two numbers. With the new pair of numbers, she repeats the process, and continues until one of the numbers on the blackboard becomes zero. What will be the sum of the numbers in Mary's notebook at that point? (4)
combinatoricsSum
TOT 2006 Fall - Senior O-Level p1 sum of numbers in Mary's notebook

Source:

2/25/2020
Three positive integers xx and yy are written on the blackboard. Mary records in her notebook the product of any two of them and reduces the third number on the blackboard by 11. With the new trio of numbers, she repeats the process, and continues until one of the numbers on the blackboard becomes zero. What will be the sum of the numbers in Mary's notebook at that point? (4)
number theory
TOT 2006 Fall - Junior A-Level p1 7-gon and 17-gon, areas

Source:

2/25/2020
Two regular polygons, a 77-gon and a 1717-gon are given. For each of them two circles are drawn, an inscribed circle and a circumscribed circle. It happened that rings containing the polygons have equal areas. Prove that sides of the polygons are equal. (3)
Heptagon17-gonareasgeometryincirclecircumcircleregular polygon