MathDB

1

Part of 2011 Tournament of Towns

Problems(7)

2011 ToT Spring Junior O p1 numbers 1-2010 are placed along a circle

Source:

3/4/2020
The numbers from 11 to 20102010 inclusive are placed along a circle so that if we move along the circle in clockwise order, they increase and decrease alternately. Prove that the difference between some two adjacent integers is even.
EvencombinatoricsDifference
Hexagon geometry

Source: Tournament towns 2011

10/3/2016
Does there exist a hexagon that can be divided into four congruent triangles by a straight cut?
geometry
2011 ToT Spring Senior O p1 faces of convex polyhedron are similar triangles

Source:

3/4/2020
The faces of a convex polyhedron are similar triangles. Prove that this polyhedron has two pairs of congruent faces.
combinatorial geometrygeometryconvex polyhedronpolyhedronsimilar triangles
2011 ToT Fall Junior O p1 circumcenter is incenter

Source:

3/22/2020
PP and QQ are points on the longest side ABAB of triangle ABCABC such that AQ=ACAQ = AC and BP=BCBP = BC. Prove that the circumcentre of triangle CPQCPQ coincides with the incentre of triangle ABCABC.
geometryCircumcenterincenterequal segments
2011 ToT Fall Junior A p1 sequence of positive integers, divisors related

Source:

3/22/2020
An integer N>1N > 1 is written on the board. Alex writes a sequence of positive integers, obtaining new integers in the following manner: he takes any divisor greater than 11 of the last number and either adds it to, or subtracts it from the number itself. Is it always (for all N>1N > 1) possible for Alex to write the number 20112011 at some point?
divisornumber theory
2011 ToT Fall Senior A p1 points with different distances

Source:

3/22/2020
Pete has marked several (three or more) points in the plane such that all distances between them are different. A pair of marked points A,BA,B will be called unusual if AA is the furthest marked point from BB, and BB is the nearest marked point to AA (apart from AA itself). What is the largest possible number of unusual pairs that Pete can obtain?
combinatorial geometrycombinatoricspointsdistance
heads and tails 2 player game with n coins

Source: Tournament of Towns 2011 oral p1

5/19/2020
There are nn coins in a row. Two players take turns picking a coin and flipping it. The location of the heads and tails should not repeat. Loses the one who can not make a move. Which of player can always win, no matter how his opponent plays?
combinatoricsgamegame strategy