7
Part of 2012 Tournament of Towns
Problems(4)
2012 ToT Spring Junior A p7 angle between 3 incenters
Source:
3/4/2020
Let be an altitude of an equilateral triangle . Let be the incentre of triangle , and let and be the incentres of triangles and respectively. Determine .
incenterangleEquilateralaltitude
2012 ToT Spring Senior A p7 a pile of 100 pebbles
Source:
3/5/2020
Konstantin has a pile of pebbles. In each move, he chooses a pile and splits it into two smaller ones until he gets piles each with a single pebble.
(a) Prove that at some point, there are piles containing a total of exactly pebbles.
(b) Prove that at some point, there are piles containing a total of exactly pebbles.
(c) Prove that Konstantin may proceed in such a way that at no point, there are piles containing a total of exactly pebbles.
combinatorics
2012 ToT Fall Junior A p7 sum of digits = 2012
Source:
3/22/2020
Peter and Paul play the following game. First, Peter chooses some positive integer with the sum of its digits equal to . Paul wants to determine this number, he knows only that the sum of the digits of Peter’s number is . On each of his moves Paul chooses a positive integer and Peter tells him the sum of the digits of . What is the minimal number of moves in which Paul can determine Peter’s number for sure?
sum of digitsnumber theory
2012 ToT Fall Senior A p7 1 000 000$ soldiers in a line
Source:
3/22/2020
There are soldiers in a line. The sergeant splits the line into segments (the length of different segments may be different) and permutes the segments (not changing the order of soldiers in each segment) forming a new line. The sergeant repeats this procedure several times (splits the new line in segments of the same lengths and permutes them in exactly the same way as the first time). Every soldier originally from the first segment recorded the number of performed procedures that took him to return to the first segment for the first time. Prove that at most of these numbers are different.
combinatorics