MathDB

2

Part of 2004 Tournament Of Towns

Problems(6)

TOT 2004 Fall - Junior O-Level p 2 n consecutive with prime sum

Source:

2/25/2020
Find all possible values of n≥1n \ge 1 for which there exist nn consecutive positive integers whose sum is a prime number.
combinatoricsconsecutiveprimeSum
TOT 2004 Spring - Junior A-Level p2 max checkers on 8x8 checkerboard

Source:

2/25/2020
What is the maximal number of checkers that can be placed on an 8×88\times 8 checkerboard so that each checker stands on the middle one of three squares in a row diagonally, with exactly one of the other two squares occupied by another checker?
combinatorics
A box containing balls

Source: Tournament of towns, Junior B-Level paper, Fall 2004

12/25/2004
A box contains red, green, blue, and white balls, 111 balls in all. If you take out 100 balls without looking, then there will always be 4 balls of different colors among them. What is the smallest number of balls you must take out without looking to guarantee that among them there will always be balls of at least 3 different colors?
combinatorics unsolvedcombinatorics
Incircle and equilateral triangle

Source: Tournament of towns, Junior A-Level paper, Fall 2004

12/25/2004
The incircle of the triangle ABC touches the sides BC, AC, and AB at points A', B', and C', respectively. It is known that AA'=BB'=CC'. Does the triangle ABC have to be equilateral? (I am very interested in ingenious solution of this problem, because I found an ugly one using Stewart's theorem and tons of algebra during the contest).
geometrytrigonometrygeometry solved
TOT 2004 Fall - Senior O-Level p2 box contains 100 red, blue, white balls

Source:

2/25/2020
A box contains red, blue, and white balls, 100100 balls in total. It is known that among any 2626 of them there are always 1010 balls of the same color. Find the minimal number NN such that among any NN balls there are always 3030 balls of the same color.
Coloringcombinatorics
Pile of stones and two persons

Source: Tournament of towns, Senior A-Level paper, Fall 2004

12/25/2004
Two persons are playing the following game. They have a pile of stones and take turns removing stones from it, with the first player taking the first turn. At each turn, the first player removes either 1 or 10 stones from the pile, and the second player removes either m or n stones. The player who can not make his move loses. It is known that for any number of stones in the pile, the first player can always win (regardless of the second player's moves). What are the possible values of m and n?
combinatorics unsolvedcombinatorics