MathDB

3

Part of 2004 Tournament Of Towns

Problems(7)

3 buckets

Source: Tournament of Towns Spring 2004 Junior O #3

5/22/2014
Bucket AA contains 3 litres of syrup. Bucket BB contains nn litres of water. Bucket CC is empty. We can perform any combination of the following operations: - Pour away the entire amount in bucket XX, - Pour the entire amount in bucket XX into bucket YY, - Pour from bucket XX into bucket YY until buckets YY and ZZ contain the same amount.
(a) How can we obtain 10 litres of 30% syrup if n=20n = 20? (b) Determine all possible values of nn for which the task in (a) is possible.
modular arithmeticalgebra unsolvedalgebra
TOT 2004 Spring - Junior A-Level p3 increases or decreases by n %

Source:

2/25/2020
Each day, the price of the shares of the corporation “Soap Bubble, Limited” either increases or decreases by nn percent, where nn is an integer such that 0<n<1000 < n < 100. The price is calculated with unlimited precision. Does there exist an nn for which the price can take the same value twice?
combinatoricsalgebra
TOT 2004 Spring - Senior O-Level p3 perimeter and diagonals convex ABCD

Source:

2/25/2020
Perimeter of a convex quadrilateral is 20042004 and one of its diagonals is 10011001. Can another diagonal be 11 ? 22 ? 10011001 ?
geometryperimeterdiagonals
TOT 2004 Spring - Senior A-Level p3 max projection of triangular pyramid

Source:

2/25/2020
The perpendicular projection of a triangular pyramid on some plane has the largest possible area. Prove that this plane is parallel to either a face or two opposite edges of the pyramid.
3D geometrygeometryprojectionpyramid
Mr. Ivanov and mr. Petrov

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

12/25/2004
We have a number of towns, with bus routes between some of them (each bus route goes from a town to another town without any stops). It is known that you can get from any town to any other by bus (possibly changing buses several times). Mr. Ivanov bought one ticket for each of the bus routes (a ticket allows single travel in either direction, but not returning on the same route). Mr. Petrov bought n tickets for each of the bus routes. Both Ivanov and Petrov started at town A. Ivanov used up all his tickets without buying any new ones and finished his travel at town B. Petrov, after using some of his tickets, got stuck at town X: he can not leave it without buying a new ticket. Prove that X is either A or B.
combinatorics unsolvedcombinatorics
Knights on the chessboard

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

12/25/2004
What is the maximal number of knights that can be placed on the usual 8x8 chessboard so that each of then threatens at most 7 others?
combinatorics unsolvedcombinatorics
Polynomials (very bad name for a topic, I know...)

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

12/25/2004
P(x) and Q(x) are polynomials of positive degree such that for all x P(P(x))=Q(Q(x)) and P(P(P(x)))=Q(Q(Q(x))). Does this necessarily mean that P(x)=Q(x)?
algebrapolynomialalgebra solved