MathDB

7

Part of 2002 Tournament Of Towns

Problems(4)

Another existence question

Source: Tournament of Towns,Spring 2002, Senior A Level, P7

5/14/2014
Do there exist irrational numbers a,ba,b both greater than 11, such that ⌊am⌋≠⌊bn⌋\lfloor{a^m}\rfloor\neq \lfloor{b^n}\rfloor for all m,n∈Nm,n\in\mathbb{N} ?
floor functionnumber theory proposednumber theory
dominoes and chains

Source: Tournament of Towns,Spring 2002, Junior A Level, P7

5/13/2014
Some domino pieces are placed in a chain according to standard rules. In each move, we may remove a sub-chain with equal numbers at its ends, turn the whole sub-chain around, and put it back in the same place. Prove that for every two legal chains formed from the same pieces and having the same numbers at their ends, we can transform one to another in a finite sequence of moves.
combinatorics unsolvedcombinatorics
A power grid on a lattice

Source: Tournament of Towns, Fall 2002, Junior A Level, P7

5/17/2014
[*] A power grid with the shape of a 3×33\times 3 lattice with 1616 nodes (vertices of the lattice) joined by wires (along the sides of squares. It may have happened that some of the wires have burned out. In one test technician can choose any two nodes and check if electrical current circulates between them (i.e there is a chain of intact wires joining the chosen nodes) . Technicial knows that current will circulate from any node to another node. What is the least number of tests required to demonstrate this? [*] Previous problem for the grid of 5×55\times 5 lattice.
ceiling functioncombinatorics proposedcombinatorics
A power grid on a lattice(once more)

Source: Tournament of Towns, Fall 2002, Senior A Level, P7

5/17/2014
[*] A power grid with the shape of a 3×33\times 3 lattice with 1616 nodes (vertices of the lattice) joined by wires (along the sides of squares. It may have happened that some of the wires have burned out. In one test technician can choose any two nodes and check if electrical current circulates between them (i.e there is a chain of intact wires joining the chosen nodes) . Technicial knows that current will circulate from any node to another node. What is the least number of tests required to demonstrate this? [*] Previous problem for the grid of 7×77\times 7 lattice.
combinatorics proposedcombinatorics