MathDB

5

Part of 2003 Tournament Of Towns

Problems(7)

Tiling a 2003 × 2003 board by 1 × 2 and 1 × 3 rectangles

Source: Tournament of Towns Spring 2003 - Junior O-Level - Problem 5

6/14/2011
Is it possible to tile 2003×20032003 \times 2003 board by 1×21 \times 2 dominoes placed horizontally and 1×31 \times 3 rectangles placed vertically?
geometryrectangleanalytic geometrygeometric transformationreflectioncomplex numberscombinatorics proposed
Largest number of squares on 9 × 9 square with the property

Source:

6/14/2011
What is the largest number of squares on 9×99 \times 9 square board that can be cut along their both diagonals so that the board does not fall apart into several pieces?
geometryrectangle
Cut a rectangle into pieces and rearrange them into a square

Source: Tournament of Towns Spring 2003 - Senior O-Level - Problem 5

6/14/2011
Prove that one can cut a×ba \times b rectangle, b2<a<b\frac{b}{2} < a < b, into three pieces and rearrange them into a square (without overlaps and holes).
geometryrectanglecombinatorics proposedcombinatorics
Does Mary have a winning strategy?

Source: Tournament of Towns Spring 2003 - Senior A-Level - Problem 5

6/15/2011
Prior to the game John selects an integer greater than 100100.
Then Mary calls out an integer dd greater than 11. If John's integer is divisible by dd, then Mary wins. Otherwise, John subtracts dd from his number and the game continues (with the new number). Mary is not allowed to call out any number twice. When John's number becomes negative, Mary loses. Does Mary have a winning strategy?
modular arithmeticcombinatorics unsolvedcombinatorics
Moving checkers by a given rule to finally reverse the order

Source: ToT 2003-JO-5

6/18/2011
2525 checkers are placed on 2525 leftmost squares of 1×N1 \times N board. Checker can either move to the empty adjacent square to its right or jump over adjacent right checker to the next square if it is empty. Moves to the left are not allowed. Find minimal NN such that all the checkers could be placed in the row of 2525 successive squares but in the reverse order.
combinatorics unsolvedcombinatorics
Placing paper in one layer

Source: ToT 2003 SO-5

6/26/2011
A paper tetrahedron is cut along some of so that it can be developed onto the plane. Could it happen that this development cannot be placed on the plane in one layer?
geometry3D geometrytetrahedroncombinatorics unsolvedcombinatorics
Beautiful inequality regarding angles in a square.

Source: ToT 2003-JA-5

6/19/2011
A point OO lies inside of the square ABCDABCD. Prove that the difference between the sum of angles OAB,OBC,OCD,ODAOAB, OBC, OCD , ODA and 180∘180^{\circ} does not exceed 45∘45^{\circ}.
inequalitiesgeometry unsolvedgeometry