MathDB

1

Part of 2008 Tournament Of Towns

Problems(7)

TT2008 Junior O-Level - P1

Source:

9/4/2010
Each of ten boxes contains a di fferent number of pencils. No two pencils in the same box are of the same colour. Prove that one can choose one pencil from each box so that no two are of the same colour.
combinatorics proposedcombinatorics
TT2008 Junior A-Level - P1

Source:

9/4/2010
100100 Queens are placed on a 100×100100 \times 100 chessboard so that no two attack each other. Prove that each of four 50×5050 \times 50 corners of the board contains at least one Queen.
combinatorics unsolvedcombinatorics
TT2008 Senior O-Level - P1

Source:

9/4/2010
Alex distributes some cookies into several boxes and records the number of cookies in each box. If the same number appears more than once, it is recorded only once. Serge takes one cookie from each box and puts them on the first plate. Then he takes one cookie from each box that is still non-empty and puts the cookies on the second plate. He continues until all the boxes are empty. Then Serge records the number of cookies on each plate. Again, if the same number appears more than once, it is recorded only once. Prove that Alex's record contains the same number of numbers as Serge's record.
combinatorics unsolvedcombinatorics
TT2008 Senior A-Level - P1

Source:

9/4/2010
A square board is divided by lines parallel to the board sides (77 lines in each direction, not necessarily equidistant ) into 6464 rectangles. Rectangles are colored into white and black in alternating order. Assume that for any pair of white and black rectangles the ratio between area of white rectangle and area of black rectangle does not exceed 2.2. Determine the maximal ratio between area of white and black part of the board. White (black) part of the board is the total sum of area of all white (black) rectangles.
geometryrectangleratiocombinatorics proposedcombinatorics
2008 ToT Spring Junior O P1 equal segments , convex hexagon related

Source:

2/26/2020
In the convex hexagon ABCDEF,AB,BCABCDEF, AB, BC and CDCD are respectively parallel to DE,EFDE, EF and FAFA. If AB=DEAB = DE, prove that BC=EFBC = EF and CD=FACD = FA.
geometryhexagonequal segments
2008 ToT Spring Junior A P1 product of consecutive, add 2 digits, perfect square

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3/7/2020
An integer NN is the product of two consecutive integers. (a) Prove that we can add two digits to the right of this number and obtain a perfect square. (b) Prove that this can be done in only one way if N>12N > 12
number theoryconsecutivePerfect Square
2008 ToT Spring Senior A P1 triangle dissected into several triangles, angles

Source:

3/7/2020
A triangle has an angle of measure θ\theta. It is dissected into several triangles. Is it possible that all angles of the resulting triangles are less than θ\theta, if (a) θ=70o\theta = 70^o ? (b) θ=80o\theta = 80^o ?
anglescombinatorics