MathDB

2

Part of 2002 Tournament Of Towns

Problems(7)

Is this dissection possible?

Source: Tournament of Towns,Spring 2002, Junior O Level, P2

5/13/2014
Can any triangle be cut into four convex figures: a triangle, a quadrilateral, a pentagon, a hexagon?
geometry proposedgeometry
A game with black and white chips

Source: Tournament of Towns,Spring 2002, Junior A Level, P2

5/13/2014
A game is played on a 23×2323\times 23 board. The first player controls two white chips which start in the bottom left and top right corners. The second player controls two black ones which start in bottom right and top left corners. The players move alternately. In each move, a player moves one of the chips under control to a square which shares a side with the square the chip is currently in. The first player wins if he can bring the white chips to squares which share a side with each other. Can the second player prevent the first player from winning?
geometryrectanglecombinatorics proposedcombinatorics
The Reflection triangle and a collinearity

Source: Tournament of Towns,Spring 2002, Senior O Level, P2

5/14/2014
ΔABC\Delta ABC and its mirror reflection ΔA′B′C′\Delta A^{\prime}B^{\prime}C^{\prime} is arbitrarily placed on the plane. Prove the midpoints of AA′,BB′,CC′AA^{\prime},BB^{\prime},CC^{\prime} are collinear.
geometrygeometric transformationreflectionrotationgeometry proposed
Arbitrarily close points

Source: Tournament of Towns,Spring 2002, Senior A Level, P2

5/14/2014
Does there exist points A,BA,B on the curve y=x3y=x^3 and on y=x3+∣x∣+1y=x^3+|x|+1 respectively such that distance between A,BA,B is less than 1100\frac{1}{100} ?
limitalgebra solvedalgebra
Mary and her number

Source: Tournament of Towns, Fall 2002, Junior O Level, P2

5/15/2014
John and Mary select a natural number each and tell that to Bill. Bill wrote their sum and product in two papers hid one paper and showed the other to John and Mary. John looked at the number (which was 20022002 ) and declared he couldn't determine Mary's number. Knowing this Mary also said she couldn't determine John's number as well. What was Mary's Number?
number theory proposednumber theory
A bound on side length

Source: Tournament of Towns, Fall 2002, Senior A Level, P2

5/17/2014
A cube is cut by a plane such that the cross section is a pentagon. Show there is a side of the pentagon of length ℓ\ell such that the inequality holds: ∣ℓ−1∣>15 |\ell-1|>\frac{1}{5}
geometry3D geometryinequalitiesgeometry proposed
Catalogued species

Source: Tournament of Towns, Fall 2002, Junior A Level, P2

5/16/2014
All the species of plants existing in Russia are catalogued (numbered by integers from 22 to 20002000 ; one after another, without omissions or repetitions). For any pair of species the gcd of their catalogue numbers was calculated and recorded but the catalogue numbers themselves were lost. Is it possible to restore the catalogue numbers from the data in hand?
number theorygreatest common divisorcombinatorics proposedcombinatorics