MathDB

5

Part of 2010 Tournament Of Towns

Problems(8)

Finding least number that can be left...

Source:

2/7/2011
101101 numbers are written on a blackboard: 12,22,32,⋯ ,10121^2, 2^2, 3^2, \cdots, 101^2. Alex choses any two numbers and replaces them by their positive difference. He repeats this operation until one number is left on the blackboard. Determine the smallest possible value of this number.
invariant
N horsemen riding along a circle road for N=3 and N=10

Source:

2/8/2011
NN horsemen are riding in the same direction along a circular road. Their speeds are constant and pairwise distinct. There is a single point on the road where the horsemen can surpass one another. Can they ride in this fashion for arbitrarily long time? Consider the cases: (a)N=3;(a) N = 3; (b)N=10.(b) N = 10.
algebra unsolvedalgebra
Rotating segment by 45 degrees and interchanging endpoints.

Source:

2/9/2011
A needle (a segment) lies on a plane. One can rotate it 45∘45^{\circ} round any of its endpoints. Is it possible that after several rotations the needle returns to initial position with the endpoints interchanged?
rotationgeometrygeometric transformationanalytic geometryinvariantgeometry unsolved
33 horsemen riding along a circular road.

Source:

2/9/2011
3333 horsemen are riding in the same direction along a circular road. Their speeds are constant and pairwise distinct. There is a single point on the road where the horsemen can surpass one another. Can they ride in this fashion for arbitrarily long time ?
summer programMathcampnumber theoryleast common multiplelinear algebramatrixgreatest common divisor
Circle with 2N pairwise equidistant points on it.

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2/12/2011
A circle is divided by 2N2N points into 2N2N arcs of length 11. These points are joined in pairs to form NN chords. Each chord divides the circle into two arcs, the length of each being an even integer. Prove that NN is even.
combinatorics unsolvedcombinatorics
Prove that sum of fractions in pentagon is less then 2

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2/13/2011
For each side of a given pentagon, divide its length by the total length of all other sides. Prove that the sum of all the fractions obtained is less than 2.
inequalitiestriangle inequalityinequalities unsolved
Maximal number of matches for the winner of the tournament.

Source:

2/19/2011
In a tournament with 5555 participants, one match is played at a time, with the loser dropping out. In each match, the numbers of wins so far of the two participants differ by not more than 11. What is the maximal number of matches for the winner of the tournament?
combinatorics unsolvedcombinatorics
Circumcenter of AOC belongs to BD

Source:

2/12/2011
The quadrilateral ABCDABCD is inscribed in a circle with center OO. The diagonals ACAC and BDBD do not pass through OO. If the circumcentre of triangle AOCAOC lies on the line BDBD, prove that the circumcentre of triangle BODBOD lies on the line ACAC.
geometrycircumcircle