MathDB

4

Part of 2010 Tournament Of Towns

Problems(8)

Arranging participants in a specific way.

Source:

2/7/2011
At the math contest each participant met at least 33 pals who he/she already knew. Prove that the Jury can choose an even number of participants (more than two) and arrange them around a table so that each participant be set between these who he/she knows.
combinatorics unsolvedcombinatorics
Find n such that S(n)=100, S(n^3)=100^3, S is sum of digits.

Source:

2/7/2011
Can it happen that the sum of digits of some positive integer nn equals 100100 while the sum of digits of number n3n^3 equals 1003100^3?
number theory unsolvednumber theory
Finding polynomial P(x) given P(2) and P(P(2))

Source:

2/9/2011
Assume that P(x)P(x) is a polynomial with integer non negative coefficients, different from constant. Baron Munchausen claims that he can restore P(x)P(x) provided he knows the values of P(2)P(2) and P(P(2))P(P(2)) only. Is the baron's claim valid?
algebrapolynomialalgebra unsolvedBaron Munchausen
Partitioning 5000 movie fans into two kind of groups.

Source:

2/9/2011
50005000 movie fans gathered at a convention. Each participant had watched at least one movie. The participants should be split into discussion groups of two kinds. In each group of the fi rst kind, the members would discuss a movie they all watched. In each group of the second kind, each member would tell about the movie that no one else in this group had watched. Prove that the chairman can always split the participants into exactly 100 groups. (A group consisting of one person is allowed; in this case this person submits a report).
inductioncombinatorics unsolvedcombinatorics
Students knowing English, German and French.

Source:

2/12/2011
In a school, more than 90%90\% of the students know both English and German, and more than 90%90\% percent of the students know both English and French. Prove that more than 90%90\% percent of the students who know both German and French also know English.
percentratiocombinatorics proposedcombinatorics
Dividing rectangle in to dominoes and drawing diagonals...

Source:

2/13/2011
A rectangle is divided into 2×12\times 1 and 1×21\times 2 dominoes. In each domino, a diagonal is drawn, and no two diagonals have common endpoints. Prove that exactly two corners of the rectangle are endpoints of these diagonals.
geometryrectangleanalytic geometrygeometric transformationreflectioncombinatorics unsolvedcombinatorics
Total area of black cells >= Total area of white cells

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2/19/2011
A square board is dissected into n2n^2 rectangular cells by n−1n-1 horizontal and n−1n-1 vertical lines. The cells are painted alternately black and white in a chessboard pattern. One diagonal consists of nn black cells which are squares. Prove that the total area of all black cells is not less than the total area of all white cells.
geometryinductionrectangle
Winning strategy for the second wizard...

Source:

2/19/2011
Two dueling wizards are at an altitude of 100100 above the sea. They cast spells in turn, and each spell is of the form "decrease the altitude by aa for me and by bb for my rival" where aa and bb are real numbers such that 0<a<b0 < a < b. Different spells have different values for aa and bb. The set of spells is the same for both wizards, the spells may be cast in any order, and the same spell may be cast many times. A wizard wins if after some spell, he is still above water but his rival is not. Does there exist a set of spells such that the second wizard has a guaranteed win, if the number of spells is (a)(a) finite; (b)(b) in finite?