MathDB

2

Part of 2010 Tournament Of Towns

Problems(8)

Cutting a piece of cake...

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2/7/2011
Karlson and Smidge divide a cake in a shape of a square in the following way. First, Karlson places a candle on the cake (chooses some interior point). Then Smidge makes a straight cut from the candle to the boundary in the direction of his choice. Then Karlson makes a straight cut from the candle to the boundary in the direction perpendicular to Smidge's cut. As a result, the cake is split into two pieces; Smidge gets the smaller one. Smidge wants to get a piece which is no less than a quarter of the cake. Can Karlson prevent Smidge from getting the piece of that size?
geometry unsolvedgeometry
Finding ratio OM/PC if BO=BP in triangle ABC.

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2/7/2011
Let MM be the midpoint of side ACAC of the triangle ABCABC. Let PP be a point on the side BCBC. If OO is the point of intersection of APAP and BMBM and BO=BPBO = BP, determine the ratio OMPC\frac{OM}{PC} .
ratiogeometry unsolvedgeometry
Prove that f(x)=x^2

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2/9/2011
Let f(x)f(x) be a function such that every straight line has the same number of intersection points with the graph y=f(x)y = f(x) and with the graph y=x2y = x^2. Prove that f(x)=x2.f(x) = x^2.
functionconicsparabolaalgebra unsolvedalgebra
Diving into equal masses for rational or irrational a

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2/9/2011
Alex has a piece of cheese. He chooses a positive number a and cuts the piece into several pieces one by one. Every time he choses a piece and cuts it in the same ratio 1:a1 : a. His goal is to divide the cheese into two piles of equal masses. Can he do it if (a)a(a) a is irrational? (b)a(b) a is rational, a≠1?a \neq 1?
rationumber theory unsolvednumber theory
Angle bisector bisects area of bicentric quadrilateral.

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2/12/2011
In a quadrilateral ABCDABCD with an incircle, AB=CD;BC<ADAB = CD; BC < AD and BCBC is parallel to ADAD. Prove that the bisector of ∠C\angle C bisects the area of ABCDABCD.
geometryparallelogramgeometry unsolved
Dividing line segment in given ratio with an instrument.

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2/12/2011
Pete has an instrument which can locate the midpoint of a line segment, and also the point which divides the line segment into two segments whose lengths are in a ratio of n:(n+1)n : (n + 1), where nn is any positive integer. Pete claims that with this instrument, he can locate the point which divides a line segment into two segments whose lengths are at any given rational ratio. Is Pete right?
rationumber theoryrelatively primegeometry proposedgeometry
Perpendicular diagonals and equality with sum of inradii.

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2/19/2011
The diagonals of a convex quadrilateral ABCDABCD are perpendicular to each other and intersect at the point OO. The sum of the inradii of triangles AOBAOB and CODCOD is equal to the sum of the inradii of triangles BOCBOC and DOADOA. (a)(a) Prove that ABCDABCD has an incircle. (b)(b) Prove that ABCDABCD is symmetric about one of its diagonals.
geometryperimeterinradiusgeometry proposed
Among 2n cyclists, each has had at least n^2 meetings.

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2/19/2011
At a circular track, 2n2n cyclists started from some point at the same time in the same direction with different constant speeds. If any two cyclists are at some point at the same time again, we say that they meet. No three or more of them have met at the same time. Prove that by the time every two cyclists have met at least once, each cyclist has had at least n2n^2 meetings.
combinatorics unsolvedcombinatorics