MathDB

3

Part of 2010 Tournament Of Towns

Problems(8)

Finding the measurement of angle with compass only...

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2/7/2011
An angle is given in a plane. Using only a compass, one must find out (a)(a) if this angle is acute. Find the minimal number of circles one must draw to be sure. (b)(b) if this angle equals 31∘31^{\circ}.(One may draw as many circles as one needs).
geometryrectangleexterior anglegeometry unsolved
Finding maximum sum of consecutive ten numbers in circle.

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2/8/2011
Each of 999999 numbers placed in a circular way is either 11 or −1-1. (Both values appear). Consider the total sum of the products of every 1010 consecutive numbers. (a)(a) Find the minimal possible value of this sum. (b)(b) Find the maximal possible value of this sum.
invariantinequalities unsolvedinequalities
Covering regular octahedron with regular hexagons...

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2/9/2011
Is it possible to cover the surface of a regular octahedron by several regular hexagons without gaps and overlaps? (A regular octahedron has 66 vertices, each face is an equilateral triangle, each vertex belongs to 44 faces.)
geometry3D geometryoctahedrongeometry unsolved
Can we get 2010 by trigonometric functions applied on 1?

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2/9/2011
Consider a composition of functions sin⁡,cos⁡,tan⁡,cot⁡,arcsin⁡,arccos⁡,arctan⁡,arccos⁡\sin, \cos, \tan, \cot, \arcsin, \arccos, \arctan, \arccos, applied to the number 11. Each function may be applied arbitrarily many times and in any order. (ex: sin⁡cos⁡arcsin⁡cos⁡sin⁡⋯1\sin \cos \arcsin \cos \sin\cdots 1). Can one obtain the number 20102010 in this way?
functiontrigonometryalgebra unsolvedalgebra
Rolling 1x1x1 cube on a 8x8 board with one face outside.

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2/12/2011
A 1×1×11\times 1\times 1 cube is placed on an 8×88\times 8 chessboard so that its bottom face coincides with a square of the chessboard. The cube rolls over a bottom edge so that the adjacent face now lands on the chessboard. In this way, the cube rolls around the chessboard, landing on each square at least once. Is it possible that a particular face of the cube never lands on the chessboard?
geometry3D geometrycombinatorics unsolvedcombinatorics
Proving that each cyclist has had at least 25 meetings.

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2/13/2011
At a circular track, 1010 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 2525 meetings.
floor functioncombinatorics unsolvedcombinatorics
Speed of car is 90% of cruiser.

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2/19/2011
From a police station situated on a straight road in nite in both directions, a thief has stolen a police car. Its maximal speed equals 9090% of the maximal speed of a police cruiser. When the theft is discovered some time later, a policeman starts to pursue the thief on a cruiser. However, he does not know in which direction along the road the thief has gone, nor does he know how long ago the car has been stolen. Is it possible for the policeman to catch the thief?
ratioabsolute value
Prove that sum of fractions in polygon is less then 2

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5/21/2010
For each side of a given polygon, divide its length by the total length of all other sides. Prove that the sum of all the fractions obtained is less than 22.
inequalitiesinequalities unsolved