MathDB

3

Part of 2007 Tournament Of Towns

Problems(10)

Least number of hooks on a chessboard (ToT 2007-Spring-O3)

Source:

9/2/2011
What is the least number of rooks that can be placed on a standard 8×88 \times 8 chessboard so that all the white squares are attacked? (A rook also attacks the square it is on, in addition to every other square in the same row or column.)
combinatorics
Can the numbers be the same? (ToT 2007-Spring-A3)

Source:

9/2/2011
Anna's number is obtained by writing down 2020 consecutive positive integers, one after another in arbitrary order. Bob's number is obtained in the same way, but with 2121 consecutive positive integers. Can they obtain the same number?
modular arithmetic
Prove that the line AA' passes through the midpoint of BB'

Source: Tournament of Towns 2007 - Spring - Senior O-Level - P3

9/2/2011
BB is a point on the line which is tangent to a circle at the point AA. The line segment ABAB is rotated about the centre of the circle through some angle to the line segment A′B′A'B'. Prove that the line AA′AA' passes through the midpoint of BB′BB'.
rotationgeometrygeometric transformationreflectiongeometry unsolved
The equation f(x)=a has an even number of solutions

Source: Tournament of Towns 2007 - Spring - Senior A-Level - P3

9/3/2011
Let f(x)f(x) be a polynomial of nonzero degree. Can it happen that for any real number aa, an even number of real numbers satisfy the equation f(x)=af(x) = a?
algebrapolynomialalgebra proposed
DEF is equilateral => ABC is also equilateral?

Source: Tournament of Towns 2007 - Fall - Junior O-Level - P3

9/3/2011
DD is the midpoint of the side BCBC of triangle ABCABC. EE and FF are points on CACA and ABAB respectively, such that BEBE is perpendicular to CACA and CFCF is perpendicular to ABAB. If DEFDEF is an equilateral triangle, does it follow that ABCABC is also equilateral?
geometrycircumcirclegeometry proposed
The strategy to get out of the circle

Source: Tournament of Towns 2007 - Fall - Junior A-Level - P3

9/3/2011
Michael is at the centre of a circle of radius 100100 metres. Each minute, he will announce the direction in which he will be moving. Catherine can leave it as is, or change it to the opposite direction. Then Michael moves exactly 11 metre in the direction determined by Catherine. Does Michael have a strategy which guarantees that he can get out of the circle, even though Catherine will try to stop him?
combinatorics unsolvedcombinatorics
Folding a Tirangle along a Line

Source: Fall 2007 Tournament of Towns Junior P-Level #3

3/31/2015
A triangle with sides a,b,ca, b, c is folded along a line ℓ\ell so that a vertex CC is on side cc. Find the segments on which point CC divides cc, given that the angles adjacent to ℓ\ell are equal.
(2 points)
geometry
Give a construction in which AC · BC is minimum

Source: Tournament of Towns 2007 - Fall - Senior O-Level - P3

9/4/2011
Give a construction by straight-edge and compass of a point CC on a line ℓ\ell parallel to a segment ABAB, such that the product AC⋅BCAC \cdot BC is minimum.
geometrygeometry proposed
Determine all increasing arithmetic progressions

Source: Tournament of Towns 2007 - Fall - Senior A-Level - P3

9/4/2011
Determine all finite increasing arithmetic progressions in which each term is the reciprocal of a positive integer and the sum of all the terms is 11.
number theory proposednumber theory
Winning Strategy

Source: Fall 2007 Tournament of Towns Senior P-Level #3

3/31/2015
Two players in turns color the squares of a 4×44 \times 4 grid, one square at the time. Player loses if after his move a square of 2×22\times2 is colored completely. Which of the players has the winning strategy, First or Second?
(3 points)