MathDB

5

Part of 2019 Tournament Of Towns

Problems(6)

Game theory(relatively hard,help required)

Source: Tournament of towns,juniors

3/13/2019
A magician and his assistent are performing the following trick.There is a row of 12 empty closed boxes. The magician leaves the room, and a person from the audience hides a coin in each of two boxes of his choice, so that the assistent knows which boxes contain coins. The magician returns, and the assistant is allowed to open one box that does not contain a coin. Next, the magician selects 4 boxes, which are simultaneously opened. The goal of the magician is to open both boxes that contain coins. Devise a method that will allow the magician and his assistant to always succesfully perform the trick.
Game TheoryMagicianalgorithmscombinatorics
1 to n^2 numbers on n x n table , same remainder by n in diff. rows + columns

Source: Tournament of Towns, Junior A-Level , Spring 2019 p5

5/13/2020
One needs to ffll the cells of an n×nn\times n table (n>1n > 1) with distinct integers from 11 to n2n^2 so that every two consecutive integers are placed in cells that share a side, while every two integers with the same remainder if divided by nn are placed in distinct rows and distinct columns. For which nn is this possible?
(Alexandr Gribalko)
square gridnumbers in a tabletablecombinatorics
orthogonal projection of a tetrahedron onto a plane is a trapezoid

Source: Tournament of Towns, Senior A-Level , Spring 2019 p5

5/13/2020
The orthogonal projection of a tetrahedron onto a plane containing one of its faces is a trapezoid of area 11, which has only one pair of parallel sides. a) Is it possible that the orthogonal projection of this tetrahedron onto a plane containing another its face is a square of area 11? b) The same question for a square of area 1/20191/2019.
(Mikhail Evdokimov)
orthogonal projectionprojection3D geometrygeometrytetrahedrontrapezoid
sequence with sum 2018, no sum of a set of consecutive = 40, max length?

Source: Tournament of Towns, Senior O-Level , Spring 2019 p5

5/11/2020
Consider a sequence of positive integers with total sum 20192019 such that no number and no sum of a set of consecutive num bers is equal to 4040. What is the greatest possible length of such a sequence?
(Alexandr Shapovalov)
SequencealgebraSumpositive integers
unrestricted supply of bricks 1 x 1 x 3, 3 cubes 1x1x1, box m x n x k

Source: Tournament of Towns, Junior O-Level , Fall 2019 p5

4/19/2020
Basil has an unrestricted supply of straight bricks 1×1×31 \times 1 \times 3 and Γ-shape bricks made of three cubes 1×1×11\times 1\times 1. Basil filled a whole box m×n×km \times n \times k with these bricks, where m,nm, n and kk are integers greater than 11. Prove that it was sufficient to use only Γ-shape bricks.
(Mikhail Evdokimov)
combinatoricsTilingtiles
mn and (m + 1)(n + 1) are perfect squares , nice pair of pos. integers

Source: Tournament of Towns, Junior A-Level , Fall 2019 p5

4/20/2020
Let us say that the pair (m,n)(m, n) of two positive different integers m and n is nice if mnmn and (m+1)(n+1)(m + 1)(n + 1) are perfect squares. Prove that for each positive integer m there exists at least one n>mn > m such that the pair (m,n)(m, n) is nice.
(Yury Markelov)
Perfect SquaresPerfect Squarenumber theory