MathDB

5

Part of 1999 Tournament Of Towns

Problems(7)

a square is cut into 100 rectangles by 9 + 9 lines parallel to it's sided

Source: Tournament Of Towns Spring 1999 Junior 0 Level p5

5/7/2020
A square is cut into 100100 rectangles by 99 straight lines parallel to one of the sides and 99 lines parallel to another. If exactly 99 of the rectangles are actually squares, prove that at least two of these 99 squares are of the same size .
(V Proizvolov)
geometryrectangleSquaressquarecombinatorial geometrycombinatorics
TOT 1999 Spring AJ5 T lies on angle bisector, incircle related

Source:

5/11/2020
The sides ABAB and ACAC are tangent at points PP and QQ, respectively, to the incircle of a triangle ABC.RABC. R and SS are the midpoints of the sides ACAC and BCBC, respectively, and TT is the intersection point of the lines PQPQ and RSRS. Prove that TT lies on the bisector of the angle BB of the triangle.
(M Evdokimov)
geometryangle bisectorincircle
TOT 1999 Spring OS5 game on a 9 x 9 board, guarantee a score of at least B

Source:

5/11/2020
Two people play a game on a 9×99 \times 9 board. They move alternately. On each move, the first player draws a cross in an empty cell, and the second player draws a nought in an empty cell. When all 8181 cells are filled, the number KK of rows and columns in which there are more crosses and the number HH of rows and columns in which there are more noughts are counted. The score for the first player is the difference B=K−HB = K- H. Find a value of BB such that the first player can guarantee a score of at least BB, while the second player can hold the first player's score to at most B, regardless how the opponent plays.
(A Kanel)
combinatoricsgamegame strategysquare tabletable
TOT 1999 Spring AS5 sequence with parity of 1s in binary representation

Source:

5/11/2020
For every non-negative integer ii, define the number M(i)M(i) as follows: write ii down as a binary number, so that we have a string of zeroes and ones, if the number of ones in this string is even, then set M(i)=0M(i) = 0, otherwise set M(i)=1M(i) = 1. (The first terms of the sequence M(i)M(i), i=0,1,2,...i = 0, 1, 2, ... are 0,1,1,0,1,0,0,1,...0, 1, 1, 0, 1, 0, 0, 1,... ) (a) Consider the finite sequence M(O),M(1),...,M(1000)M(O), M(1), . . . , M(1000) . Prove that there are at least 320320 terms in this sequence which are equal to their neighbour on the right : M(i)=M(i+1)M(i) = M(i + 1 ) . (b) Consider the finite sequence M(O),M(1),...,M(1000000)M(O), M(1), . . . , M(1000000) . Prove that the number of terms M(i)M(i) such that M(i)=M(i+7)M(i) = M(i +7) is at least 450000450000.
(A Kanel)
number theorydecimal representationBinarypower of 2Sequence
TOT 1999 Autumn OJ5 divide a 6x6 chessboard into 18 rectangles (1x2 or 2x1)

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5/11/2020
Is it possible to divide a 6×66 \times 6 chessboard into 1818 rectangles, each either 1×21 \times 2 or 2×12 \times 1, and to draw exactly one diagonal on each rectangle such that no two of these diagonals have a common endpoint?
(A Shapovalov)
rectanglecombinatorial geometrycombinatoricsTiling
TOT 1999 Autumn AS3 number in sequence repeated >=100 times

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5/11/2020
Tireless Thomas and Jeremy construct a sequence. At the beginning there is one positive integer in the sequence. Then they successively write new numbers in the sequence in the following way: Thomas obtains the next number by adding to the previous number one of its (decimal) digits, while Jeremy obtains the next number by subtracting from the previous number one of its digits. Prove that there is a number in this sequence which will be repeated at least 100100 times.
(A Shapovalov)
combinatoricsSequenceDigits
TOT 1999 Autumn OS5 divide a 8x8 chessboard into 32 rectangles (1x2 or 2x1)

Source:

5/11/2020
Is it possible to divide a 8×88 \times 8 chessboard into 3232 rectangles, each either 1×21 \times 2 or 2×12 \times 1, and to draw exactly one diagonal on each rectangle such that no two of these diagonals have a common endpoint?
(A Shapovalov)
combinatorial geometrycombinatoricsrectangleTiling