MathDB

4

Part of 2013 Tournament of Towns

Problems(7)

2013 ToT Spring Junior O p4, 8 rooks on a 8x8 chessboard

Source:

3/4/2020
Eight rooks are placed on a 8\times 8 chessboard, so that no two rooks attack one another. All squares of the board are divided between the rooks as follows. A square where a rook is placed belongs to it. If a square is attacked by two rooks then it belongs to the nearest rook; in case these two rooks are equidistant from this square each of them possesses a half of the square. Prove that every rook possesses the equal area.
ChessboardRookcombinatorics
2013 ToT Spring Junior A p4, 1000 nonzero real numbers painted bw

Source:

3/4/2020
On a circle, there are 10001000 nonzero real numbers painted black and white in turn. Each black number is equal to the sum of two white numbers adjacent to it, and each white number is equal to the product of two black numbers adjacent to it. What are the possible values of the total sum of 10001000 numbers?
ColoringSumProductalgebracombinatorics
2013 ToT Spring Senior O p4 100 stones have a sticker showing its true weight

Source:

3/5/2020
Each of 100100 stones has a sticker showing its true weight. No two stones weight the same. Mischievous Greg wants to rearrange stickers so that the sum of the numbers on the stickers for any group containing from 11 to 9999 stones is different from the true weight of this group. Is it always possible?
weighingscombinatorics
2013 ToT Spring Senior A p4 1, 2,...,100 are written on a circle

Source:

3/5/2020
Integers 1,2,...,1001, 2,...,100 are written on a circle, not necessarily in that order. Can it be that the absolute value of the di erence between any two adjacent integers is at least 3030 and at most 5050?
combinatorial geometrySum
2013 ToT Fall Junior O p4 KL = BC when <ALK +< LKB = 60^o

Source:

3/22/2020
Let ABCABC be an isosceles triangle. Suppose that points KK and LL are chosen on lateral sides ABAB and ACAC respectively so that AK=CLAK = CL and ∠ALK+∠LKB=60o\angle ALK + \angle LKB = 60^o. Prove that KL=BCKL = BC.
geometryisoscelesequal segmentsangles
2013 ToT Fall Junior A p4 square inside polygons in a 8x8 chessboard

Source:

3/22/2020
There is a 8×88\times 8 table, drawn in a plane and painted in a chess board fashion. Peter mentally chooses a square and an interior point in it. Basil can draws any polygon (without self-intersections) in the plane and ask Peter whether the chosen point is inside or outside this polygon. What is the minimal number of questions suffcient to determine whether the chosen point is black or white?
Chessboardcombinatoricssquare tableColoring
2013 ToT Fall Senior O p4 every n=sum offinite no of cubes of distinct integers

Source:

3/22/2020
Is it true that every integer is a sum of finite number of cubes of distinct integers?
Sumperfect cubenumber theoryInteger