MathDB

2

Part of 1998 Tournament Of Towns

Problems(7)

TOT 1998 Spring OJ2 chess king tours entire 8x8 chessboard

Source:

5/11/2020
A chess king tours an entire 8×88\times 8 chess board, visiting each square exactly once and returning at last to his starting position. Prove that he made an even number of diagonal moves.
(V Proizvolov)
ChessboardcombinatoricsEven
TOT 1998 Spring AJ2 MD = MG inside parallelogram, <MAD = <AMO

Source:

5/11/2020
ABCDABCD is a parallelogram. A point MM is found on the side ABAB or its extension such that ∠MAD=∠AMO\angle MAD = \angle AMO where OO is the intersection point of the diagonals of the parallelogram. Prove that MD=MGMD = MG.
(M Smurov)
geometryparallelogramequal segmentsequal angles
TOT 1998 Spring OS2 sum of product of digits of all 3-digit numbers

Source:

5/11/2020
For every four-digit number, we take the product of its four digits. Then we add all of these products together . What is the result?
( G Galperin)
number theorySumProductDigits4-digit
TOT 1998 Spring AS2 square of side 1 is divided into rectangles

Source:

5/11/2020
A square of side 11 is divided into rectangles . We choose one of the two smaller sides of each rectangle (if the rectangle is a square, then we choose any of the four sides) . Prove that the sum of the lengths of all the chosen sides is at least 11 .
(Folklore)
combinatoricscombinatorial geometryTilingRectanglesgeometryrectangle
Digits in a perfect square

Source: ToT Junior O Level Autumn 1998

10/5/2008
The units-digit of the square of an integer is 9 and the tens-digit of this square is 0. Prove that the hundreds-digit is even.
number theory proposednumber theory
TOT 1998 Autumn AJ2 8x8 square , blue, 2x1dominoes

Source:

5/11/2020
John and Mary each have a white 8×88 \times 8 square divided into 1×11 \times 1 cells. They have painted an equal number of cells on their respective squares in blue. Prove that one can cut up each of the two squares into 2×12 \times 1 dominoes so that it is possible to reassemble John's dominoes into a new square and Mary's dominoes into another square with the same pattern of blue cells.
(A Shapovalov)
combinatoricsColoringcombinatorial geometryTiling
TOT 1998 Autumn OS2 n concurrent equal paper disks

Source:

5/11/2020
On the plane are nn paper disks of radius 11 whose boundaries all pass through a certain point, which lies inside the region covered by the disks. Find the perimeter of this region.
(P Kozhevnikov)
combinatorial geometrycombinatoricsdiscsperimeterconcurrent