MathDB

3

Part of 1998 Tournament Of Towns

Problems(8)

TOT 1998 Spring OJ3 OM/ON = AB/CD

Source:

5/11/2020
ABAB and CDCD are segments lying on the two sides of an angle whose vertex is OO. AA is between OO and BB, and CC is between OO and DD . The line connecting the midpoints of the segments ADAD and BCBC intersects ABAB at MM and CDCD at NN. Prove that OMON=ABCD\frac{OM}{ON}=\frac{AB}{CD}
(V Senderov)
ratiogeometrymidpointsangle
TOT 1998 Spring AJ3 six dice, 6dgit number divisible by 7

Source:

5/11/2020
Six dice are strung on a rigid wire so that the wire passes through two opposite faces of each die. Each die can be rotated independently of the others. Prove that it is always possible to rotate the dice and then place the wire horizontally on a table so that the six-digit number formed by their top faces is divisible by 77. (The faces of a die are numbered from 11 to 66, the sum of the numbers on opposite faces is always equal to 77.)
(G Galperin)
combinatoricsdivisibledivides
TOT 1998 Spring OS3 max colours to pain 8x8 chessboard

Source:

5/11/2020
What is the maximum number of colours that can be used to paint an 8×88 \times 8 chessboard so that every square is painted in a single colour, and is adjacent , horizontally, vertically but not diagonally, to at least two other squares of its own colour?
(A Shapovalov)
ChessboardColoringcombinatorics
TOT 1998 Spring AS3 1,2, 22,23 ,..., 210 written on blackboard.

Source:

5/11/2020
(a) The numbers 1,2,4,8,16,32,64,1281 , 2, 4, 8, 1 6 , 32, 64, 1 28 are written on a blackboard. We are allowed to erase any two numbers and write their difference instead (this is always a non-negative number). After this procedure has been repeated seven times, only a single number will remain. Could this number be 9797? (b) The numbers 1,2,22,23,...,2101 , 2, 22, 23 , . . . , 210 are written on a blackboard. We are allowed to erase any two numbers and write their difference instead (this is always a non-negative number) . After this procedure has been repeated ten times, only a single number will remain. What values could this number have?
(A.Shapovalov)
combinatorics
Points on a triangle

Source: ToT Junior O Level Autumn 1998

10/5/2008
In a triangle ABC ABC the points A′ A', B′ B' and C′ C' lie on the sides BC BC, CA CA and AB AB, respectively. It is known that \angle AC'B' \equal{} \angle B'A'C, \angle CB'A' \equal{} \angle A'C'B and \angle BA'C' \equal{} \angle C'B'A. Prove that A′ A', B′ B' and C′ C' are the midpoints of the corresponding sides.
geometry proposedgeometry
TOT 1998 Autumn AJ3 quadrilateral is tangential, circles, tangents

Source:

5/11/2020
Segment ABAB intersects two equal circles, is parallel to the line joining their centres, and all the points of intersection of the segment and the circles lie between AA and BB. From the point AA tangents to the circle nearest to AA are drawn, and from the point BB tangents to the circle nearest to BB are also drawn. It turns out that the quadrilateral formed by the four tangents extended contains both circles. Prove that a circle can be drawn so that it touches all four sides of the quadrilateral.
(P Kozhevnikov)
geometrytangentialcirclesTangents
TOT 1998 Autumn AS3, magic 3x3, sum of product in rows=sum of product in columns

Source:

5/11/2020
Nine numbers are arranged in a square table: a1   a2   a3a_1 \,\,\, a_2 \,\,\,a_3 b1   b2   b3b_1 \,\,\,b_2 \,\,\,b_3 c1   c2   c3c_1\,\,\, c_2 \,\,\,c_3 . It is known that the six numbers obtained by summing the rows and columns of the table are equal: a1+a2+a3=b1+b2+b3=c1+c2+c3=a1+b1+c1=a2+b2+c2=a3+b3+c3a_1 + a_2 + a_3 = b_1 + b_2 + b_3 = c_1 + c_2 + c_3 = a_1 + b_1 + c_1 = a_2 + b_2 + c_2 = a_3 + b_3 + c_3 . Prove that the sum of products of numbers in the rows is equal to the sum of products of numbers in the columns: a1b1c1+a2b2c2+a3b3c3=a1a2a3+b1b2b3+c1c2c3a_1 b_1 c_1 + a_2 b_2c_2 + a_3b_3c_3 = a_1a_2a_3 + b_1 b_2 b_3 + c_1 c_2c_3 .
(V Proizvolov)
tangentialmagic squaretablesquare table
TOT 1998 Autumn OS3 Knight moves on 17 marked cells of 8x8 chessboard

Source:

5/11/2020
On an 8×88 \times 8 chessboard, 1717 cells are marked. Prove that one can always choose two cells among the marked ones so that a Knight will need at least three moves to go from one of the chosen cells to the other.
(R Zhenodarov)
combinatoricsChessboard