1980 Tournament Of Towns

Part of Tournament Of Towns

Subcontests

(6)

1

TOT 006 1980 Spring S3 30 vectors in space, <45^o angle

30

non-zero vectors in

3

dimensional space. Prove that among these there are two such that the angle between them is less than

45^o

1

TOT 005 1980 Spring J5 S5 regions by line segments of length 18, area

A finite set of line segments, of total length

18

, belongs to a square of unit side length (we assume that the square includes its boundary and that a line segment includes its end points). The line segments are parallel to the sides of the square and may intersect one another. Prove that among the regions into which the square is divided by the line segments, at least one of these must have area not less than

0.01

.
(A Berzinsh, Riga)

1

TOT 004 1980 Spring J4 S4 sum of areas in a grid NxN of random convex ABCD

We are given convex quadrilateral

ABCD

. Each of its sides is divided into

N

line segments of equal length. The points of division of side

AB

are connected with the points of division of side

CD

by straight lines (which we call the first set of straight lines), and the points of division of side BC are connected with the points of division of side

DA

by straight lines (which we call the second set of straight lines) as shown in the diagram, which illustrates the case

N = 4

N^2

smaller quadrilaterals. From these we choose

N

quadrilaterals in such a way that any two are at least divided by one line from the first set and one line from the second set. Prove that the sum of the areas of these chosen quadrilaterals is equal to the area of

ABCD

N

.
(A Andjans, Riga) http://4.bp.blogspot.com/-8Qqk4r68nhU/XVco29-HzzI/AAAAAAAAKgo/UY8mXxg7tD0OrS6bEnoAw7Vuf31BuOE8wCK4BGAYYCw/s1600/TOT%2B1980%2BSpring%2BJ4.png

1

TOT 003 1980 Spring J3 permutations of 2,...,102 divisible by k for all k

If permutations of the numbers

2, 3,4,..., 102

a_i,a_2, a_3,...,a_{101}

, find all such permutations in which

a_k

is divisible by

k

k

1

TOT 002 1980 Spring J2 S2 delete a row in NxN array of numbers

N \times N

array of numbers, all rows are different (two rows are said to be different even if they differ only in one entry). Prove that there is a column which can be deleted in such a way that the resulting rows will still be different.
(A Andjans, Riga)

1

TOT 001 1980 Spring J1 S1 red and blue points on circumference

On the circumference of a circle there are red and blue points. One may add a red point and change the colour of both its neighbours (to the other colour) or remove a red point and change the colour of both its previous neighbours. Initially there are two red points. Prove that there is no sequence of allowed operations which leads to the configuration consisting of two blue points.
(K Kazarnovskiy, Moscow)