2017 Tournament Of Towns

Part of Tournament Of Towns

Subcontests

(6)

1

Dominos Overwhelming

1\times 2

dominoes are placed on an

8 \times 8

chessboard without overlapping. They may partially stick out from the chessboard but the center of each domino must be strictly inside the chessboard (not on its border). Place on the chessboard in such a way: a) at least

40

dominoes, (3 points) b) at least

41

dominoes, (3 points) c) more than

41

dominoes. (6 points)
(Mikhail Evdokimov)

1

Hopping grasshoppers

A grasshopper can jump along a checkered strip for

8, 9

10

cells in any direction. A natural number

n

is called jumpable if the grasshopper can start from some cell of a strip of length

n

and visit every cell exactly once. Find at least one non-jumpable number

n > 50

.
(Egor Bakaev)

1

Control Weights

There is a set of control weights, each of them weighs a non-integer number of grams. Any integer weight from

1

40

g can be balanced by some of these weights (the control weights are on one balance pan, and the measured weight on the other pan).What is the least possible number of the control weights?
(Alexandr Shapovalov)

1

Inscribable Hexagon

All the sides of the convex hexagon

ABCDEF

are equal. In addition,

AD = BE = CF

. Prove that a circle can be inscribed into this hexagon.
(Boyan Obukhov)

1

Inequalities on sums of powers

From given positive numbers, the following infinite sequence is defined:

a_1

is the sum of all original numbers,

a_2

is the sum of the squares of all original numbers,

a_3

is the sum of the cubes of all original numbers, and so on (

a_k

is the sum of the

k

-th powers of all original numbers). a) Can it happen that

a_1 > a_2 > a_3 > a_4 > a_5

a_5 < a_6 < a_7 < \ldots

? (4 points) b) Can it happen that

a_1 < a_2 < a_3 < a_4 < a_5

a_5 > a_6 > a_7 > \ldots

? (4 points)
(Alexey Tolpygo)

1

A Chess Tournament

A chess tournament had 10 participants. Each round, the participants split into pairs, and each pair played a game. In total, each participant played with every other participant exactly once, and in at least half of the games both the players were from the same town. Prove that during each round there was a game played by two participants from the same town.
(Boris Frenkin)