1996 Moscow Mathematical Olympiad

Part of Moscow Mathematical Olympiad

Subcontests

(6)

1

Moscow 1996 G8 P5 - Rook on a Chess Board

A rook stands in a corner of an

n

n

chess board. For what

n

, moving alternately along horizontals and verticals, can the rook visit all the cells of the board and return to the initial corner after

n^2

moves? (A cell is visited only if the rook stops on it, those that the rook “flew over” during the move are not counted as visited.)
Proposed by A. Spivak

1

Moscow 1996 G8 P6 - Students and Problems

Eight students solved

8

problems.
a) It turned out that each problem was solved by

5

students. Prove that there are two students such that each problem is solved by at least one of them.
b) If it turned out that each problem was solved by

4

students, it can happen that there is no pair of students such that each problem is solved by at least one of them. (Give an example.)
Proposed by S. Tokarev

1

Moscow 1996 G8 P4 - Equilateral triangle

Consider an equilateral triangle

\triangle ABC

K

L

BC

into three equal parts, the point

M

divides the leg

AC

1:2

, counting from the vertex

A

\angle AKM+\angle ALM=30^{\circ}

.
Proposed by V. Proizvolov

1

Moscow 1996 G8 P3 - Gardeners and Nodes

At the nodes of graph paper, gardeners live; everywhere around them grow flowers. Each flower is to be taken care of by the three gardeners nearest to it. One of the gardeners wishes to know which are the flowers (s)he has to take care of. Sketch the plot of these gardeners.
Proposed by I. F. Sharygin

1

Moscow 1996 G8 P2 - 10 iron weights

Along a circle, 10 iron weights have been placed. Between every two weights there is a brass ball. The mass of each ball is equal to the difference of the masses of its neighboring weights. Prove that it is possible to divide the balls among two pans so as to make the balance in equilibrium.
Proposed by V. Proizvolov

1

Moscow 1996 G8 P1 - Is it true that a=b?

It is known that

a+\frac{b^2}{a}=b+\frac{a^2}{b}

. Is it true that

a=b

a

b

are nonzero real numbers?
Proposed by R.Fedorov