1961 Leningrad Math Olympiad

Part of Saint Petersburg Mathematical Olympiad

Subcontests

(3)

1

1961 Leningrad Math Olympiad - Grade 8

8.1 Construct a quadrilateral using side lengths and distances between the midpoints of the diagonals.
8.2 It is known that

a,b

\sqrt{a}+\sqrt{b}

are rational numbers. Prove that then

\sqrt{a}

\sqrt{b}

are rational.
8.3 / 9.2 Solve equation

x^3 - [x]=3

8.4 Prove that if in a triangle the angle bisector of the vertex, bisects the angle between the median and the altitude, then the triangle either isosceles or right. .
8.5 Given

n

x_1, x_2, . . . , x_n

, each of which is equal to

+1

-1

. At the same time

x_1x_2 + x_2x_3 + . . . + x_{n-1}x_n + x_nx_1 = 0 .

n

is divisible by

4

.
8.6 There are

n

points marked on the circle, and it is known that for of any two, one of the arcs connecting them has a measure less than

120^0

.Prove that all points lie on an arc of size

120^0

.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983442_1961_leningrad_math_olympiad]here.

1

1961 Leningrad Math Olympiad - Grade 7

7.1. / 6.5 Prove that out of any six people there will always be three pairs of acquaintances or three pairs of strangers.
7.2 Given a circle

O

K

, as well as a line

L

. Construct a segment of given length parallel to

L

and such that its ends lie on

O

K

respectively
7.3 The three-digit number

\overline{abc}

is divisible by

37

. Prove that the sum of the numbers

\overline{bca}

\overline{cab}

is also divisible by

37

. (typo corrected)
7.4. Point

C

is the midpoint of segment

AB

. On an arbitrary ray drawn from point

C

and not lying on line

AB

, three consecutive points

P

M

Q

PM=MQ

AP+BQ>2CM

. https://cdn.artofproblemsolving.com/attachments/f/3/a8031007f5afc31a8b5cef98dd025474ac0351.png
7.5. Given

2n+1

different objects. Prove that you can choose an odd number of objects from them in as many ways as an even number.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c3983442_1961_leningrad_math_olympiad]here.

1

1961 Leningrad Math Olympiad - Grade 6

6.1. Three workers can do some work. Second and the third can together complete it twice as fast as the first, the first and the third can together complete it three times faster than the second. At what time since the first and second can do this job faster than the third?
6.2. Prove that the greatest common divisor of the sum of two numbers and their least common multiple is equal to their greatest common divisor the numbers themselves.
6.3. There were 20 schoolchildren at the consultation and 20 problems were dealt with. It turned out that each student solved two problems and each problem was solved by two schoolchildren. Prove that it is possible to organize the analysis in this way tasks so that everyone solves one problem and all tasks are solved. [hide=original wording] Наконсультациибыло20школьниковиразбиралось20задач. Оказалось, что каждый школьник решил две задачи и каждую задачу решило два школьника. Докажите, что можно так организовать разбор задач, чтобыкаждыйрассказалоднузадачуивсезадачибылирассказаны.
6.4.Two people Α and Β must get from point Μ to point Ν,located 15 km from M. On foot they can move at a speed of 6 km/h. In addition, they have a bicycle at their disposal, on which υou can drive at a speed of 15 km/h. A and B depart from Μ at the same time, A walks, and B rides a bicycle until meeting pedestrian C, going from N to M. Then B walks and C rides a bicycle to meeting with A, hands him a bicycle, on which he arrives at N. When must pedestrian C leave Nfor A and B to arrive at N simultaneously if he walks at the same speed as A and B?
6.5./ 7.1 Prove that out of any six people there will always be three pairs of acquaintances or three pairs of strangers.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983442_1961_leningrad_math_olympiad]here.