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National and Regional Contests
Russia Contests
Saint Petersburg Mathematical Olympiad
1963 Leningrad Math Olympiad
1963 Leningrad Math Olympiad
Part of
Saint Petersburg Mathematical Olympiad
Subcontests
(3)
grade 8
1
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1963 Leningrad Math Olympiad - Grade 8
8.1 On the median drawn from the vertex of the triangle to the base, point
A
A
A
is taken. The sum of the distances from
A
A
A
to the sides of the triangle is equal to
s
s
s
. Find the distances from
A
A
A
to the sides if the lengths of the sides are equal to
x
x
x
and
y
y
y
. 8.2 Fraction
0
,
a
b
c
.
.
.
0, abc...
0
,
ab
c
...
is composed according to the following rule:
a
a
a
and
c
c
c
are arbitrary digits, and each next digit is equal to the remainder of the sum of the previous two digits when divided by
10
10
10
. Prove that this fraction is purely periodic. 8.3 Two convex polygons with
m
m
m
and
n
n
n
sides are drawn on the plane (
m
>
n
m>n
m
>
n
). What is the greatest possible number of parts, they can break the plane? 8.4 The sum of three integers that are perfect squares is divisible by
9
9
9
. Prove that among them, there are two numbers whose difference is divisible by
9
9
9
. 8.5 / 9.5 Given
k
+
2
k+2
k
+
2
integers. Prove that among them there are two integers such that either their sum or their difference is divisible by
2
k
2k
2
k
. 8.6 A right angle rotates around its vertex. Find the locus of the midpoints of the segments connecting the intersection points sides of an angle and a given circle. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here.
grade 7
1
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1963 Leningrad Math Olympiad - Grade 7
7.1 . The area of the quadrilateral is
3
3
3
cm
2
^2
2
, and the lengths of its diagonals are
6
6
6
cm and
2
2
2
cm. Find the angle between the diagonals. 7.2 Prove that the number
1
+
2
3456789
1 + 2^{3456789}
1
+
2
3456789
is composite. 7.3
20
20
20
people took part in the chess tournament. The participant who took clear (undivided)
19
19
19
th place scored
9.5
9.5
9.5
points. How could they distribute points among other participants? 7.4 The sum of the distances between the midpoints of opposite sides of a quadrilateral is equal to its semi-perimeter. Prove that this quadrilateral is a parallelogram. 7.5
40
40
40
people travel on a bus without a conductor passengers carrying only coins in denominations of
10
10
10
,
15
15
15
and
20
20
20
kopecks. Total passengers have
49
49
49
coins. Prove that passengers will not be able to pay the required amount of money to the ticket office and pay each other correctly. (Cost of a bus ticket in 1963 was 5 kopecks.) 7.6 Some natural number
a
a
a
is divided with a remainder by all natural numbers less than
a
a
a
. The sum of all the different (!) remainders turned out to be equal to
a
a
a
. Find
a
a
a
. 7.7 Two squares were cut out of a chessboard. In what case is it possible and in what case not to cover the remaining squares of the board with dominoes (i.e., figures of the form
2
×
1
2\times 1
2
×
1
) without overlapping? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here.
grade 6
1
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1963 Leningrad Math Olympiad - Grade 6
6.1 Two people went from point A to point B. The first one walked along highway at a speed of 5 km/h, and the second along a path at a speed of 4 km/h. The first of them arrived at point B an hour later and traveled 6 kilometers more. Find the distance from A to B along the highway. 6.2. A pedestrian walks along the highway at a speed of 5 km/hour. Along this highway in both directions at the same speed Buses run, meeting every 5 minutes. At 12 o'clock the pedestrian noticed that the buses met near him and, Continuing to walk, he began to count those oncoming and overtaking buses. At 2 p.m., buses met near him again. It turned out that during this time the pedestrian encountered 4 buses more than overtook him. Find the speed of the bus 6.3. Prove that the difference
4
3
43
−
1
7
17
43^{43} - 17^{17}
4
3
43
−
1
7
17
is divisible by
10
10
10
. 6.4. Two squares are cut out of the chessboard on the border of the board. When is it possible and when is it not possible to cover with the remaining squares of the board? shapes of the view without overlay? 6.5. The distance from city A to city B (by air) is 30 kilometers, from B to C - 80 kilometers, from C to D - 236 kilometers, from D to E - 86 kilometers, from E to A- 40 kilometers. Find the distance from E to C. 6.6. Is it possible to write down the numbers from
1
1
1
to
1963
1963
1963
in a series so that any two adjacent numbers and any two numbers located one after the other were mutually prime? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here.