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Contests
National and Regional Contests
Russia Contests
Saint Petersburg Mathematical Olympiad
1966 Leningrad Math Olympiad
1966 Leningrad Math Olympiad
Part of
Saint Petersburg Mathematical Olympiad
Subcontests
(3)
grade 8
1
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1966 Leningrad Math Olympiad - Grade 8
8.1 / 7.4 What number needs to be put in place * so that the next the problem had a unique solution: “There are n straight lines on the plane, intersecting at * points. Find n.” ? 8.2 / 7.3 Prove that for any natural number
n
n
n
the number
n
(
2
n
+
1
)
(
3
n
+
1
)
.
.
.
(
1966
n
+
1
)
n(2n+1)(3n+1)...(1966n + 1)
n
(
2
n
+
1
)
(
3
n
+
1
)
...
(
1966
n
+
1
)
is divisible by every prime number less than
1966
1966
1966
. 8.3 / 7.6 There are
n
n
n
points on the plane so that any triangle with vertices at these points has an area less than
1
1
1
. Prove that all these points can be enclosed in a triangle of area
4
4
4
. 8.4 Prove that the sum of all divisors of the number
n
2
n^2
n
2
is odd. 8.5 A quadrilateral has three obtuse angles. Prove that the larger of its two diagonals emerges from the vertex of an acute angle. 8.6 Numbers
x
1
,
x
2
,
.
.
.
x_1, x_2, . . .
x
1
,
x
2
,
...
are constructed according to the following rule:
x
1
=
2
,
x
2
=
(
x
1
5
+
1
)
/
5
x
1
,
x
3
=
(
x
2
5
+
1
)
/
5
x
2
,
.
.
.
x_1 = 2, x_2 = (x^5_1 + 1)/5x_1, x_3 = (x^5_2 + 1)/5x_2, ...
x
1
=
2
,
x
2
=
(
x
1
5
+
1
)
/5
x
1
,
x
3
=
(
x
2
5
+
1
)
/5
x
2
,
...
Prove that no matter how much we continued this construction, all the resulting numbers will be no less
1
/
5
1/5
1/5
and no more than
2
2
2
. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988082_1966_leningrad_math_olympiad]here.
grade 7
1
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1966 Leningrad Math Olympiad - Grade 7
7.1 / 6.3 All integers from 1 to 1966 are written on the board. Allowed is to erase any two numbers by writing their difference instead. Prove that repeating such an operation many times cannot ensure that There are only zeros left on the board. 7.2 Prove that the radius of a circle is equal to the difference between the lengths of two chords, one of which subtends an arc of
1
/
10
1/10
1/10
of a circle, and the other subtends an arc in
3
/
10
3/10
3/10
of a circle. 7.3 Prove that for any natural number
n
n
n
the number
n
(
2
n
+
1
)
(
3
n
+
1
)
.
.
.
(
1966
n
+
1
)
n(2n+1)(3n+1)...(1966n + 1)
n
(
2
n
+
1
)
(
3
n
+
1
)
...
(
1966
n
+
1
)
is divisible by every prime number less than
1966
1966
1966
. 7.4 What number needs to be put in place * so that the next the problem had a unique solution: “There are n straight lines on the plane, intersecting at * points. Find n.” ? 7.5 / 6.4 Black paint was sprayed onto a white surface. Prove that there are three points of the same color lying on the same line, and so, that one of the points lies in the middle between the other two. 7.6 There are
n
n
n
points on the plane so that any triangle with vertices at these points has an area less than
1
1
1
. Prove that all these points can be enclosed in a triangle of area
4
4
4
. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988082_1966_leningrad_math_olympiad]here.
grade 6
1
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1966 Leningrad Math Olympiad - Grade 6
6.1 Which number is greater
1000...001
⏟
1965
z
e
r
o
e
s
/
1000...001
⏟
1966
z
e
r
o
e
s
o
r
1000...001
⏟
1966
z
e
r
o
e
s
/
1000...001
⏟
1967
z
e
r
o
e
s
?
\underbrace{1000. . . 001}_{1965\, zeroes} / \underbrace{1000 . . . 001}_{1966\, zeroes} \,\,\, or \,\,\, \underbrace{1000. . . 001}_{1966\, zeroes} / \underbrace{1000 . . . 001}_{1967\, zeroes} \,\,?
1965
zeroes
1000...001
/
1966
zeroes
1000...001
or
1966
zeroes
1000...001
/
1967
zeroes
1000...001
?
6.2
30
30
30
teams participate in the football championship. Prove that at any moment there will be two teams that have played at this point the same number of matches. 6.3./ 7.1 All integers from
1
1
1
to
1966
1966
1966
are written on the board. Allowed is to erase any two numbers by writing their difference instead. Prove that repeating such an operation many times cannot ensure that There are only zeros left on the board. 6.4 / 7.5 Black paint was sprayed onto a white surface. Prove that there are three points of the same color lying on the same line, and so, that one of the points lies in the middle between the other two. 6.5 In a chess tournament, there are more than three chess players, and each player plays each other the same number of times. There were
26
26
26
rounds in the tournament. After the
13
13
13
th round, one of the participants discovered that he had an odd number points, and each of the other participants has an even number of points. How many chess players participated in the tournament? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988082_1966_leningrad_math_olympiad]here.