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National and Regional Contests
Russia Contests
Saint Petersburg Mathematical Olympiad
1965 Leningrad Math Olympiad
1965 Leningrad Math Olympiad
Part of
Saint Petersburg Mathematical Olympiad
Subcontests
(3)
grade 8
1
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1965 Leningrad Math Olympiad - Grade 8
8.1 A
24
×
60
24 \times 60
24
×
60
rectangle is divided by lines parallel to it sides, into unit squares. Draw another straight line so that after that the rectangle was divided into the largest possible number of parts. 8.2 Engineers always tell the truth, but businessmen always lie. F and G are engineers. A declares that, B asserts that, C asserts that, D says that, E insists that, F denies that G is an businessman. C also announces that D is a businessman. If A is a businessman, then how much total businessmen in this company? 8.3 There is a straight road through the field. A tourist stands on the road at a point ?. It can walk along the road at a speed of 6 km/h and across the field at a speed of 3 km/h. Find the locus of the points where the tourist can get there within an hour's walk. 8.4 / 7.5 Let
[
A
]
[A]
[
A
]
denote the largest integer not greater than
A
A
A
. Solve the equation:
[
(
5
+
6
x
)
/
8
]
=
(
15
x
−
7
)
/
5
[(5 + 6x)/8] = (15x-7)/5
[(
5
+
6
x
)
/8
]
=
(
15
x
−
7
)
/5
. 8.5. In some state, every two cities are connected by a road. Each road is only allowed to move in one direction. Prove that there is a city from which you can travel around everything. state, having visited each city exactly once. 8.6 Find all eights of prime numbers such that the sum of the squares of the numbers in the eight is 992 less than their quadruple product. [hide=original wording]Найдите все восьмерки простых чисел такие, что сумма квадратов чисел в восьмерке на 992 меньше, чем их учетверенное произведение. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988081_1965_leningrad_math_olympiad]here.
grade 7
1
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1965 Leningrad Math Olympiad - Grade 7
7.1 Prove that a natural number with an odd number of divisors is a perfect square. 7.2 In a triangle
A
B
C
ABC
A
BC
with area
S
S
S
, medians
A
K
AK
A
K
and
B
E
BE
BE
are drawn, intersecting at the point
O
O
O
. Find the area of the quadrilateral
C
K
O
E
CKOE
C
K
OE
. https://cdn.artofproblemsolving.com/attachments/0/f/9cd32bef4f4459dc2f8f736f7cc9ca07e57d05.png7.3 . The front tires of a car wear out after
25
,
000
25,000
25
,
000
kilometers, and the rear tires after
15
,
000
15,000
15
,
000
kilometers. When you need to swap tires so that the car can travel the longest possible distance with the same tires? 7.4 A
24
×
60
24 \times 60
24
×
60
rectangle is divided by lines parallel to it sides, into unit squares. How many parts will this rectangle be divided into if you also draw a diagonal in it? 7.5 / 8.4 Let
[
A
]
[A]
[
A
]
denote the largest integer not greater than
A
A
A
. Solve the equation:
[
(
5
+
6
x
)
/
8
]
=
(
15
x
−
7
)
/
5
[(5 + 6x)/8] = (15x-7)/5
[(
5
+
6
x
)
/8
]
=
(
15
x
−
7
)
/5
. 7.6 Black paint was sprayed onto a white surface. Prove that there are two points of the same color, the distance between which is
1965
1965
1965
meters. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988081_1965_leningrad_math_olympiad]here.
grade 6
1
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1965 Leningrad Math Olympiad - Grade 6
6.1 The bindery had 92 sheets of white paper and
135
135
135
sheets of colored paper. It took a sheet of white paper to bind each book. and a sheet of colored paper. After binding several books of white Paper turned out to be half as much as the colored one. How many books were bound? 6.2 Prove that if you multiply all the integers from
1
1
1
to
1965
1965
1965
, you get the number, the last whose non-zero digit is even. 6.3 The front tires of a car wear out after
25
,
000
25,000
25
,
000
kilometers, and the rear tires after
15
,
000
15,000
15
,
000
kilometers of travel. When should you swap tires so that they wear out at the same time? 6.4 A rectangle
19
19
19
cm
×
65
\times 65
×
65
cm is divided by straight lines parallel to its sides into squares with side 1 cm. How many parts will this rectangle be divided into if you also draw a diagonal in it? 6.5 Find the dividend, divisor and quotient in the example: https://cdn.artofproblemsolving.com/attachments/2/e/de053e7e11e712305a89d3b9e78ac0901dc775.png6.6 Odd numbers from
1
1
1
to
49
49
49
are written out in table form
1
3
5
7
9
\,\,\,1\,\,\,\,\,\, 3\,\,\,\,\,\, 5\,\,\,\,\,\, 7\,\,\,\,\,\, 9
1
3
5
7
9
11
13
15
17
19
11\,\,\, 13\,\,\, 15\,\,\, 17\,\,\, 19
11
13
15
17
19
21
23
25
27
29
21\,\,\, 23\,\,\, 25\,\,\, 27\,\,\, 29
21
23
25
27
29
31
33
35
37
39
31\,\,\, 33\,\,\, 35\,\,\, 37\,\,\, 39
31
33
35
37
39
41
43
45
47
49
41\,\,\, 43\,\,\, 45\,\,\, 47\,\,\, 49
41
43
45
47
49
5
5
5
numbers are selected, any two of which are not on the same line or in one column. What is their sum? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988081_1965_leningrad_math_olympiad]here.