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Contests
National and Regional Contests
Russia Contests
Saint Petersburg Mathematical Olympiad
1962 Leningrad Math Olympiad
1962 Leningrad Math Olympiad
Part of
Saint Petersburg Mathematical Olympiad
Subcontests
(4)
grade 8
1
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1962 Leningrad Math Olympiad - Grade 8
8.1 Four circles are placed on planes so that each one touches the other two externally. Prove that the points of tangency lie on one circle. https://cdn.artofproblemsolving.com/attachments/9/8/883a82fb568954b09a4499a955372e2492dbb8.png8.2. Let the integers
a
a
a
and
b
b
b
be represented as
x
2
−
5
y
2
x^2-5y^2
x
2
−
5
y
2
, where
x
x
x
and
y
y
y
are integer numbers. Prove that the number
a
b
ab
ab
can also be presented in this form. 8.3 Solve the equation
x
(
x
+
d
)
(
x
+
2
d
)
(
x
+
3
d
)
=
a
x(x + d)(x + 2d)(x + 3d) = a
x
(
x
+
d
)
(
x
+
2
d
)
(
x
+
3
d
)
=
a
. 8.4 / 9.1 Let
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
,
m
+
n
+
p
=
1
m+n+p=1
m
+
n
+
p
=
1
. Prove that
−
1
≤
a
m
+
b
n
+
c
p
≤
1
-1 \le am + bn + cp \le 1
−
1
≤
am
+
bn
+
c
p
≤
1
8.5 Inscribe a triangle with the largest area in a semicircle. 8.6 Three circles of the same radius intersect at one point. Prove that the other three points intersections lie on a circle of the same radius. https://cdn.artofproblemsolving.com/attachments/4/7/014952f2dcf0349d54b07230e45a42c242a49d.png8.7 Find the circle of smallest radius that contains a given triangle. 8.8 / 9.2 Given a polynomial
x
2
n
+
a
1
x
2
n
−
2
+
a
2
x
2
n
−
4
+
.
.
.
+
a
n
−
1
x
2
+
a
n
,
x^{2n} +a_1x^{2n-2} + a_2x^{2n-4} + ... + a_{n-1}x^2 + a_n,
x
2
n
+
a
1
x
2
n
−
2
+
a
2
x
2
n
−
4
+
...
+
a
n
−
1
x
2
+
a
n
,
which is divisible by
x
−
1
x-1
x
−
1
. Prove that it is divisible by
x
2
−
1
x^2-1
x
2
−
1
. 8.9 Prove that for any prime number
p
p
p
other than
2
2
2
and from
5
5
5
, there is a natural number
k
k
k
such that only ones are involved in the decimal notation of the number
p
k
pk
p
k
.. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983459_1962_leningrad_math_olympiad]here.
7.5*
1
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3 colors for 49 maps on a circle
The circle is divided into
49
49
49
areas so that no three areas touch at one point. The resulting “map” is colored in three colors so that no two adjacent areas have the same color. The border of two areas is considered to be colored in both colors. Prove that on the circle there are two diametrically opposite points, colored in one color.
grade 7
1
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1962 Leningrad Math Olympiad - Grade 7
7.1. Prove that from the sides of an arbitrary quadrilateral you can fold a trapezoid. 7.2 / 6.2 The numbers
A
A
A
and
B
B
B
are relatively prime. What common divisors can have the numbers
A
+
B
A+B
A
+
B
and
A
−
B
A-B
A
−
B
? 7.3. / 6.4
15
15
15
magazines lie on the table, completely covering it. Prove that it is possible to remove eight of them so that the remaining magz cover at least
7
/
15
7/15
7/15
of the table area. 7.4 In a six-digit number that is divisible by
7
7
7
, the last digit has been moved to the beginning. Prove that the resulting number is also divisible at
7
7
7
. [url=https://artofproblemsolving.com/community/c6h3391057p32066818]7.5* (asterisk problems in separate posts) 7.6 On sides
A
B
AB
A
B
and
B
C
BC
BC
of triangle
A
B
C
ABC
A
BC
, are constructed squares
A
B
D
E
ABDE
A
B
D
E
and
B
C
K
L
BCKL
BC
K
L
with centers
O
1
O_1
O
1
and
O
2
O_2
O
2
.
M
1
M_1
M
1
and
M
2
M_2
M
2
are midpoints of segments
D
L
DL
D
L
and
A
C
AC
A
C
. Prove that
O
1
M
1
O
2
M
2
O_1M_1O_2M_2
O
1
M
1
O
2
M
2
is a square. https://cdn.artofproblemsolving.com/attachments/8/1/8aa816a84c5ac9de78b396096cf718063de390.pngPS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983459_1962_leningrad_math_olympiad]here.
grade 6
1
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1962 Leningrad Math Olympiad - Grade 6
6.1 Three people with one double seater motorbike simultaneously headed from city A to city B . How should they act so that time, for which the last of them will get to , was the smallest? Determine this time. Pedestrian speed - 5 km/h, motorcycle speed - 45 km/h, distance from A to B is equal to 60 kilometers . 6.2 / 7.2 The numbers
A
A
A
and
B
B
B
are relatively prime. What common divisors can have the numbers
A
+
B
A+B
A
+
B
and
A
−
B
A-B
A
−
B
? 6.3. A person's age in
1962
1962
1962
was one more than the sum of digits of the year of his birth. How old is he? 6.4. / 7.3
15
15
15
magazines lie on the table, completely covering it. Prove that it is possible to remove eight of them so that the remaining magz cover at least
7
/
15
7/15
7/15
of the table area. 6.5. Prove that a
201
×
201
201 \times 201
201
×
201
chessboard can be bypassed by moving a chess knight, visiting each square exactly once. 6.6. Can an integer whose last two digits are odd be the square of another integer? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983459_1962_leningrad_math_olympiad]here.