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National and Regional Contests
Russia Contests
Saint Petersburg Mathematical Olympiad
1967 Leningrad Math Olympiad
1967 Leningrad Math Olympiad
Part of
Saint Petersburg Mathematical Olympiad
Subcontests
(3)
grade 8
1
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1967 Leningrad Math Olympiad - Grade 8
8.1
x
x
x
and
y
y
y
are the roots of the equation
t
2
−
c
t
−
c
=
0
t^2-ct-c=0
t
2
−
c
t
−
c
=
0
. Prove that holds the inequality
x
3
+
y
3
+
(
x
y
)
3
≥
0.
x^3 + y^3 + (xy)^3 \ge 0.
x
3
+
y
3
+
(
x
y
)
3
≥
0.
8.2. Two circles touch internally at point
A
A
A
. Through a point
B
B
B
of the inner circle, different from
A
A
A
, a tangent to this circle intersecting the outer circle at points C and
D
D
D
. Prove that
A
B
AB
A
B
is a bisector of angle
C
A
D
CAD
C
A
D
. https://cdn.artofproblemsolving.com/attachments/2/8/3bab4b5c57639f24a6fd737f2386a5e05e6bc7.png 8.3 Prove that
2
3
100
+
1
2^{3^{100}} + 1
2
3
100
+
1
is divisible by
3
101
3^{101}
3
101
. 8.4 / 7.5 An entire arc of circle is drawn through the vertices
A
A
A
and
C
C
C
of the rectangle
A
B
C
D
ABCD
A
BC
D
lying inside the rectangle. Draw a line parallel to
A
B
AB
A
B
intersecting
B
C
BC
BC
at point
P
P
P
,
A
D
AD
A
D
at point
Q
Q
Q
, and the arc
A
C
AC
A
C
at point
R
R
R
so that the sum of the areas of the figures
A
Q
R
AQR
A
QR
and
C
P
R
CPR
CPR
is the smallest. https://cdn.artofproblemsolving.com/attachments/1/4/9b5a594f82a96d7eff750e15ca6801a5fc0bf1.png 8.5 In a certain group of people, everyone has one enemy and one Friend. Prove that these people can be divided into two companies so that in every company there will be neither enemies nor friends. 8.6 Numbers
a
1
,
a
2
,
.
.
.
,
a
100
a_1, a_2, . . . , a_{100}
a
1
,
a
2
,
...
,
a
100
are such that
a
1
−
2
a
2
+
a
3
≤
0
a_1 - 2a_2 + a_3 \le 0
a
1
−
2
a
2
+
a
3
≤
0
a
2
−
2
a
3
+
a
4
≤
0
a_2-2a_3 + a_ 4 \le 0
a
2
−
2
a
3
+
a
4
≤
0
.
.
.
...
...
a
98
−
2
a
99
+
a
100
≤
0
a_{98}-2a_{99 }+ a_{100} \le 0
a
98
−
2
a
99
+
a
100
≤
0
and at the same time
a
1
=
a
100
≥
0
a_1 = a_{100}\ge 0
a
1
=
a
100
≥
0
. Prove that all these numbers are non-negative. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988083_1967_leningrad_math_olympiad]here.
grade 7
1
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1967 Leningrad Math Olympiad - Grade 7
7.1 Construct a trapezoid given four sides. 7.2 Prove that
(
1
+
x
+
x
2
+
.
.
.
+
x
100
)
(
1
+
x
102
)
−
102
x
101
≥
0.
(1 + x + x^2 + ...+ x^{100})(1 + x^{102}) - 102x^{101} \ge 0 .
(
1
+
x
+
x
2
+
...
+
x
100
)
(
1
+
x
102
)
−
102
x
101
≥
0.
7.3 In a quadrilateral
A
B
C
D
ABCD
A
BC
D
,
M
M
M
is the midpoint of AB,
N
N
N
is the midpoint of
C
D
CD
C
D
. Lines
A
D
AD
A
D
and BC intersect
M
N
MN
MN
at points
P
P
P
and
Q
Q
Q
, respectively. Prove that if
∠
B
Q
M
=
∠
A
P
M
\angle BQM = \angle APM
∠
BQM
=
∠
A
PM
, then
B
C
=
A
D
BC=AD
BC
=
A
D
. https://cdn.artofproblemsolving.com/attachments/a/2/1c3cbc62ee570a823b5f3f8d046da9fbb4b0f2.png7.4 / 6.4 Each of the eight given different natural numbers less than
16
16
16
. Prove that among their pairwise differences there is at least at least three are the same. 7.5 / 8.4 An entire arc of circle is drawn through the vertices
A
A
A
and
C
C
C
of the rectangle
A
B
C
D
ABCD
A
BC
D
lying inside the rectangle. Draw a line parallel to
A
B
AB
A
B
intersecting
B
C
BC
BC
at point
P
P
P
,
A
D
AD
A
D
at point
Q
Q
Q
, and the arc
A
C
AC
A
C
at point
R
R
R
so that the sum of the areas of the figures
A
Q
R
AQR
A
QR
and
C
P
R
CPR
CPR
is the smallest. https://cdn.artofproblemsolving.com/attachments/1/4/9b5a594f82a96d7eff750e15ca6801a5fc0bf1.png7.6 / 6.5 The distance AB is 100 km. From A and B , cyclists simultaneously ride towards each other at speeds of 20 km/h and 30 km/hour accordingly. Together with the first A, a fly flies out with speed 50 km/h, she flies until she meets the cyclist from B, after which she turns around and flies back until she meets the cyclist from A, after which turns around, etc. How many kilometers will the fly fly in the direction from A to B until the cyclists meet? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988083_1967_leningrad_math_olympiad]here.
grade 6
1
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1967 Leningrad Math Olympiad - Grade 6
6.1 The capacities of cubic vessels are in the ratio 1:8:27 and the volumes of liquid poured into them are 1: 2: 3. After this, from the first to a certain amount of liquid was poured into the second vessel, and then from the second in the third so that in all three vessels the liquid level became the same. After this, 128 4/7 liters were poured from the first vessel into the second, and from the second in the first back so much that the height of the liquid column in the first vessel became twice as large as in the second. It turned out that in the first vessel there were 100 fewer liters than at first. How much liquid was initially in each vessel? 6.2 How many times a day do all three hands on a clock coincide, including the second hand? 6.3. Prove that in Leningrad there are two people who have the same number of familiar Leningraders. 6.4 / 7.4 Each of the eight given different natural numbers less than
16
16
16
. Prove that among their pairwise differences there is at least at least three are the same. 6.5 / 7.6 The distance AB is 100 km. From A and B , cyclists simultaneously ride towards each other at speeds of 20 km/h and 30 km/hour accordingly. Together with the first A, a fly flies out with speed 50 km/h, she flies until she meets the cyclist from B, after which she turns around and flies back until she meets the cyclist from A, after which turns around, etc. How many kilometers will the fly fly in the direction from A to B until the cyclists meet? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988083_1967_leningrad_math_olympiad]here.