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National and Regional Contests
Russia Contests
Saint Petersburg Mathematical Olympiad
1968 Leningrad Math Olympiad
1968 Leningrad Math Olympiad
Part of
Saint Petersburg Mathematical Olympiad
Subcontests
(4)
8.6*
1
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10-digit numbers with digits 1-3, written
All
10
10
10
-digit numbers consisting of digits
1
,
2
1, 2
1
,
2
and
3
3
3
are written one under the other. Each number has one more digit added to the right.
1
1
1
,
2
2
2
or
3
3
3
, and it turned out that to the number
111...11
111. . . 11
111...11
added
1
1
1
to the number
222...22
222. . . 22
222...22
was assigned
2
2
2
, and the number
333...33
333. . . 33
333...33
was assigned
3
3
3
. It is known that any two numbers that differ in all ten digits have different digits assigned to them. Prove that the assigned column of numbers matches with one of the ten columns written earlier.
grade 8
1
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1968 Leningrad Math Olympiad - Grade 8
8.1 In the parallelogram
A
B
C
D
ABCD
A
BC
D
, the diagonal
A
C
AC
A
C
is greater than the diagonal
B
D
BD
B
D
. The point
M
M
M
on the diagonal
A
C
AC
A
C
is such that around the quadrilateral
B
C
D
M
BCDM
BC
D
M
one can circumscribe a circle. Prove that
B
D
BD
B
D
is the common tangent of the circles circumscribed around the triangles
A
B
M
ABM
A
BM
and
A
D
M
ADM
A
D
M
. https://cdn.artofproblemsolving.com/attachments/b/3/9f77ff1f2198c201e5c270ec5b091a9da4d0bc.png8.2
A
A
A
is an odd integer,
x
x
x
and
y
y
y
are roots of equation
t
2
+
A
t
−
1
=
0
t^2+At-1=0
t
2
+
A
t
−
1
=
0
. Prove that
x
4
+
y
4
x^4 + y^4
x
4
+
y
4
and
x
5
+
y
5
x^5+ y^5
x
5
+
y
5
are coprime integer numbers. 8.3 A regular triangle is reflected symmetrically relative to one of its sides. The new triangle is again reflected symmetrically about one of its sides. This is repeated several times. It turned out that the resulting triangle coincides with the original one. Prove that an even number of reflections were made. 8.4 /7.6 Several circles are arbitrarily placed in a circle of radius
3
3
3
, the sum of their radii is
25
25
25
. Prove that there is a straight line that intersects at least
9
9
9
of these circles. 8.5 All two-digit numbers that do not end in zero are written one after another so that each subsequent number begins with that the same digit with which the previous number ends. Prove that you can do this and find the sum of the largest and smallest of all multi-digit numbers that can be obtained in this way. [url=https://artofproblemsolving.com/community/c6h3390996p32049528]8,6* (asterisk problems in separate posts) PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988084_1968_leningrad_math_olympiad]here.
grade 7
1
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1968 Leningrad Math Olympiad - Grade 7
7.1 A rectangle that is not a square is inscribed in a square. Prove that its semi-perimeter is equal to the diagonal of the square. 7.2 Find five numbers whose pairwise sums are 0, 2, 4,5, 7, 9, 10, 12, 14, 17. 7.3 In a
1000
1000
1000
-digit number, all but one digit is a five. Prove that this number is not a perfect square. 7.4 / 6.5 Several teams took part in the volleyball tournament. Team A is considered stronger than team B if either A beat B or there is a team C such that A beat C, and C beat B. Prove that if team T is the winner of the tournament, then it is the strongest the rest of the teams. 7.5 In a pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
,
K
K
K
is the midpoint of
A
B
AB
A
B
,
L
L
L
is the midpoint of
B
C
BC
BC
,
M
M
M
is the midpoint of
C
D
CD
C
D
,
N
N
N
is the midpoint of
D
E
DE
D
E
,
P
P
P
is the midpoint of
K
M
KM
K
M
,
Q
Q
Q
is the midpoint of
L
N
LN
L
N
. Prove that the segment
P
Q
PQ
PQ
is parallel to side
A
E
AE
A
E
and is equal to its quarter. https://cdn.artofproblemsolving.com/attachments/2/5/be8e9b0692d98115dbad04f960e8a856dc593f.png7.6 / 8.4 Several circles are arbitrarily placed in a circle of radius
3
3
3
, the sum of their radii is
25
25
25
. Prove that there is a straight line that intersects at least
9
9
9
of these circles. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988084_1968_leningrad_math_olympiad]here.
grade 6
1
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1968 Leningrad Math Olympiad - Grade 6
6.1 The student bought a briefcase, a fountain pen and a book. If the briefcase cost 5 times cheaper, the fountain pen was 2 times cheaper, and the book was 2 1/2 times cheaper cheaper, then the entire purchase would cost 2 rubles. If the briefcase was worth 2 times cheaper, a fountain pen is 4 times cheaper, and a book is 3 times cheaper, then the whole the purchase would cost 3 rubles. How much does it really cost? ´ 6.2. Which number is greater:
888...88
⏟
19
d
i
g
i
t
s
⋅
333...33
⏟
68
d
i
g
i
t
s
o
r
444...44
⏟
19
d
i
g
i
t
s
⋅
666...67
⏟
68
d
i
g
i
t
s
?
\underbrace{888...88}_{19 \, digits} \cdot \underbrace{333...33}_{68 \, digits} \,\,\, or \,\,\, \underbrace{444...44}_{19 \, digits} \cdot \underbrace{666...67}_{68 \, digits} \, ?
19
d
i
g
i
t
s
888...88
⋅
68
d
i
g
i
t
s
333...33
or
19
d
i
g
i
t
s
444...44
⋅
68
d
i
g
i
t
s
666...67
?
6.3 Distance between Luga and Volkhov 194 km, between Volkhov and Lodeynoye Pole 116 km, between Lodeynoye Pole and Pskov 451 km, between Pskov and Luga 141 km. What is the distance between Pskov and Volkhov? 6.4 There are
4
4
4
objects in pairs of different weights. How to use a pan scale without weights Using five weighings, arrange all these objects in order of increasing weights? 6.5 . Several teams took part in the volleyball tournament. Team A is considered stronger than team B if either A beat B or there is a team C such that A beat C, and C beat B. Prove that if team T is the winner of the tournament, then it is the strongest the rest of the teams. 6.6 In task 6.1, determine what is more expensive: a briefcase or a fountain pen. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988084_1968_leningrad_math_olympiad]here.