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National and Regional Contests
Russia Contests
Saint Petersburg Mathematical Olympiad
1969 Leningrad Math Olympiad
1969 Leningrad Math Olympiad
Part of
Saint Petersburg Mathematical Olympiad
Subcontests
(3)
grade 8
1
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1969 Leningrad Math Olympiad - Grade 8
[url=https://artofproblemsolving.com/community/c893771h1861957p12597232]8.1 The point
E
E
E
lies on the base
[
A
D
]
[AD]
[
A
D
]
of the trapezoid
A
B
C
D
ABCD
A
BC
D
. The perimeters of the triangles
A
B
E
,
B
C
E
ABE, BCE
A
BE
,
BCE
and
C
D
E
CDE
C
D
E
are equal. Prove that
∣
B
C
∣
=
∣
A
D
∣
/
2
|BC| = |AD|/2
∣
BC
∣
=
∣
A
D
∣/2
8.2 In a convex pentagon, the lengths of all sides are equal. Find the point on the longest diagonal from which all sides are visible underneath angles not exceeding a right angle. [url=https://artofproblemsolving.com/community/c893771h1862007p12597620]8.3 Every city in the certain state is connected by airlines with no more than with three other ones, but one can get from every city to every other city changing a plane once only or directly. What is the maximal possible number of the cities? [url=https://artofproblemsolving.com/community/c893771h1861966p12597273]8.4*/7.4* (asterisk problems in separate posts) [url=https://artofproblemsolving.com/community/c893771h1862002p12597605]8.5 Four different three-digit numbers starting with the same digit have the property that their sum is divisible by three of them without a remainder. Find these numbers. [url=https://artofproblemsolving.com/community/c893771h1861967p12597280]8.6 Given a finite sequence of zeros and ones, which has two properties: a) if in some arbitrary place in the sequence we select five digits in a row and also select five digits in any other place in a row, then these fives will be different (they may overlap); b) if you add any digit to the right of the sequence, then property (a) will no longer hold true. Prove that the first four digits of our sequence coincide with the last four. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988085_1969_leningrad_math_olympiad]here.
grade 7
1
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1969 Leningrad Math Olympiad - Grade 7
7.1 / 6.1 There are
8
8
8
rooks on the chessboard such that no two of them they don't hit each other. Prove that the black squares contain an even number of rooks. 7.2 The sides of triangle
A
B
C
ABC
A
BC
are extended as shown in the figure. At this
A
A
′
=
3
A
B
AA' = 3 AB
A
A
′
=
3
A
B
,,
B
B
′
=
5
B
C
BB' = 5BC
B
B
′
=
5
BC
,
C
C
′
=
8
C
A
CC'= 8 CA
C
C
′
=
8
C
A
. How many times is the area of the triangle
A
B
C
ABC
A
BC
less than the area of the triangle
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
? https://cdn.artofproblemsolving.com/attachments/9/f/06795292291cd234bf2469e8311f55897552f6.png[url=https://artofproblemsolving.com/community/c893771h1860178p12579333]7.3 Prove the equality
2
x
2
−
1
+
4
x
2
−
4
+
6
x
2
−
9
+
.
.
.
+
20
x
2
−
100
=
11
(
x
−
1
)
(
x
+
10
)
+
11
(
x
−
2
)
(
x
+
9
)
+
.
.
.
+
11
(
x
−
10
)
(
x
+
1
)
\frac{2}{x^2-1}+\frac{4}{x^2-4} +\frac{6}{x^2-9}+...+\frac{20}{x^2-100} =\frac{11}{(x-1)(x+10)}+\frac{11}{(x-2)(x+9)}+...+\frac{11}{(x-10)(x+1)}
x
2
−
1
2
+
x
2
−
4
4
+
x
2
−
9
6
+
...
+
x
2
−
100
20
=
(
x
−
1
)
(
x
+
10
)
11
+
(
x
−
2
)
(
x
+
9
)
11
+
...
+
(
x
−
10
)
(
x
+
1
)
11
[url=https://artofproblemsolving.com/community/c893771h1861966p12597273]7.4* / 8.4 * (asterisk problems in separate posts) 7.5 . The collective farm consists of
4
4
4
villages located in the peaks of square with side
10
10
10
km. It has the means to conctruct 28 kilometers of roads . Can a collective farm build such a road system so that was it possible to get from any village to any other? 7.6 / 6.6 Two brilliant mathematicians were told in natural terms number and were told that these numbers differ by one. After that they take turns asking each other the same question: “Do you know my number?" Prove that sooner or later one of them will answer positively. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988085_1969_leningrad_math_olympiad]here.
grade 6
1
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1969 Leningrad Math Olympiad - Grade 6
6.1 / 7.1 There are
8
8
8
rooks on the chessboard such that no two of them they don't hit each other. Prove that the black squares contain an even number of rooks. 6.2 The natural numbers are arranged in a
3
×
3
3 \times 3
3
×
3
table. Kolya and Petya crossed out 4 numbers each. It turned out that the sum of the numbers crossed out by Petya is three times the sum numbers crossed out by Kolya. What number is left uncrossed? \begin{tabular}{|c|c|c|}\hline 4 & 12 & 8 \\ \hline 13 & 24 & 14 \\ \hline 7 & 5 & 23 \\ \hline \end{tabular} 6.3 Misha and Sasha left at noon on bicycles from city A to city B. At the same time, I left from B to A Vanya. All three travel at constant but different speeds. At one o'clock Sasha was exactly in the middle between Misha and Vanya, and at half past one Vanya was in the middle between Misha and Sasha. When Misha will be exactly in the middle between Sasha and Vanya? 6.4 There are
35
35
35
piles of nuts on the table. Allowed to add one nut at a time to any
23
23
23
piles. Prove that by repeating this operation, you can equalize all the heaps. 6.5 There are
64
64
64
vertical stripes on the round drum, and each stripe you need to write down a six-digit number from digits
1
1
1
and
2
2
2
so that all the numbers were different and any two adjacent ones differed in exactly one discharge. How to do this? 6.6 / 7.6 Two brilliant mathematicians were told in natural terms number and were told that these numbers differ by one. After that they take turns asking each other the same question: “Do you know my number?" Prove that sooner or later one of them will answer positively. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988085_1969_leningrad_math_olympiad]here.