MathDB
Problems
Contests
National and Regional Contests
Russia Contests
Saint Petersburg Mathematical Olympiad
1964 Leningrad Math Olympiad
1964 Leningrad Math Olympiad
Part of
Saint Petersburg Mathematical Olympiad
Subcontests
(3)
grade 8
1
Hide problems
1964 Leningrad Math Olympiad - Grade 8
8.1 Find all primes
p
,
q
p,q
p
,
q
and
r
r
r
such that
p
q
r
=
5
(
p
+
q
+
r
)
.
pqr= 5(p + q + r).
pq
r
=
5
(
p
+
q
+
r
)
.
8.2 Prove that if
a
b
‾
/
b
c
‾
=
a
/
c
\overline{ab}/\overline{bc} = a/c
ab
/
b
c
=
a
/
c
, then
a
b
b
.
.
.
b
b
‾
/
b
b
.
.
.
b
b
c
‾
=
a
/
c
\overline{abb...bb}/\overline{bb...bbc} = a/c
abb
...
bb
/
bb
...
bb
c
=
a
/
c
(each number has
n
n
n
digits).8.3 / 9.1 Construct a triangle with perimeter, altitude and angle at the base. 8.4. / 9.4 Prove that the square of the sum of N distinct non-zero squares of integers is also the sum of
N
N
N
squares of non-zero integers. 8.5. In the quadrilateral
A
B
C
D
ABCD
A
BC
D
the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
are drawn. Prove that if the circles inscribed in
A
B
C
ABC
A
BC
and
A
D
C
ADC
A
D
C
touch each other each other, then the circles inscribed in
B
A
D
BAD
B
A
D
and in
B
C
D
BCD
BC
D
also touch each other. 8.6 / 9.6 If the numbers
A
A
A
and
n
n
n
are coprime, then there are integers
X
X
X
and
Y
Y
Y
such that
∣
X
∣
<
n
|X| <\sqrt{n}
∣
X
∣
<
n
,
∣
Y
∣
<
n
|Y| <\sqrt{n}
∣
Y
∣
<
n
and
A
X
−
Y
AX-Y
A
X
−
Y
is divided by
n
n
n
. Prove it. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983461_1964_leningrad_math_olympiad]here.
grade 7
1
Hide problems
1964 Leningrad Math Olympiad - Grade 7
7.1 Given a convex
n
n
n
-gon all of whose angles are obtuse. Prove that the sum of the lengths of the diagonals in it is greater than the sum of the lengths of the sides. 7.2 Find all integer values for
x
x
x
and
y
y
y
such that
x
4
+
4
y
4
x^4 + 4y^4
x
4
+
4
y
4
is a prime number. (typo corrected) 7.3. Given a triangle
A
B
C
ABC
A
BC
. Parallelograms
A
B
K
L
ABKL
A
B
K
L
,
B
C
M
N
BCMN
BCMN
and
A
C
F
G
ACFG
A
CFG
are constructed on the sides, Prove that the segments
K
N
KN
K
N
,
M
F
MF
MF
and
G
L
GL
G
L
can form a triangle. https://cdn.artofproblemsolving.com/attachments/a/f/7a0264b62754fafe4d559dea85c67c842011fc.png7.4 / 6.2 Prove that a
10
×
10
10 \times 10
10
×
10
chessboard cannot be covered with
25
25
25
figures like https://cdn.artofproblemsolving.com/attachments/0/4/89aafe1194628332ec13ad1c713bb35cbefff7.png. 7.5 Find the greatest number of different natural numbers, each of which is less than
50
50
50
, and every two of which are coprime. 7.6. Given a triangle
A
B
C
ABC
A
BC
.
D
D
D
and
E
E
E
are the midpoints of the sides
A
B
AB
A
B
and
B
C
BC
BC
. Point
M
M
M
lies on
A
C
AC
A
C
,
M
E
>
E
C
ME > EC
ME
>
EC
. Prove that
M
D
<
A
D
MD < AD
M
D
<
A
D
. https://cdn.artofproblemsolving.com/attachments/e/c/1dd901e0121e5c75a4039d21b954beb43dc547.pngPS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983461_1964_leningrad_math_olympiad]here.
grade 6
1
Hide problems
1964 Leningrad Math Olympiad - Grade 6
6.1 Three shooters - Anilov, Borisov and Vorobiev - made
6
6
6
each shots at one target and scored equal points. It is known that Anilov scored
43
43
43
points in the first three shots, and Borisov scored
43
43
43
points with the first shot knocked out 3 points. How many points did each shooter score per shot? if there was one hit in 50, two in 25, three in 20, three in 10, two in 5, in 3 - two, in 2 - two, in 1 - three? https://cdn.artofproblemsolving.com/attachments/a/1/4abb71f7bccc0b9d2e22066ec17c31ef139d6a.png6.2 / 7.4 Prove that a
10
×
10
10 \times 10
10
×
10
chessboard cannot be covered with
25
25
25
figures like https://cdn.artofproblemsolving.com/attachments/0/4/89aafe1194628332ec13ad1c713bb35cbefff7.png. 6.3 The squares of a chessboard contain natural numbers such that each is equal to the arithmetic mean of its neighbors. Sum of numbers standing in the corners of the board is
16
16
16
. Find the number standing on the field
e
2
e2
e
2
. 6.4 There is a table
100
×
100
100 \times 100
100
×
100
. What is the smallest number of letters which can be arranged in its cells so that no two are identical the letters weren't next to each other? 6.5 The pioneer detachment is lined up in a rectangle. In each rank the tallest is noted, and from these pioneers the most short. In each row, the lowest one is noted, and from them is selected the tallest. Which of these two pioneers is taller? (This means that the two pioneers indicated are the highest of the low and the lowest of tall - must be different) 6.6 Find the product of three numbers whose sum is equal to the sum of their squares, equal to the sum of their cubes and equal to
1
1
1
. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983461_1964_leningrad_math_olympiad]here.