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Problems
Contests
National and Regional Contests
Moldova Contests
Chisinau City MO
1976 Chisinau City MO
1976 Chisinau City MO
Part of
Chisinau City MO
Subcontests
(14)
129
1
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Chisinau MO p129 1976 X f(x+\pi)=f(x)/(3f(x)-1)
The function
f
(
x
)
f (x)
f
(
x
)
satisfies the relation
f
(
x
+
π
)
=
f
(
x
)
3
f
(
x
)
−
1
f(x+\pi)=\frac{f(x)}{3f(x) -1}
f
(
x
+
π
)
=
3
f
(
x
)
−
1
f
(
x
)
for any real number
x
x
x
. Prove that the function
f
(
x
)
f (x)
f
(
x
)
is periodic.
132
1
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Chisinau MO p132 1976 X AB/CD = AO x BO / CO x DO in tangential ABCD
Let
O
O
O
be the center of a circle inscribed in a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
and
∣
A
B
∣
=
a
|AB|= a
∣
A
B
∣
=
a
,
∣
C
D
∣
=
|CD|=
∣
C
D
∣
=
c. Prove that
a
c
=
A
O
⋅
B
O
C
O
⋅
D
O
.
\frac{a}{c}=\frac{AO\cdot BO}{CO\cdot DO}.
c
a
=
CO
⋅
D
O
A
O
⋅
BO
.
133
1
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Chisinau MO p133 1976 X # inside triangle inside square, area ineq.
A triangle with a parallelogram inside was placed in a square. Prove that the area of a parallelogram is not more than a quarter of a square.
131
1
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Chisinau MO p131 1976 X sum x_i^2 <= - nab if sum x_i=0
The sum of the real numbers
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2, ...,x_n
x
1
,
x
2
,
...
,
x
n
belonging to the segment
[
a
,
b
]
[a, b]
[
a
,
b
]
is equal to zero. Prove that
x
1
2
+
x
2
2
+
.
.
.
+
x
n
2
≤
−
n
a
b
.
x_1^2+ x_2^2+ ...+x_n^2 \le - nab.
x
1
2
+
x
2
2
+
...
+
x
n
2
≤
−
nab
.
130
1
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Chisinau MO p130 1976 X |f (x) - f (y) | <= | x - y|^a , a>1. constant
Prove that the function
f
(
x
)
f (x)
f
(
x
)
satisfying the relation
∣
f
(
x
)
−
f
(
y
)
∣
≤
∣
x
−
y
∣
a
|f (x) - f (y) | \le | x - y|^a
∣
f
(
x
)
−
f
(
y
)
∣
≤
∣
x
−
y
∣
a
for any real numbers
x
,
y
x, y
x
,
y
and some number
a
>
1
a> 1
a
>
1
is constant.
127
1
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Chisinau MO p127 1976 IX convex 1976-gon divided into 1975 triangles
The convex
1976
1976
1976
-gon is divided into
1975
1975
1975
triangles. Prove that there is a straight line separating one of these triangles from the rest.
126
1
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Chisinau MO p126 1976 IX P (n + km) is divisible by m for any integer k
Let
P
(
x
)
P (x)
P
(
x
)
be a polynomial with integer coefficients and
P
(
n
)
=
m
P (n) =m
P
(
n
)
=
m
for some integers
n
,
m
n, m
n
,
m
(
m
≠
10
m \ne 10
m
=
10
). Prove that
P
(
n
+
k
m
)
P (n + km)
P
(
n
+
km
)
is divisible by
m
m
m
for any integer
k
k
k
.
125
1
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Chisinau MO p125 1976 IX 20 different books on mathematics and physics
From twenty different books on mathematics and physics, sets are made containing
5
5
5
books on mathematics and
5
5
5
books on physics each. How many math books should there be for the largest number of possible sets?
124
1
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Chisinau MO p124 1976 IX system 3x3 a=b^2-c^2
Find
3
3
3
numbers, each of which is equal to the square of the difference of the other two.
123
1
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Chisinau MO p123 1976 VIII triangles from 5 points
Five points are given on the plane. Prove that among all the triangles with vertices at these points there are no more than seven acute-angled ones.
122
1
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Chisinau MO p122 1976 VIII cyclic criterion, 4 areas in prime numbers
The diagonals of some convex quadrilateral are mutually perpendicular and divide the quadrangle into
4
4
4
triangles, the areas of which are expressed by prime numbers. Prove that a circle can be inscribed in this quadrilateral.
121
1
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Chisinau MO p121 1976 VIII integer P(x) has no integer roots
Prove that the polynomial
P
(
x
)
P (x)
P
(
x
)
with integer coefficients, taking odd values for
x
=
0
x = 0
x
=
0
and
x
=
1
x= 1
x
=
1
, has no integer roots.
120
1
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Chisinau MO p120 1976 VIII product of \sqrt[m]{m}-\sqrt[k]{k}
Find the product of all numbers of the form
m
m
−
k
k
\sqrt[m]{m}-\sqrt[k]{k}
m
m
−
k
k
,
m
,
k
m ,k
m
,
k
are natural numbers satisfying the inequalities
1
≤
k
<
m
≤
n
1 \le k < m \le n
1
≤
k
<
m
≤
n
, where
n
>
3
n> 3
n
>
3
.
119
1
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Chisinau MO p119 1976 VIII Serpent Gorynych has 1976 heads
The Serpent Gorynych has
1976
1976
1976
heads. The fabulous hero can cut down
33
,
21
,
17
33, 21, 17
33
,
21
,
17
or
1
1
1
head with one blow of the sword, but at the same time, the Serpent grows, respectively,
48
,
0
,
14
48, 0, 14
48
,
0
,
14
or
349
349
349
heads. If all the heads are cut off, then no new heads will grow. Will the hero be able to defeat the Serpent?