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Problems
Contests
National and Regional Contests
Moldova Contests
Chisinau City MO
1975 Chisinau City MO
1975 Chisinau City MO
Part of
Chisinau City MO
Subcontests
(22)
117
1
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Chisinau MO p117 1975 R2 X tan^2 80^o root of x^3 - 33x^2 + 27x - 33 = 0
Prove that the numbers
tan
2
2
0
o
,
tan
2
4
0
o
,
tan
2
8
0
o
\tan^2 20^o, \tan^2 40^o,\tan^2 80^o
tan
2
2
0
o
,
tan
2
4
0
o
,
tan
2
8
0
o
are the roots of the equation
x
3
−
33
x
2
+
27
x
−
33
=
0
x^3 - 33x^2 + 27x - 33 = 0
x
3
−
33
x
2
+
27
x
−
33
=
0
.
116
1
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Chisinau MO p116 1975 R2 X xsin a+ysin b+z sin c=0 only 1 integer solution
The sides of a triangle are equal to
2
,
3
,
4
\sqrt2, \sqrt3, \sqrt4
2
,
3
,
4
and its angles are
α
,
β
,
γ
\alpha, \beta, \gamma
α
,
β
,
γ
, respectively. Prove that the equation
x
sin
α
+
y
sin
β
+
z
sin
γ
=
0
x\sin \alpha + y\sin \beta + z\sin \gamma = 0
x
sin
α
+
y
sin
β
+
z
sin
γ
=
0
has exactly one solution in integers
x
,
y
,
z
x, y, z
x
,
y
,
z
.
113
1
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Chisinau MO p113 1975 R2 X odd if n <(44 + \sqrt{1975})^100 <n + 1
Prove that any integer
n
n
n
satisfying the inequality
n
<
(
44
+
1975
)
1
00
<
n
+
1
n <(44 + \sqrt{1975})^100 <n + 1
n
<
(
44
+
1975
)
1
00
<
n
+
1
is odd.
111
1
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Chisinau MO p111 1975 R2 IX 6 concyclic wanted, squares on sides of triangle
Three squares are constructed on the sides of the triangle to the outside. What should be the angles of the triangle so that the six vertices of these squares, other than the vertices of the triangle, lie on the same circle?
110
1
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Chisinau MO p110 1975 R2 IX property of centrally symmetric convex octagon
Prove that any centrally symmetric convex octagon has a diagonal passing through the center of symmetry that is not parallel to any of its sides.
106
1
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Chisinau MO p106 1975 R2 VIII square construction
Construct a square from four points, one on each side.
105
1
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Chisinau MO p105 1975 R2 VIII A, B, M, K conyclic if AB=BC=CD
Let
M
M
M
be the point of intersection of the diagonals, and
K
K
K
be the point of intersection of the bisectors of the angles
B
B
B
and
C
C
C
of the convex quadrilateral
A
B
C
D
ABCD
A
BC
D
. Prove that points
A
,
B
,
M
,
K
A, B, M, K
A
,
B
,
M
,
K
lie on the same circle if the following relation holds:
∣
A
B
∣
=
∣
B
C
∣
=
∣
C
D
∣
|AB|=|BC|=|CD|
∣
A
B
∣
=
∣
BC
∣
=
∣
C
D
∣
103
1
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Chisinau MO p103 1975 R2 VIII 1/2-1/3+1/4-...+1/1974 - 1/1975 <2/5
Prove the inequality:
1
2
−
1
3
+
1
4
−
1
5
+
.
.
.
+
1
1974
−
1
1975
<
2
5
\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{1974}-\frac{1}{1975}<\frac{2}{5}
2
1
−
3
1
+
4
1
−
5
1
+
...
+
1974
1
−
1975
1
<
5
2
104
1
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Chisinau MO p104 1975 R2 VIII x^2+y^2>=2\sqrt2 (x-y) if xy = 1
Prove that
x
2
+
y
2
≥
2
2
(
x
−
y
)
x^2+y^2 \ge 2\sqrt2 (x-y)
x
2
+
y
2
≥
2
2
(
x
−
y
)
if
x
y
=
1
xy = 1
x
y
=
1
102
1
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Chisinau MO p102 1975 R2 VIII 2 player game, number divisible by 9
Two people write a
2
k
2k
2
k
-digit number, using only the numbers
1
,
2
,
3
,
4
1, 2, 3, 4
1
,
2
,
3
,
4
and
5
5
5
. The first number on the left is written by the first of them, the second - the second, the third - the first, etc. Can the second one achieve this so that the resulting number is divisible by
9
9
9
, if the first seeks to interfere with it? Consider the cases
k
=
10
k = 10
k
=
10
and
k
=
15
k = 15
k
=
15
.
101
1
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Chisinau MO p101 1975 R2 VIII 2 numbers in k+1 whose diff. divisible by k
Prove that among any
k
+
1
k + 1
k
+
1
natural numbers there are two numbers whose difference is divisible by
k
k
k
.
100
1
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Chisinau MO p100 1975 R1 X sin^3x+sin^32x +sin^33x=(sinx+sin2x+sin3x)^3
Solve the equation:
sin
3
x
+
sin
3
2
x
+
sin
3
3
x
=
(
sin
x
+
sin
2
x
+
sin
3
x
)
3
\sin ^3x+\sin ^32x+\sin ^33x=(\sin x + \sin 2x + \sin 3x)^3
sin
3
x
+
sin
3
2
x
+
sin
3
3
x
=
(
sin
x
+
sin
2
x
+
sin
3
x
)
3
.
99
1
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Chisinau MO p99 1975 R1 X sin 54^o - sin 18^o = 0.5
Prove the equality:
sin
5
4
o
−
sin
1
8
o
=
0.5
\sin 54^o -\sin 18^o = 0.5
sin
5
4
o
−
sin
1
8
o
=
0.5
97
1
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Chisinau MO p97 1975 R1 X min of (x-1) (x -2) (x -3) (x - 4) + 10
Find the smallest value of the expression
(
x
−
1
)
(
x
−
2
)
(
x
−
3
)
(
x
−
4
)
+
10
(x-1) (x -2) (x -3) (x - 4) + 10
(
x
−
1
)
(
x
−
2
)
(
x
−
3
)
(
x
−
4
)
+
10
.
94
1
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Chisinau MO p94 1975 R1 IX locus of vertex of equilateral
A straight line
ℓ
\ell
ℓ
and a point
A
A
A
outside of it are given on the plane. Find the locus of the vertices
C
C
C
of the equilateral triangle
A
B
C
ABC
A
BC
, the vertex
B
B
B
of which lies on the straight line
ℓ
\ell
ℓ
.
93
1
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Chisinau MO p93 1975 R1 IX (a^2+b^2+c^2)^2=2(a^4+b^4+c^4) if a+b+c=0
Prove that
(
a
2
+
b
2
+
c
2
)
2
=
2
(
a
4
+
b
4
+
c
4
)
(a^2 + b^2 + c^2)^ 2 = 2 (a^4 + b^4 + c^4)
(
a
2
+
b
2
+
c
2
)
2
=
2
(
a
4
+
b
4
+
c
4
)
if
a
+
b
+
c
=
0
a + b + c = 0
a
+
b
+
c
=
0
.
92
1
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Chisinau MO p92 1975 R1 IX x^2-y^2=105
Solve in natural numbers the equation
x
2
−
y
2
=
105
x^2-y^2=105
x
2
−
y
2
=
105
.
90
1
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Chisinau MO p90 1975 R1 VIII right triangle construction, using 2 medians
Construct a right-angled triangle along its two medians, starting from the acute angles.
89
1
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Chisinau MO p89 1975 R1 VIII circle criterion as a closed line
A closed line on a plane is such that any quadrangle inscribed in it has the sum of opposite angles equal to
18
0
o
180^o
18
0
o
. Prove that this line is a circle.
88
1
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Chisinau MO p88 1975 R1 VIII 0.123456789101112... not periodic
Prove that the fraction
0.123456789101112...
0.123456789101112...
0.123456789101112...
is not periodic.
87
1
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Chisinau MO p87 1975 R1 VIII 2 numbers among 100 whose diff. divisible by 99
Prove that among any
100
100
100
natural numbers there are two numbers whose difference is divisible by
99
99
99
.
86
1
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Chisinau MO p86 1975 R1 VIII sign of \sqrt{4+\sqrt7}-\sqrt{4-\sqrt7}-\sqrt2
What is the number
x
=
4
+
7
−
4
−
7
−
2
x =\sqrt{4+\sqrt7}-\sqrt{4-\sqrt7}-\sqrt2
x
=
4
+
7
−
4
−
7
−
2
, positive, negative or zero?