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Problems
Contests
National and Regional Contests
Moldova Contests
Chisinau City MO
1977 Chisinau City MO
1977 Chisinau City MO
Part of
Chisinau City MO
Subcontests
(10)
135
1
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Chisinau MO p135 1977 VIII x=1978 - 1977/(1978 - ...)
Solve the equation:
x
=
1978
−
1977
1978
−
1977
.
.
.
.
.
.
1977
1978
−
1977
x
x=1978 - \dfrac{1977}{1978 - \dfrac{1977}{\frac{...}{...\dfrac{1977}{1978 -\dfrac{1977}{x}}}}}{}
x
=
1978
−
1978
−
...
1978
−
x
1977
1977
...
1977
1977
140
1
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Chisinau MO p140 1977 IX sum nC_{n}^{n} =n x 2 ^{n-1}
Prove the identities:
C
n
1
+
2
C
n
2
+
3
C
n
3
+
.
.
.
+
n
C
n
n
=
n
⋅
2
n
−
1
C_{n}^{1}+2C_{n}^{2}+3C_{n}^{3}+...+nC_{n}^{n}=n\cdot 2 ^{n-1}
C
n
1
+
2
C
n
2
+
3
C
n
3
+
...
+
n
C
n
n
=
n
⋅
2
n
−
1
C
n
1
−
2
C
n
2
+
3
C
n
3
+
.
.
.
−
(
−
1
)
n
−
1
n
C
n
n
=
0
C_{n}^{1}-2C_{n}^{2}+3C_{n}^{3}+...-(-1)^{n-1}nC_{n}^{n}=0
C
n
1
−
2
C
n
2
+
3
C
n
3
+
...
−
(
−
1
)
n
−
1
n
C
n
n
=
0
150
1
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Chisinau MO p150 1977 X area bounded by curves y=1-x^2, |x|=1-|y|.
Find the area of the figure bounded by the curves
y
=
1
−
x
2
y=1-x^2
y
=
1
−
x
2
,
∣
x
∣
=
1
−
∣
y
∣
.
|x|=1-|y|.
∣
x
∣
=
1
−
∣
y
∣.
153
1
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Chisinau MO p153 1977 X tan (\pi / 3^n) is irrational
Prove that the number
tan
π
3
n
\tan \frac{\pi}{3^n}
tan
3
n
π
is irrational for any natural
n
n
n
.
146
1
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Chisinau MO p146 1977 IX convex polygon iff property
Prove that
n
n
n
(
≥
4
\ge 4
≥
4
) points of the plane are vertices of a convex
n
n
n
-gon if and only if any
4
4
4
of them are vertices of a convex quadrilateral.
139
1
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Chisinau MO p139 1977 VIII angle bisector, \beta^2=ac- b^2ac/(a+c)^2
Let
β
\beta
β
be the length of the bisector of angle
B
B
B
, and
a
′
,
c
′
a', c'
a
′
,
c
′
be the lengths of the segments into which this bisector divides the side
A
C
AC
A
C
of the triangle
A
B
C
ABC
A
BC
. Prove the relation
β
2
=
a
c
−
a
′
c
′
\beta^2 = ac-a'c'
β
2
=
a
c
−
a
′
c
′
and derive from this the formula
β
2
=
a
c
−
b
2
a
c
(
a
+
c
)
2
\beta^2=ac-\frac{b^2ac}{(a+c)^2}
β
2
=
a
c
−
(
a
+
c
)
2
b
2
a
c
.
138
1
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Chisinau MO p138 1977 VIII isosceles, <A=40^o , BC:AD=\sqrt3
In an isosceles triangle
B
A
C
BAC
B
A
C
(
∣
A
C
∣
=
∣
A
B
∣
| AC | = | AB |
∣
A
C
∣
=
∣
A
B
∣
) , point
D
D
D
is marked on the side
A
C
AC
A
C
. Determine the angles of the triangle
B
D
C
BDC
B
D
C
if
∠
A
=
4
0
o
\angle A = 40^o
∠
A
=
4
0
o
and
∣
B
C
∣
:
∣
A
D
∣
=
3
|BC|: |AD|= \sqrt3
∣
BC
∣
:
∣
A
D
∣
=
3
.
137
1
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Chisinau MO p137 1977 VIII angle in 4 equal parts
Determine the angles of a triangle in which the median, bisector and altitude, drawn from one vertex, divide this angle into four equal parts.
136
1
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Chisinau MO p136 1977 VIII property of R as union of non-empty sets A, B
We represent the number line
R
R
R
as the union of two non-empty sets
A
,
B
A, B
A
,
B
different from
R
R
R
. Prove that one of the sets
A
,
B
A, B
A
,
B
does not have the following property: the difference of any elements of the set belongs to the same set.
134
1
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Chisinau MO p134 1977 VIII no 35351 in sequence 1,8,22,43,...
Where is the number
35351
35 351
35351
in the sequence
1
,
8
,
22
,
43
,
.
.
.
1, 8, 22, 43,...
1
,
8
,
22
,
43
,
...
?