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Problems
Contests
National and Regional Contests
Moldova Contests
Chisinau City MO
1979 Chisinau City MO
1979 Chisinau City MO
Part of
Chisinau City MO
Subcontests
(15)
183
1
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Chisinau MO p183 1979 X sin^3 a cos 3a + cos^3 a sin 3a=3/4 sin4a
Prove the identity
sin
3
a
cos
3
a
+
cos
3
a
sin
3
a
=
3
4
sin
4
a
.
\sin^3 a \cos 3a + \cos^3 a \sin 3a=\frac{3}{4}\sin 4a.
sin
3
a
cos
3
a
+
cos
3
a
sin
3
a
=
4
3
sin
4
a
.
182
1
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Chisinau MO p182 1979 X section of cube by a plane not a regular pentagon
Prove that a section of a cube by a plane cannot be a regular pentagon.
181
1
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Chisinau MO p181 1979 X collinear wanted, combo geo
Prove that if every line connecting any two points of some finite set of points of the plane contains at least one more point of this set, then all points of the set lie on one straight line.
180
1
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Chisinau MO p180 1979 X trinomial, |f '(0) | <= 8 if |f(x) | <=1 for0<x<=1
It is known that for
0
≤
x
≤
1
0\le x \le 1
0
≤
x
≤
1
the square trinomial
f
(
x
)
f (x)
f
(
x
)
satisfies the condition
∣
f
(
x
)
∣
≤
1
|f(x) | \le 1
∣
f
(
x
)
∣
≤
1
. Show that
∣
f
′
(
0
)
∣
≤
8.
| f '(0) | \le 8.
∣
f
′
(
0
)
∣
≤
8.
179
1
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Chisinau MO p179 1979 X x^2 + y^2 = 1979
Prove that the equation
x
2
+
y
2
=
1979
x^2 + y^2 = 1979
x
2
+
y
2
=
1979
has no integer solutions.
178
1
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Chisinau MO p178 1979 IX concyclic feet of altitudes and medians
Prove that the bases of the altitudes and medians of an acute-angled triangle lie on the same circle.
177
1
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Chisinau MO p177 1979 IX cut a square into five squares
Is it possible to cut a square into five squares?
176
1
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Chisinau MO p176 1979 IX x^2+1 = cos x
Indicate all the roots of the equation
x
2
+
1
=
cos
x
x^2+1 = \cos x
x
2
+
1
=
cos
x
.
174
1
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Chisinau MO p174 1979 IX any odd a, exista b such 2^b-1 divisible by a
Prove that for any odd number
a
a
a
there exists an integer
b
b
b
such that
2
b
−
1
2^b-1
2
b
−
1
is divisible by
a
a
a
.
175
1
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Chisinau MO p175 1979 IX 1/a + 1/b + 1/c >= 9 id a+b+c=1
Prove that if the sum of positive numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
is equal to
1
1
1
, then
1
a
+
1
b
+
1
c
≥
9.
\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge 9.
a
1
+
b
1
+
c
1
≥
9.
170
1
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Chisinau MO p170 1979 VIII a_{k-1}+ a_{k+1}<=2a_k
The numbers
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
(
n
≥
3
n\ge 3
n
≥
3
) satisfy the relations
a
1
=
a
n
=
0
,
a
k
−
1
+
a
k
+
1
≤
2
a
k
(
k
=
2
,
3
,
.
.
.
,
n
−
1
)
a_1=a_n = 0, a_{k-1}+ a_{k+1}\le 2a_k \,\,\, (k = 2, 3,..., n-1)
a
1
=
a
n
=
0
,
a
k
−
1
+
a
k
+
1
≤
2
a
k
(
k
=
2
,
3
,
...
,
n
−
1
)
Prove that the numbers
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
are non-negative.
173
1
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Chisinau MO p173 1979 VIII regular pentagon criterion
The inner angles of the pentagon inscribed in the circle are equal to each other. Prove that this pentagon is regular.
171
1
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Chisinau MO p171 1979 VIII ax^2 + (a+b) x + b= x has no roots
Are there numbers
a
,
b
a, b
a
,
b
such that
∣
a
−
b
∣
≤
1979
| a -b |\le 1979
∣
a
−
b
∣
≤
1979
and the equation
a
x
2
+
(
a
+
b
)
x
+
b
=
x
ax^2 + (a + b) x + b = x
a
x
2
+
(
a
+
b
)
x
+
b
=
x
has no roots?
169
1
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Chisinau MO p169 1979 VIII x^8+1/x^8 is integer if x+1/x integer
Prove that the number
x
8
+
1
x
8
x^8+\frac{1}{x^8}
x
8
+
x
8
1
is an integer if
x
+
1
x
x+\frac{1}{x }
x
+
x
1
is an integer.
172
1
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Chisinau MO p22 1949-56 VIII-IX bisector of right angle, bisects another angle
Show that in a right-angled triangle the bisector of the right angle divides into equal parts the angle between the altitude and the median, drawn from the same vertex.