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Contests
National and Regional Contests
Moldova Contests
Chisinau City MO
1949-56 Chisinau City MO
1949-56 Chisinau City MO
Part of
Chisinau City MO
Subcontests
(60)
56
1
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Chisinau MO p56 1949-56 X system 2x2 (x+y)/xy +xy/(x+y)= a+1/a
Solve the system of equations
{
x
+
y
x
y
+
x
y
x
+
y
=
a
+
1
a
x
−
y
x
y
+
x
y
x
−
y
=
c
+
1
c
\begin{cases} \dfrac{x+y}{xy}+\dfrac{xy}{x+y}= a+ \dfrac{1}{a}\\ \\\dfrac{x-y}{xy}+\dfrac{xy}{x-y}= c+ \dfrac{1}{c}\end{cases}
⎩
⎨
⎧
x
y
x
+
y
+
x
+
y
x
y
=
a
+
a
1
x
y
x
−
y
+
x
−
y
x
y
=
c
+
c
1
62
1
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Chisinau MO p62 1949-56 X volume of tetrahedron is constant
On two intersecting lines
ℓ
1
\ell_1
ℓ
1
and
ℓ
2
\ell_2
ℓ
2
, segments
A
B
AB
A
B
and
C
D
CD
C
D
of a given length are selected, respectively. Prove that the volume of the tetrahedron
A
B
C
D
ABCD
A
BC
D
does not depend on the position of the segments
A
B
AB
A
B
and
C
D
CD
C
D
on the lines
ℓ
1
\ell_1
ℓ
1
and
ℓ
2
\ell_2
ℓ
2
.
61
1
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Chisinau MO p61 1949-56 X locus of projections, on all planes
Find the locus of the projections of a given point on all planes containing another point
B
B
B
.
60
1
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Chisinau MO p60 1949-56 X sum of the distances, regular tetrahedron
Show that the sum of the distances from any point of a regular tetrahedron to its faces is equal to the height of this tetrahedron.
59
1
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Chisinau MO p59 1949-56 X cos ^2 A + cos ^2 B + cos ^2 C=1
Show that triangle
A
B
C
ABC
A
BC
is right-angled if its angles satisfy the ratio
cos
2
A
+
cos
2
B
+
c
o
s
2
C
=
1
\cos^2A + \cos ^2B +\ cos ^2C=1
cos
2
A
+
cos
2
B
+
co
s
2
C
=
1
.
58
1
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Chisinau MO p58 1949-56 X no of lines, m points out of n points
On the plane
n
n
n
points are chosen so that exactly
m
m
m
of them lie on one straight line and no three points not included in these
m
m
m
points lie on one straight line. What is the number of all lines, each of which contains at least two of these points?
57
1
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Chisinau MO p57 1949-56 X | z |- 2 = 1 + 2 i
Solve the equation:
∣
z
∣
−
2
=
1
+
2
i
| z |- 2 = 1 + 2 i
∣
z
∣
−
2
=
1
+
2
i
, where
∣
r
∣
| r |
∣
r
∣
is the modulus of a complex number
z
z
z
.
55
1
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Chisinau MO p55 1949-56 X (5-x)^4+ (x-2)^ 4 = 17, (a - x) ^4+ (x - b)^4 = c
Find the real roots of the equation
(
5
−
x
)
4
+
(
x
−
2
)
4
=
17
(5-x)^4+ (x-2)^ 4 = 17
(
5
−
x
)
4
+
(
x
−
2
)
4
=
17
and the real roots of a more general equation
(
a
−
x
)
4
+
(
x
−
b
)
4
=
c
(a - x) ^4+ (x - b)^4 = c
(
a
−
x
)
4
+
(
x
−
b
)
4
=
c
54
1
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Chisinau MO p54 1949-56 X x^2/3 + 48/x^3 =10 (x/3- 4 /x)
Solve the equation:
x
2
3
+
48
x
3
=
10
(
x
3
−
4
x
)
\frac{x^2}{3}+\frac{48}{x^3}=10 \left(\frac{x}{3}-\frac{4 }{x} \right)
3
x
2
+
x
3
48
=
10
(
3
x
−
x
4
)
53
1
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Chisinau MO p53 1949-56 X \sqrt[3]{a+\sqrt{x}}+\sqrt[3]{a-\sqrt{x}}=\sqrt[3]{b}
Solve the equation:
a
+
x
3
+
a
−
x
3
=
b
3
\sqrt[3]{a+\sqrt{x}}+\sqrt[3]{a-\sqrt{x}}=\sqrt[3]{b}
3
a
+
x
+
3
a
−
x
=
3
b
52
1
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Chisinau MO p52 1949-56 X 4^n < (2n+1)C_{2n}^n
Prove that for any natural number
n
n
n
the following inequality holds
4
n
<
(
2
n
+
1
)
C
2
n
n
4^n < (2n+1)C_{2n}^n
4
n
<
(
2
n
+
1
)
C
2
n
n
35
1
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Chisinau MO p35 1949-56 VIII-IX 2/(c+a)=1/(b+c)+1/(a+b) if 2b^2= a^2+ c^2
The numbers
a
2
,
b
2
,
c
2
a^2, b^2, c^2
a
2
,
b
2
,
c
2
form an arithmetic progression. Show that the numbers
1
b
+
c
,
1
c
+
a
,
1
a
+
b
\frac{1}{b+c},\frac{1}{c+a},\frac{1}{a+b}
b
+
c
1
,
c
+
a
1
,
a
+
b
1
also form arithmetic progression.
51
1
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Chisinau MO p51 1949-56 VIII-IX no of roots of sinx = lgx by graph
Determine graphically the number of roots of the equation
sin
x
=
lg
x
\sin x = \lg x
sin
x
=
l
g
x
.
50
1
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Chisinau MO p50 1949-56 VIII-IX ctg(a/2)}> 1 + ctga for 0 <a < pi /2
Prove the inequality:
c
t
g
a
2
>
1
+
c
t
g
a
ctg \frac{a}{2}> 1 + ctg a
c
t
g
2
a
>
1
+
c
t
g
a
for
0
<
a
<
π
2
0 <a <\frac{\pi}{2}
0
<
a
<
2
π
49
1
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Chisinau MO p49 1949-56 VIII-IX cos (pi/7) cos (4pi/7) cos (5pi/7) =1/8
Prove the identity:
cos
π
7
⋅
cos
4
π
7
⋅
cos
5
π
7
=
1
8
\cos \frac{\pi}{7} \cdot \cos \frac{4\pi}{7} \cdot \cos \frac{5\pi}{7} = \frac{1}{8}
cos
7
π
⋅
cos
7
4
π
⋅
cos
7
5
π
=
8
1
48
1
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Chisinau MO p48 1949-56 VIII-IX sin^3 a+cos^3 a =? if sina+cosa = m
Calculate
sin
3
a
+
cos
3
a
\sin^3 a + \cos^3 a
sin
3
a
+
cos
3
a
if you know that
sin
a
+
cos
a
=
m
\sin a+ \cos a = m
sin
a
+
cos
a
=
m
.
47
1
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Chisinau MO p47 1949-56 VIII-IX a^4+b^4+c^4=a^2b^2+b^2c^2+c^2a^2.
Determine the type of triangle if the lengths of its sides
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfy the relation
a
4
+
b
4
+
c
4
=
a
2
b
2
+
b
2
c
2
+
c
2
a
2
a^4 + b^4 + c^4 = a^2b^2 + b^2c^2 + c^2a^2
a
4
+
b
4
+
c
4
=
a
2
b
2
+
b
2
c
2
+
c
2
a
2
46
1
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Chisinau MO p46 1949-56 VIII-IX locus, constant PA/PB
Determine the locus of points, for whom the ratio of the distances to two given points has a constant value.
45
1
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Chisinau MO p45 1949-56 VIII-IX locus, equal tangents to 2 circles
Determine the locus of points, from which the tangent segments to two given circles are equal.
44
1
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Chisinau MO p44 1949-56 VIII-IX locus, constant PA^-PB^2
Determine the locus of points, for each of which the difference between the squares of the distances to two given points is a constant value.
43
1
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Chisinau MO p43 1949-56 VIII-IX equal lengths of arcs wanted
On the radius
O
A
OA
O
A
of a certain circle, as on the diameter, a circle is constructed. A ray is drawn from the center
O
O
O
, intersecting the larger and smaller circles at points
B
B
B
and
C
C
C
, respectively. Show that the lengths of arcs
A
B
AB
A
B
and
A
C
AC
A
C
are equal.
42
1
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Chisinau MO p42 1949-56 VIII-IXtrapezoid, isosceles triangle inscribed
A trapezoid and an isosceles triangle are inscribed in a circle. The larger base of the trapezoid is the diameter of the circle, and the sides of the triangle are parallel to the sides of the trapezoid. Show that the trapezoid and the triangle have equal areas.
41
1
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Chisinau MO p41 1949-56 VIII-IX bisectors of a rectangle form a square
Prove that the bisectors of the angles of a rectangle, extended to their mutual intersection, form a square.
40
1
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Chisinau MO p40 1949-56 VIII-IX system 3x3 log_{2} x + log_{4} y +log_{4} z =2
Solve the system of equations:
{
log
2
x
+
log
4
y
+
log
4
z
=
2
log
3
y
+
log
9
z
+
log
9
x
=
2
log
4
z
+
log
16
x
+
log
16
y
=
2
\begin{cases} \log_{2} x + \log_{4} y + \log_{4} z =2 \\ \log_{3} y + \log_{9} z + \log_{9} x =2 \\ \log_{4} z + \log_{16} x + \log_{16} y =2\end{cases}
⎩
⎨
⎧
lo
g
2
x
+
lo
g
4
y
+
lo
g
4
z
=
2
lo
g
3
y
+
lo
g
9
z
+
lo
g
9
x
=
2
lo
g
4
z
+
lo
g
16
x
+
lo
g
16
y
=
2
39
1
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Chisinau MO p39 1949-56 VIII-IX log_{x} 2 log_{2x} 2 = log_{4x} 2
Solve the equation:
log
x
2
⋅
log
2
x
2
=
log
4
x
2
\log_{x} 2 \cdot \log_{2x} 2 = \log_{4x} 2
lo
g
x
2
⋅
lo
g
2
x
2
=
lo
g
4
x
2
.
38
1
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Chisinau MO p38 1949-56 VIII-IX compare log_{3} 7 with log_{ 1/3} (1/7)
Which is more
log
3
7
\log_3 7
lo
g
3
7
or
log
1
3
1
7
\log_{\frac{1}{3}} \frac{1}{7}
lo
g
3
1
7
1
?
37
1
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Chisinau MO p37 1949-56 VIII-IX sum nx+(n-1)x^2+...+2x^{n-1}+x^n
Calculate the sum:
n
x
+
(
n
−
1
)
x
2
+
.
.
.
+
2
x
n
−
1
+
x
n
nx+(n-1)x^2+...+2x^{n-1}+x^n
n
x
+
(
n
−
1
)
x
2
+
...
+
2
x
n
−
1
+
x
n
36
1
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Chisinau MO p36 1949-56 VIII-IX sum 1+ 2q + 3q^2 +...+nq^{n-1}
Calculate the sum:
1
+
2
q
+
3
q
2
+
.
.
.
+
n
q
n
−
1
1+ 2q + 3q^2 +...+nq^{n-1}
1
+
2
q
+
3
q
2
+
...
+
n
q
n
−
1
34
1
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Chisinau MO p34 1949-56 VIII-IX construction by altitude , median, bisector
Construct a triangle by its altitude , median and angle bisector originating from one vertex.
33
1
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Chisinau MO p33 1949-56 VIII-IX triangle construction, 2 feet of altitudes, side
Construct a triangle, the base of which lies on the given line, and the feet of the altitudes, drawn on the sides, coincide with the given points.
32
1
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Chisinau MO p32 1949-56 VIII-IXlocus of midpoints of equal segments, 90^o
Determine the locus of points that are the midpoints of segments of equal length, the ends of which lie on the sides of a given right angle.
31
1
Hide problems
Chisinau MO p31 1949-56 VIII-IX locus of midpoints of chords
Find the locus of the points that are the midpoints of the chords of the secant to the given circle and passing through a given point.
30
1
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Chisinau MO p30 1949-56 VIII-IX computational in a trapezoid
Through the point of intersection of the diagonals of the trapezoid, a straight line is drawn parallel to its bases. Determine the length of the segment of this straight line, enclosed between the lateral sides of the trapezoid, if the lengths of the bases of the trapezoid are equal to
a
a
a
and
b
b
b
.
29
1
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Chisinau MO p29 1949-56 VIII-IX products from a point on circumcircle
Let
M
M
M
be an arbitrary point of a circle circumscribed around an acute-angled triangle
A
B
C
ABC
A
BC
. Prove that the product of the distances from the point
M
M
M
to the sides
A
C
AC
A
C
and
B
C
BC
BC
is equal to the product of the distances from
M
M
M
to the side
A
B
AB
A
B
and to the tangent to the circumscribed circle at point
C
C
C
.
28
1
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Chisinau MO p28 1949-56 VIII-IX 2\sqrt{(p-b)(p-c)} <= a
Prove the inequality
2
(
p
−
b
)
(
p
−
c
)
≤
a
2\sqrt{(p-b)(p-c)}\le a
2
(
p
−
b
)
(
p
−
c
)
≤
a
, where
a
,
b
,
c
a, b, c
a
,
b
,
c
are the lengths of the sides, and
p
p
p
is the semiperimeter of some triangle..
27
1
Hide problems
Chisinau MO p27 1949-56 VIII-IX similar right triangles
The areas of two right-angled triangles have ratio equal to the squares of their hypotenuses. Show that these triangles are similar.
26
1
Hide problems
Chisinau MO p26 1949-56 VIII-IX 2 median, 1 atltitude, congruence
Formulate a criterion for the conguence of triangles by two medians and an altitude.
25
1
Hide problems
Chisinau MO p25 1949-56 VIII-IX angle bisectors in triangle of feet of altitudes
Show that the straight lines passing through the feet of the altitudes of an acute-angled triangle form a triangle in which the altitudes of the original triangle are angle bisectors.
24
1
Hide problems
Chisinau MO p24 1949-56 VIII-IX similar triangles by segment ,feet of altitudes
Show that a line passing through the feet of two altitudes of an acute-angled triangle cuts off a similar triangle.
23
1
Hide problems
Chisinau MO p23 1949-56 VIII-IX distance wanted, interior point of 60^o
Inside the angle
A
B
C
ABC
A
BC
of
6
0
o
60^o
6
0
o
, point
O
O
O
is selected, which is located at distances from the sides of the angle
a
a
a
and
b
b
b
, respectively. Determine the distance from the top of the angle to this point.
21
1
Hide problems
Chisinau MO p21 1949-56 VIII-IX inverse of Pythagorean Theorem
The sides of the triangle
A
B
C
ABC
A
BC
satisfy the relation
c
2
=
a
2
+
b
2
c^2 = a^2 + b^2
c
2
=
a
2
+
b
2
. Show that angle
C
C
C
is right.
20
1
Hide problems
Chisinau MO p20 1949-56 VIII-IX A-> B 50km/h , B-> A 30km/h, average
From point
A
A
A
to point
B
B
B
, the car drove at a speed of
50
50
50
km / h, and from
B
B
B
to
A
A
A
, at a speed of
30
30
30
km / h. What was the average vehicle speed?
19
1
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Chisinau MO p19 1949-56 VIII-IX tallest kid
The schoolchildren sat down on chairs located in transverse and longitudinal rows. The tallest student was chosen from each transverse row, and the lowest was chosen among them. Then the lowest student was selected from each longitudinal row, and the tallest was chosen among them. Which of these two students is higher?
18
1
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Chisinau MO p18 1949-56 VIII-IX b^2x^2+(b^2+c^2-a^2)x+c^2 = 0 imaginary roots
Prove that if the numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
are the lengths of the sides of some nondegenerate triangle, then the equation
b
2
x
2
+
(
b
2
+
c
2
−
a
2
)
x
+
c
2
=
0
b^2x^2 + (b^2 + c^2 - a^2) x + c^2 = 0
b
2
x
2
+
(
b
2
+
c
2
−
a
2
)
x
+
c
2
=
0
has imaginary roots.
17
1
Hide problems
Chisinau MO p17 1949-56 VIII-IX x^2+px+q+(x+a)(2x+p)=0 has real roots for any a
Prove that if the roots of the equation
x
2
+
p
x
+
q
=
0
x^2 + px + q = 0
x
2
+
p
x
+
q
=
0
are real, then for any real number
a
a
a
the roots of the equation
x
2
+
p
x
+
q
+
(
x
+
a
)
(
2
x
+
p
)
=
0
x^2 + px + q + (x + a) (2x + p) = 0
x
2
+
p
x
+
q
+
(
x
+
a
)
(
2
x
+
p
)
=
0
are also real.
15
1
Hide problems
Chisinau MO p15 1949-56 VIII-IX system 3x3 , xy/(x+y) = 2/5
Solve the system of equations:
{
x
y
x
+
y
=
12
5
y
z
y
+
z
=
18
5
z
x
z
+
y
=
36
13
\begin{cases} \dfrac{xy}{x+y}=\dfrac{12}{5}\\ \\ \dfrac{yz}{y+z}=\dfrac{18}{5} \\ \\ \dfrac{zx}{z+y}=\dfrac{36}{13} \end{cases}
⎩
⎨
⎧
x
+
y
x
y
=
5
12
y
+
z
yz
=
5
18
z
+
y
z
x
=
13
36
14
1
Hide problems
Chisinau MO p14 1949-56 VIII-IX given 1/a+1/b+1/c=1/(a+b+c), sum of two =0
Prove that if the numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
are related by the relation
1
a
+
1
b
+
1
c
=
1
a
+
b
+
c
\frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}= \frac{1}{a+b+c}
a
1
+
b
1
+
c
1
=
a
+
b
+
c
1
then the sum of some two of them is equal to zero.
13
1
Hide problems
Chisinau MO p13 1949-56 VIII-IX factor (a+b+c)^3- a^3 -b^3 -c^3
Factor the polynomial
(
a
+
b
+
c
)
3
−
a
3
−
b
3
−
c
3
(a+b+c)^3- a^3 -b^3 -c^3
(
a
+
b
+
c
)
3
−
a
3
−
b
3
−
c
3
12
1
Hide problems
Chisinau MO p12 1949-56 VIII-IX factor bc (b+c) +ca (c-a)-ab(a + b)
Factor the polynomial
b
c
(
b
+
c
)
+
c
a
(
c
−
a
)
−
a
b
(
a
+
b
)
bc (b+c) +ca (c-a)-ab(a + b)
b
c
(
b
+
c
)
+
c
a
(
c
−
a
)
−
ab
(
a
+
b
)
.
11
1
Hide problems
Chisinau MO p11 1949-56 VIII-IX factor x^3+x^2z+xyz+y^2z-y^3
Factor the polynomial
x
3
+
x
2
z
+
x
y
z
+
y
2
z
−
y
3
x^3+x^2z+xyz+y^2z-y^3
x
3
+
x
2
z
+
x
yz
+
y
2
z
−
y
3
.
10
1
Hide problems
Chisinau MO p10 1949-56 VIII-IX rational 1/(\sqrt[3]{4}+\sqrt[3]{2}+2})
Get rid of irrationality in the denominator of a fraction
1
4
3
+
2
3
+
2
\frac{1}{\sqrt[3]{4}+\sqrt[3]{2}+2}
3
4
+
3
2
+
2
1
.
9
1
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Chisinau MO p9 1949-56 VIII-IX n(n^2 + 5) is divisible by 6
Prove that for any integer
n
n
n
the number
n
(
n
2
+
5
)
n (n^2 + 5)
n
(
n
2
+
5
)
is divisible by
6
6
6
.
8
1
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Chisinau MO p8 1949-56 VIII-IX sum of 2squares of integers mod 4 not 3
Prove that the remainder of dividing the sum of two squares of integers by
4
4
4
is different from
3
3
3
.
7
1
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Chisinau MO p7 1949-56 VIII-IX if n! not divisible by n + 1, then n+1 is prime
Prove that if the product
1
⋅
2
⋅
.
.
.
⋅
n
1\cdot 2\cdot ...\cdot n
1
⋅
2
⋅
...
⋅
n
(
n
>
3
n> 3
n
>
3
) is not divisible by
n
+
1
n + 1
n
+
1
, then
n
+
1
n + 1
n
+
1
is prime.
6
1
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Chisinau MO p6 1949-56 VIII-IX remainder of perfect square mod 3 not2
Prove that the remainder of dividing the square of an integer by
3
3
3
is different from
2
2
2
.
5
1
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Chisinau MO p5 1949-56 VIII-IX perfect square cannot end with two fives
Prove that the square of any integer cannot end with two fives.
4
1
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Chisinau MO p4 1949-56 VIII-IX product of 4 consecutive + 1 =perfect square
Prove that the product of four consecutive integers plus
1
1
1
is a perfect square.
3
1
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Chisinau MO p3 1949-56 VIII-IX N = 10 ...050...01 not a perfect cube
Prove that the number
N
=
10...050...01
N = 10 ...050...01
N
=
10...050...01
(1, 49 zeros, 5 , 99 zeros, 1) is a not cube of an integer.
2
1
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Chisinau MO p2 1949-56 VIII-IX last digit of 777^{777}
What is the last digit of
77
7
777
777^{777}
77
7
777
?
1
1
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Chisinau MO p1 1949-56 VIII-IX crossing every 15th no from 1-100 in a circle
The numbers
1
,
2
,
.
.
.
,
1000
1, 2, ..., 1000
1
,
2
,
...
,
1000
are written out in a row along a circle. Starting from the first, every fifteenth number in the circle is crossed out
(
1
,
16
,
31
,
.
.
.
)
(1, 16, 31, ...)
(
1
,
16
,
31
,
...
)
, in this case, the crossed out numbers are still taken into account at each new round of the circle. How many numbers are left uncrossed?