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Problems
Contests
National and Regional Contests
Cuba Contests
Cuba MO
2001 Cuba MO
2001 Cuba MO
Part of
Cuba MO
Subcontests
(9)
9
1
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[CEGF] = [BDG]
In triangle
A
B
C
ABC
A
BC
, right at
C
C
C
, let
F
F
F
be the intersection point of the altitude
C
D
CD
C
D
with the angle bisector
A
E
AE
A
E
and
G
G
G
be the intersection point of
E
D
ED
E
D
with
B
F
BF
BF
. Prove that the area of the quadrilateral
C
E
G
F
CEGF
CEGF
is equal to the area of the triangle
B
D
G
BDG
B
D
G
.
8
1
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x + cos x = 1
Find all real solutions of the equation
x
+
c
o
s
x
=
1
x + cos x = 1
x
+
cos
x
=
1
.
7
1
Hide problems
x^{19} + x^{17} = x^{16 }+ x^7 + a
Prove that the equation
x
19
+
x
17
=
x
16
+
x
7
+
a
x^{19} + x^{17} = x^{16 }+ x^7 + a
x
19
+
x
17
=
x
16
+
x
7
+
a
for any
a
∈
R
a \in R
a
∈
R
has at least two imaginary roots
6
1
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\frac{a}{a+c} + \frac{b}{b+a} > \frac{c}{b+c} , ax^2 - 4bx + 4c = 0
The roots of the equation
a
x
2
−
4
b
x
+
4
c
=
0
ax^2 - 4bx + 4c = 0
a
x
2
−
4
b
x
+
4
c
=
0
with
a
>
0
a > 0
a
>
0
belong to interval
[
2
,
3
]
[2, 3]
[
2
,
3
]
. Prove that: a)
a
≤
b
≤
c
<
a
+
b
.
a \le b \le c < a + b.
a
≤
b
≤
c
<
a
+
b
.
b)
a
a
+
c
+
b
b
+
a
>
c
b
+
c
.
\frac{a}{a+c} + \frac{b}{b+a} > \frac{c}{b+c} .
a
+
c
a
+
b
+
a
b
>
b
+
c
c
.
5
1
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a = ( p +\sqrt{p^2 + q} )^2 is irrational
Let
p
p
p
and
q
q
q
be two positive integers such that
1
≤
q
≤
p
1 \le q \le p
1
≤
q
≤
p
. Also let
a
=
(
p
+
p
2
+
q
)
2
a = \left( p +\sqrt{p^2 + q} \right)^2
a
=
(
p
+
p
2
+
q
)
2
. a) Prove that the number
a
a
a
is irrational. b) Show that
{
a
}
>
0.75
\{a\} > 0.75
{
a
}
>
0.75
.
4
1
Hide problems
ellipse, tangents of 4 points 2001 Cuba MO 2.4
The tangents at four different points of an arc of a circle less than
18
0
o
180^o
18
0
o
intersect forming a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
. Prove that two of the vertices belong to an ellipse whose foci to the other two vertices.
1
2
Hide problems
numbers in 3 x3 board
In each square of a
3
×
3
3 \times 3
3
×
3
board a real number is written. The element of the
i
i
i
-th row and the
j
j
j
-th column is equal to abso;uteof the difference of the sum of the elements of column
j
j
j
and the sum of the elements of row
i
i
i
. Prove that every element of the board is equal to the sum or difference of two other elements on the board.
f(x) f(k - x) > 0
Let
f
f
f
be a linear function such that
f
(
0
)
=
−
5
f(0) = -5
f
(
0
)
=
−
5
and
f
(
f
(
0
)
)
=
−
15
f(f(0)) = -15
f
(
f
(
0
))
=
−
15
. Find the values of
k
∈
R
k \in R
k
∈
R
for which the solutions of the inequality
f
(
x
)
⋅
f
(
k
−
x
)
>
0
f(x) \cdot f(k - x) > 0
f
(
x
)
⋅
f
(
k
−
x
)
>
0
, lie in an interval of length
2
2
2
.
3
2
Hide problems
sum of digits of m = n(2n-1) is equal to 2000.
Prove that there is no natural number n such that the sum of all the digits of the number m, where
m
=
n
(
2
n
−
1
)
m = n(2n-1)
m
=
n
(
2
n
−
1
)
is equal to
2000
2000
2000
.
(2n+1)^3-2 is sum of 3n-1 perfect squares
Let
n
n
n
be a positive integer. a) Prove that the number
(
2
n
+
1
)
3
−
(
2
n
−
1
)
3
(2n + 1)^3 - (2n - 1)^3
(
2
n
+
1
)
3
−
(
2
n
−
1
)
3
is the sum of three perfect squares. b) Prove that the number
(
2
n
+
1
)
3
−
2
(2n+1)^3-2
(
2
n
+
1
)
3
−
2
is the sum of
3
n
−
1
3n-1
3
n
−
1
perfect squares greater than
1
1
1
.
2
2
Hide problems
<BKC = <CDB if MA x MC + MA x CD = MB xMD
Let
M
M
M
be the point of intersection of the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
of the convex quadrilateral
A
B
C
D
ABCD
A
BC
D
. Let
K
K
K
be the intersection point of the extension of side
A
B
AB
A
B
(from
A
A
A
) with the bisector of the
∠
A
C
D
\angle ACD
∠
A
C
D
. If
M
A
⋅
M
C
+
M
A
⋅
C
D
=
M
B
⋅
M
D
MA \cdot MC + MA \cdot CD = MB\cdot MD
M
A
⋅
MC
+
M
A
⋅
C
D
=
MB
⋅
M
D
, prove that
∠
B
K
C
=
∠
C
D
B
\angle BKC = \angle CDB
∠
B
K
C
=
∠
C
D
B
.
AP_|_MK if MC = KD in square ABCD - 2001 Cuba MO 2.2
Let
A
B
C
D
ABCD
A
BC
D
be a square. On the sides
B
C
BC
BC
and
C
D
CD
C
D
the points
M
M
M
and
K
K
K
respectively, so that
M
C
=
K
D
MC = KD
MC
=
KD
. Let
P
P
P
the intersection point of of segments
M
D
MD
M
D
and
B
K
BK
B
K
. Prove that
A
P
⊥
M
K
AP \perp MK
A
P
⊥
M
K
.