MathDB
Problems
Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2011 Moldova Team Selection Test
2011 Moldova Team Selection Test
Part of
Moldova Team Selection Test
Subcontests
(4)
4
2
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Integer part of an expresion
Let
n
n
n
be an integer satisfying
n
≥
2
n\geq2
n
≥
2
. Find the greatest integer not exceeding the expression: E=1+\sqrt{1+\frac{2^2}{3!}}+\sqrt[3]{1+\frac{3^2}{4!}}+\dots+\+\sqrt[n]{1+\frac{n^2}{(n+1)!}}
increase a number by 1 and decrease another number by 1.
Initially, on the blackboard are written all natural numbers from
1
1
1
to
20
20
20
. A move consists of selecting
2
2
2
numbers
a
<
b
a<b
a
<
b
written on the blackboard such that their difference is at least
2
2
2
, erasing these numbers and writting
a
+
1
a+1
a
+
1
and
b
−
1
b-1
b
−
1
instead. What is the maximum numbers of moves one can perform?
3
2
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equilateral triangles build on sides of a quadrilateral
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral and
M
M
M
the midpoint of the segment
A
B
AB
A
B
. Outside of the quadrilateral are constructed the equilateral triangles
B
C
E
BCE
BCE
,
C
D
F
CDF
C
D
F
and
D
A
G
DAG
D
A
G
. Let
P
P
P
and
N
N
N
be the midpoints of the segments
G
F
GF
GF
and
E
F
EF
EF
. Prove that the triangle
M
N
P
MNP
MNP
is equilateral.
Bisectors in triangle
Let
A
B
C
ABC
A
BC
be a triangle with
∠
B
A
C
=
60
\angle BAC=60
∠
B
A
C
=
60
. Let
B
1
B_1
B
1
and
C
1
C_1
C
1
be the feet of the bisectors from
B
B
B
and
C
C
C
. Let
A
1
A_1
A
1
be the symmetrical of
A
A
A
according to line
B
1
C
1
B_1C_1
B
1
C
1
. Prove that
A
1
,
B
,
C
A_1, B, C
A
1
,
B
,
C
are colinear.
2
2
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System of equations
Find all pairs of real number
x
x
x
and
y
y
y
which simultaneously satisfy the following 2 relations:
x
+
y
+
4
=
12
x
+
11
y
x
2
+
y
2
x+y+4=\frac{12x+11y}{x^2+y^2}
x
+
y
+
4
=
x
2
+
y
2
12
x
+
11
y
y
−
x
+
3
=
11
x
−
12
y
x
2
+
y
2
y-x+3=\frac{11x-12y}{x^2+y^2}
y
−
x
+
3
=
x
2
+
y
2
11
x
−
12
y
Product equals to 1
Let
x
1
,
x
2
,
…
,
x
n
x_1, x_2, \ldots, x_n
x
1
,
x
2
,
…
,
x
n
be real positive numbers such that
x
1
⋅
x
2
⋯
x
n
=
1
x_1\cdot x_2\cdots x_n=1
x
1
⋅
x
2
⋯
x
n
=
1
. Prove the inequality
1
x
1
(
x
1
+
1
)
+
1
x
2
(
x
2
+
1
)
+
⋯
+
1
x
n
(
x
n
+
1
)
≥
n
2
\frac1{x_1(x_1+1)}+\frac1{x_2(x_2+1)}+\cdots+\frac1{x_n(x_n+1)}\geq\frac n2
x
1
(
x
1
+
1
)
1
+
x
2
(
x
2
+
1
)
1
+
⋯
+
x
n
(
x
n
+
1
)
1
≥
2
n
1
2
Hide problems
Equation with real numbers
Find all real numbers
x
,
y
x, y
x
,
y
such that:
y
+
3
x
+
2
=
23
2
+
y
2
−
49
−
16
x
y+3\sqrt{x+2}=\frac{23}2+y^2-\sqrt{49-16x}
y
+
3
x
+
2
=
2
23
+
y
2
−
49
−
16
x
Sum of all elements of a set
Natural numbers have been divided in groups as follow:
(
1
)
,
(
2
,
4
)
,
(
3
,
5
,
7
)
,
(
6
,
8
,
10
,
12
)
,
(
9
,
11
,
13
,
15
,
17
)
,
…
(1), (2, 4), (3, 5, 7), (6, 8, 10, 12), (9, 11, 13, 15, 17), \ldots
(
1
)
,
(
2
,
4
)
,
(
3
,
5
,
7
)
,
(
6
,
8
,
10
,
12
)
,
(
9
,
11
,
13
,
15
,
17
)
,
…
. Let
S
n
S_n
S
n
be the sum of the elements of the
n
n
n
th group. Prove that
S
2
n
+
1
2
n
+
1
−
S
2
n
2
n
\frac{S_{2n+1}}{2n+1}-\frac{S_{2n}}{2n}
2
n
+
1
S
2
n
+
1
−
2
n
S
2
n
is even.