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Problems
Contests
National and Regional Contests
Moldova Contests
JBMO TST - Moldova
2010 Junior Balkan Team Selection Tests - Moldova
2010 Junior Balkan Team Selection Tests - Moldova
Part of
JBMO TST - Moldova
Subcontests
(8)
1
1
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(1/a +1/b)(1/c +1/d)+1/ab +1/cd=6 /\sqrt{abcd}
The positive real numbers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
satisfy the equality
(
1
a
+
1
b
)
(
1
c
+
1
d
)
+
1
a
b
+
1
c
d
=
6
a
b
c
d
\left(\frac{1}{a}+ \frac{1}{b}\right) \left(\frac{1}{c}+ \frac{1}{d}\right) + \frac{1}{ab}+ \frac{1}{cd} = \frac{6}{\sqrt{abcd}}
(
a
1
+
b
1
)
(
c
1
+
d
1
)
+
ab
1
+
c
d
1
=
ab
c
d
6
Find the value of the
a
2
+
a
c
+
c
2
b
2
−
b
d
+
d
2
\frac{a^2+ac+c^2}{b^2-bd+d^2}
b
2
−
b
d
+
d
2
a
2
+
a
c
+
c
2
2
1
Hide problems
E= |(x-y)/(x+y)+(x^2-y^2)/(x^2+y^2)+(x^3-y^3)/(x^3+y^3)| if x + y = 3 \sqrt{xy
The positive real numbers
x
x
x
and
y
y
y
satisfy the relation
x
+
y
=
3
x
y
x + y = 3 \sqrt{xy}
x
+
y
=
3
x
y
. Find the value of the numerical expression
E
=
∣
x
−
y
x
+
y
+
x
2
−
y
2
x
2
+
y
2
+
x
3
−
y
3
x
3
+
y
3
∣
.
E=\left| \frac{x-y}{x+y}+\frac{x^2-y^2}{x^2+y^2}+\frac{x^3-y^3}{x^3+y^3}\right|.
E
=
x
+
y
x
−
y
+
x
2
+
y
2
x
2
−
y
2
+
x
3
+
y
3
x
3
−
y
3
.
7
1
Hide problems
\sqrt{(a - b) (a + c)} +\sqrt{(a - c) (a + b) } \ge 2\sqrt{a^2-bc} isosceles
In the triangle
A
B
C
ABC
A
BC
with
∣
A
B
∣
=
c
,
∣
B
C
∣
=
a
,
∣
C
A
∣
=
b
| AB | = c, | BC | = a, | CA | = b
∣
A
B
∣
=
c
,
∣
BC
∣
=
a
,
∣
C
A
∣
=
b
the relations hold simultaneously
a
≥
m
a
x
{
b
,
c
,
b
c
}
,
(
a
−
b
)
(
a
+
c
)
+
(
a
−
c
)
(
a
+
b
)
≥
2
a
2
−
b
c
a \ge max \{ b, c, \sqrt{bc}\}, \sqrt{(a - b) (a + c)} + \sqrt{(a - c) (a + b) } \ge 2\sqrt{a^2-bc}
a
≥
ma
x
{
b
,
c
,
b
c
}
,
(
a
−
b
)
(
a
+
c
)
+
(
a
−
c
)
(
a
+
b
)
≥
2
a
2
−
b
c
Prove that the triangle
A
B
C
ABC
A
BC
is isosceles.
5
1
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a/(a+b) x (a+2b)/(a+3b) < \sqrt{ a /(a+4b)}
For any strictly positive numbers
a
a
a
and
b
b
b
, prove the inequality
a
a
+
b
⋅
a
+
2
b
a
+
3
b
<
a
a
+
4
b
.
\frac{a}{a+b} \cdot \frac{a+2b}{a+3b} < \sqrt{ \frac{a}{a+4b}}.
a
+
b
a
⋅
a
+
3
b
a
+
2
b
<
a
+
4
b
a
.
6
1
Hide problems
E =\frac{a}{b + c}+\frac{b}{c + a}+\frac{c}{a + b} , right triangle, inradii
In a right triangle with the length legs
b
b
b
and
c
c
c
, and the length hypotenuse
a
a
a
, the ratio between the length of the hypotenuse and the length of the diameter of the inscribed circle does not exceed
1
+
2
1 + \sqrt2
1
+
2
. Determine the numerical value of the expression of
E
=
a
b
+
c
+
b
c
+
a
+
c
a
+
b
E =\frac{a}{b + c}+\frac{b}{c + a}+\frac{c}{a + b}
E
=
b
+
c
a
+
c
+
a
b
+
a
+
b
c
.
4
1
Hide problems
equal sum in each line and in each column, in 5x5 square
In the
25
25
25
squares of a
5
×
5
5 \times 5
5
×
5
square, zeros are initially written. in the every minute Ionel chooses two squares with a common side. If they are written in them the numbers
a
a
a
and
b
b
b
, then he writes instead the numbers
a
+
1
a + 1
a
+
1
and
b
+
1
b + 1
b
+
1
or
a
−
1
a - 1
a
−
1
and
b
−
1
b - 1
b
−
1
. Over time he noticed that the sums of the numbers in each line were equal, as well as the sums of the numbers in each column are equal. Prove that this observation was made after an even number of minutes.
3
1
Hide problems
collinearity wanted, tangents to circumcircle related
The tangent to the circle circumscribed to the triangle
A
B
C
ABC
A
BC
, taken through the vertex
A
A
A
, intersects the line
B
C
BC
BC
at the point
P
P
P
, and the tangents to the same circle, taken through
B
B
B
and
C
C
C
, intersect the lines
A
C
AC
A
C
and
A
B
AB
A
B
, respectively at the points
Q
Q
Q
and
R
R
R
. Prove that the points
P
,
Q
P, Q
P
,
Q
¸ and
R
R
R
are collinear.
8
1
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Combinatorics
What is the minimum
n
n
n
so that grid
n
x
n
nxn
n
x
n
can be covered with equal number of 2x2 squares and angle triminoes (2x2 without one square)