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Problems
Contests
National and Regional Contests
Romania Contests
JBMO TST - Romania
2003 Junior Balkan Team Selection Tests - Romania
2003 Junior Balkan Team Selection Tests - Romania
Part of
JBMO TST - Romania
Subcontests
(4)
3
3
Hide problems
\sqrt{n}+\sqrt{n+1} < \sqrt{x}+\sqrt{y} <\sqrt{4n+2}, diophantine inequality
Let
n
n
n
be a positive integer. Prove that there are no positive integers
x
x
x
and
y
y
y
such as
n
+
n
+
1
<
x
+
y
<
4
n
+
2
\sqrt{n}+\sqrt{n+1} < \sqrt{x}+\sqrt{y} <\sqrt{4n+2}
n
+
n
+
1
<
x
+
y
<
4
n
+
2
sums a + b,b + c and a + b + c are nonnegative, numbers around a circle
Five real numbers of absolute values not greater than
1
1
1
and having the sum equal to
1
1
1
are written on the circumference of a circle. Prove that one can choose three consecutively disposed numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
, such that all the sums
a
+
b
,
b
+
c
a + b,b + c
a
+
b
,
b
+
c
and
a
+
b
+
c
a + b + c
a
+
b
+
c
are nonnegative.
2 out of 2003 such their sum is not a divisor of the sum of other elements.
A set of
2003
2003
2003
positive integers is given. Show that one can find two elements such that their sum is not a divisor of the sum of the other elements.
4
3
Hide problems
one can color all the points of a plane using only 2 colors
Show that one can color all the points of a plane using only two colors such that no line segment has all points of the same color.
angle chasing inside a square ABCD, <MAB =< MBC = <BME=x
Let
E
E
E
be the midpoint of the side
C
D
CD
C
D
of a square
A
B
C
D
ABCD
A
BC
D
. Consider the point
M
M
M
inside the square such that
∠
M
A
B
=
∠
M
B
C
=
∠
B
M
E
=
x
\angle MAB = \angle MBC = \angle BME = x
∠
M
A
B
=
∠
MBC
=
∠
BME
=
x
. Find the angle
x
x
x
.
1/8 is common area of overlapping unit squares, min max of distance of centers
Two unit squares with parallel sides overlap by a rectangle of area
1
/
8
1/8
1/8
. Find the extreme values of the distance between the centers of the squares.
2
3
Hide problems
30 divides n_1^4 + n_2^4+...+n_{31}^4, among 31 primes exist 3 consecutive
Consider the prime numbers
n
1
<
n
2
<
.
.
.
<
n
31
n_1< n_2 <...< n_{31}
n
1
<
n
2
<
...
<
n
31
. Prove that if
30
30
30
divides
n
1
4
+
n
2
4
+
.
.
.
+
n
31
4
n_1^4 + n_2^4+...+n_{31}^4
n
1
4
+
n
2
4
+
...
+
n
31
4
, then among these numbers one can find three consecutive primes.
intersecting circles are seen from intersection of tangents under same angle.
Two circles
C
1
(
O
1
)
C_1(O_1)
C
1
(
O
1
)
and
C
2
(
O
2
)
C_2(O_2)
C
2
(
O
2
)
with distinct radii meet at points
A
A
A
and
B
B
B
. The tangent from
A
A
A
to
C
1
C_1
C
1
intersects the tangent from
B
B
B
to
C
2
C_2
C
2
at point
M
M
M
. Show that both circles are seen from
M
M
M
under the same angle.
a^n has an odd number of digits in the decimal representation for all n >0
Let
a
a
a
be a positive integer such that the number
a
n
a^n
a
n
has an odd number of digits in the decimal representation for all
n
>
0
n > 0
n
>
0
. Prove that the number
a
a
a
is an even power of
10
10
10
.
1
3
Hide problems
rectangle wanted, perpendicular bisectors related to a rhombus
Consider a rhombus
A
B
C
D
ABCD
A
BC
D
with center
O
O
O
. A point
P
P
P
is given inside the rhombus, but not situated on the diagonals. Let
M
,
N
,
Q
,
R
M,N,Q,R
M
,
N
,
Q
,
R
be the projections of
P
P
P
on the sides
(
A
B
)
,
(
B
C
)
,
(
C
D
)
,
(
D
A
)
(AB), (BC), (CD), (DA)
(
A
B
)
,
(
BC
)
,
(
C
D
)
,
(
D
A
)
, respectively. The perpendicular bisectors of the segments
M
N
MN
MN
and
R
Q
RQ
RQ
meet at
S
S
S
and the perpendicular bisectors of the segments
N
Q
NQ
NQ
and
M
R
MR
MR
meet at
T
T
T
. Prove that
P
,
S
,
T
P, S, T
P
,
S
,
T
and
O
O
O
are the vertices of a rectangle.
1 + 3/(a+b+c)>= 6/(ab+bc+ca) if a,b,c>0 with abc = 1
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive real numbers with
a
b
c
=
1
abc = 1
ab
c
=
1
. Prove that
1
+
3
a
+
b
+
c
≥
6
a
b
+
b
c
+
c
a
1 + \frac{3}{a+b+c}\ge \frac{6}{ab+bc+ca}
1
+
a
+
b
+
c
3
≥
ab
+
b
c
+
c
a
6
collinear wanted, rectangles ABCD, AEFG with B,E,D,G collinear given
Suppose
A
B
C
D
ABCD
A
BC
D
and
A
E
F
G
AEFG
A
EFG
are rectangles such that the points
B
,
E
,
D
,
G
B,E,D,G
B
,
E
,
D
,
G
are collinear (in this order). Let the lines
B
C
BC
BC
and
G
F
GF
GF
intersect at point
T
T
T
and let the lines
D
C
DC
D
C
and
E
F
EF
EF
intersect at point
H
H
H
. Prove that points
A
,
H
A, H
A
,
H
and
T
T
T
are collinear.