MathDB
Problems
Contests
International Contests
Junior Balkan MO
1997 Junior Balkan MO
1997 Junior Balkan MO
Part of
Junior Balkan MO
Subcontests
(5)
4
1
Hide problems
find the triangle with R(b+c)=a\sqrt{bc}
Determine the triangle with sides
a
,
b
,
c
a,b,c
a
,
b
,
c
and circumradius
R
R
R
for which
R
(
b
+
c
)
=
a
b
c
R(b+c) = a\sqrt{bc}
R
(
b
+
c
)
=
a
b
c
. Romania
2
1
Hide problems
compute the expression
Let
x
2
+
y
2
x
2
−
y
2
+
x
2
−
y
2
x
2
+
y
2
=
k
\frac{x^2+y^2}{x^2-y^2} + \frac{x^2-y^2}{x^2+y^2} = k
x
2
−
y
2
x
2
+
y
2
+
x
2
+
y
2
x
2
−
y
2
=
k
. Compute the following expression in terms of
k
k
k
:
E
(
x
,
y
)
=
x
8
+
y
8
x
8
−
y
8
−
x
8
−
y
8
x
8
+
y
8
.
E(x,y) = \frac{x^8 + y^8}{x^8-y^8} - \frac{ x^8-y^8}{x^8+y^8}.
E
(
x
,
y
)
=
x
8
−
y
8
x
8
+
y
8
−
x
8
+
y
8
x
8
−
y
8
.
Ciprus
5
1
Hide problems
to be even or to be odd,this is the matter :D
Let
n
1
n_1
n
1
,
n
2
n_2
n
2
,
…
\ldots
…
,
n
1998
n_{1998}
n
1998
be positive integers such that
n
1
2
+
n
2
2
+
⋯
+
n
1997
2
=
n
1998
2
.
n_1^2 + n_2^2 + \cdots + n_{1997}^2 = n_{1998}^2.
n
1
2
+
n
2
2
+
⋯
+
n
1997
2
=
n
1998
2
.
Show that at least two of the numbers are even.
3
1
Hide problems
Is that related with Erdosh Mordelli's theorem?
Let
A
B
C
ABC
A
BC
be a triangle and let
I
I
I
be the incenter. Let
N
N
N
,
M
M
M
be the midpoints of the sides
A
B
AB
A
B
and
C
A
CA
C
A
respectively. The lines
B
I
BI
B
I
and
C
I
CI
C
I
meet
M
N
MN
MN
at
K
K
K
and
L
L
L
respectively. Prove that
A
I
+
B
I
+
C
I
>
B
C
+
K
L
AI+BI+CI>BC+KL
A
I
+
B
I
+
C
I
>
BC
+
K
L
. Greece
1
1
Hide problems
9 points inside a square
Show that given any 9 points inside a square of side 1 we can always find 3 which form a triangle with area less than
1
8
\frac 18
8
1
. Bulgaria