MathDB
Problems
Contests
National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
1996 Greece Junior Math Olympiad
1996 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(5)
4a
1
Hide problems
2 simplifies (an + b)/(cn + d) => (ad - bc) is even (Greece 1996 junior 4a)
If the fraction
a
n
+
b
c
n
+
d
\frac{an + b}{cn + d}
c
n
+
d
an
+
b
may be simplified using
2
2
2
(as a common divisor ), show that the number
a
d
−
b
c
ad - bc
a
d
−
b
c
is even. (
a
,
b
,
c
,
d
,
n
a, b, c, d, n
a
,
b
,
c
,
d
,
n
are natural numbers and the
c
n
+
d
cn + d
c
n
+
d
different from zero).
4b
1
Hide problems
n^2 + n + p = 1996 diophantine (1996 Greece Junior 4b)
Determine whether exist a prime number
p
p
p
and natural number
n
n
n
such that
n
2
+
n
+
p
=
1996
n^2 + n + p = 1996
n
2
+
n
+
p
=
1996
.
1
1
Hide problems
(x^2 + 2x + 1)^2+...+(x^2 +1996x + 1995)^2=0 (1996 Greece Junior p1)
Solve the equation
(
x
2
+
2
x
+
1
)
2
+
(
x
2
+
3
x
+
2
)
2
+
(
x
2
+
4
x
+
3
)
2
+
.
.
.
+
(
x
2
+
1996
x
+
1995
)
2
=
0
(x^2 + 2x + 1)^2+(x^2 + 3x + 2)^2+(x^2 + 4x +3)^2+...+(x^2 + 1996x + 1995)^2= 0
(
x
2
+
2
x
+
1
)
2
+
(
x
2
+
3
x
+
2
)
2
+
(
x
2
+
4
x
+
3
)
2
+
...
+
(
x
2
+
1996
x
+
1995
)
2
=
0
3
1
Hide problems
min of 2x^4 - 2x^2y^2 + y^4 - 8x^2 + 18 (1996 Greece Junior p3)
Determine the minimum value of the expression
2
x
4
−
2
x
2
y
2
+
y
4
−
8
x
2
+
18
2x^4 - 2x^2y^2 + y^4 - 8x^2 + 18
2
x
4
−
2
x
2
y
2
+
y
4
−
8
x
2
+
18
where
x
,
y
∈
R
x, y \in R
x
,
y
∈
R
.
2
1
Hide problems
computational with midpoints and areas (Greece Junior 1996 p2)
In a triangle
A
B
C
ABC
A
BC
let
D
,
E
,
Z
,
H
,
G
D,E,Z,H,G
D
,
E
,
Z
,
H
,
G
be the midpoints of
B
C
,
A
D
,
B
D
,
E
D
,
E
Z
BC,AD,BD,ED,EZ
BC
,
A
D
,
B
D
,
E
D
,
EZ
respectively. Let
I
I
I
be the intersection of
B
E
,
A
C
BE,AC
BE
,
A
C
and let
K
K
K
be the intersection of
H
G
,
A
C
HG,AC
H
G
,
A
C
. Prove that: a)
A
K
=
3
C
K
AK=3CK
A
K
=
3
C
K
b)
H
K
=
3
H
G
HK=3HG
HK
=
3
H
G
c)
B
E
=
3
E
I
BE=3EI
BE
=
3
E
I
d)
(
E
G
H
)
=
1
32
(
A
B
C
)
(EGH)=\frac{1}{32}(ABC)
(
EG
H
)
=
32
1
(
A
BC
)
Notation
(
.
.
.
)
(...)
(
...
)
stands for area of
.
.
.
.
....
....