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Problems
Contests
National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
2005 Greece Junior Math Olympiad
2005 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
1
1
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Trapezoid
We are given a trapezoid
A
B
C
D
ABCD
A
BC
D
with
A
B
∥
C
D
AB \parallel CD
A
B
∥
C
D
,
C
D
=
2
A
B
CD=2AB
C
D
=
2
A
B
and
D
B
⊥
B
C
DB \perp BC
D
B
⊥
BC
. Let
E
E
E
be the intersection of lines
D
A
DA
D
A
and
C
B
CB
CB
, and
M
M
M
be the midpoint of
D
C
DC
D
C
. (a) Prove that
A
B
M
D
ABMD
A
BM
D
is a rhombus. (b) Prove that triangle
C
D
E
CDE
C
D
E
is isosceles. (c) If
A
M
AM
A
M
and
B
D
BD
B
D
meet at
O
O
O
, and
O
E
OE
OE
and
A
B
AB
A
B
meet at
N
,
N,
N
,
prove that the line
D
N
DN
D
N
bisects segment
E
B
EB
EB
.
2
1
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Evaluate sum
If
f
(
n
)
=
2
n
+
1
+
n
(
n
+
1
)
n
+
1
+
n
f(n)=\frac{2n+1+\sqrt{n(n+1)}}{\sqrt{n+1}+\sqrt{n}}
f
(
n
)
=
n
+
1
+
n
2
n
+
1
+
n
(
n
+
1
)
for all positive integers
n
n
n
, evaluate (a)
f
(
1
)
f(1)
f
(
1
)
, (b) the sum
A
=
f
(
1
)
+
f
(
2
)
+
.
.
.
+
f
(
400
)
A=f(1)+f(2)+...+f(400)
A
=
f
(
1
)
+
f
(
2
)
+
...
+
f
(
400
)
.
3
1
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Maximum area
Let
A
A
A
be a given point outside a given circle. Determine points
B
,
C
,
D
B, C, D
B
,
C
,
D
on the circle such that the quadrilateral
A
B
C
D
ABCD
A
BC
D
is convex and has the maximum area .
4
1
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Find nonzero integers.
Find all nonzero integers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
with
a
>
b
>
c
>
d
a>b>c>d
a
>
b
>
c
>
d
that satisfy
a
b
+
c
d
=
34
ab+cd=34
ab
+
c
d
=
34
and
a
c
−
b
d
=
19.
ac-bd=19.
a
c
−
b
d
=
19.