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Turkey Junior National Olympiad
2002 Turkey Junior National Olympiad
2002 Turkey Junior National Olympiad
Part of
Turkey Junior National Olympiad
Subcontests
(3)
3
1
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Turkey Junior Olympiad 2002, Part II - P3
Find all ordered positive integer pairs of
(
m
,
n
)
(m,n)
(
m
,
n
)
such that
2
n
−
1
2^n-1
2
n
−
1
divides
2
m
+
1
2^m+1
2
m
+
1
.
2
1
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Turkey Junior Olympiad 2002, Part II - P2
\text{ }
[asy] unitsize(11); for(int i=0; i<6; ++i) { if(i<5) draw( (i, 0)--(i,5) ); else draw( (i, 0)--(i,2) ); if(i < 3) draw((0,i)--(5,i)); else draw((0,i)--(4,i)); } [/asy] We are dividing the above figure into parts with shapes: [asy]unitsize(11); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,1)--(2,2)); draw((0,0)--(1,0)); draw((0,1)--(2,1)); draw((0,2)--(2,2)); [/asy][asy] unitsize(11); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,1)--(2,2)); draw((3,1)--(3,2)); draw((0,0)--(1,0)); draw((0,1)--(3,1)); draw((0,2)--(3,2)); [/asy] After that division, find the number of [asy]unitsize(11); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,1)--(2,2)); draw((0,0)--(1,0)); draw((0,1)--(2,1)); draw((0,2)--(2,2)); [/asy] shaped parts.
1
1
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Turkey Junior Olympiad 2002, Part II - P1
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid such that
∣
A
C
∣
=
8
|AC|=8
∣
A
C
∣
=
8
,
∣
B
D
∣
=
6
|BD|=6
∣
B
D
∣
=
6
, and
A
D
∥
B
C
AD \parallel BC
A
D
∥
BC
. Let
P
P
P
and
S
S
S
be the midpoints of
[
A
D
]
[AD]
[
A
D
]
and
[
B
C
]
[BC]
[
BC
]
, respectively. If
∣
P
S
∣
=
5
|PS|=5
∣
PS
∣
=
5
, find the area of the trapezoid
A
B
C
D
ABCD
A
BC
D
.