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Turkey Contests
Turkey Junior National Olympiad
1999 Turkey Junior National Olympiad
1999 Turkey Junior National Olympiad
Part of
Turkey Junior National Olympiad
Subcontests
(3)
3
1
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Turkey Junior Olympiad 1999, Part II - P3
Let
d
(
n
)
d(n)
d
(
n
)
denote the largest odd integer divides
n
n
n
. Calculate the sum
d
(
1
)
+
d
(
2
)
+
d
(
3
)
+
⋯
+
d
(
2
99
)
d(1)+d(2)+d(3)+\dots+d(2^{99})
d
(
1
)
+
d
(
2
)
+
d
(
3
)
+
⋯
+
d
(
2
99
)
.
2
1
Hide problems
Turkey Junior Olympiad 1999, Part II - P2
Each of integers from
1
1
1
to
20
20
20
are placed into the dots below. Two dots are adjacent, if below figure contains a line segment connecting them. Prove that how the numbers are arranged, it is possible to find an adjacent pair such that the difference between the numbers written on them is greater than
3
3
3
.[asy]real u=0.25cm; for(int i = 0; i < 4; ++i) { real v = u*(i+1); pair P1 = dir(90+0*72)*(0,v); pair P2 = dir(90+1*72)*(0,v); pair P3 = dir(90+2*72)*(0,v); pair P4 = dir(90+3*72)*(0,v); pair P5 = dir(90+4*72)*(0,v); dot(P1);dot(P2); dot(P3);dot(P4);dot(P5); path p = P1--P2--P3--P4--P5--cycle; draw(p); } [/asy]
1
1
Hide problems
Turkey Junior Olympiad 1999, Part II - P1
The chord
[
C
D
]
[CD]
[
C
D
]
is parallel to the diameter
[
A
B
]
[AB]
[
A
B
]
of a circle with center
O
O
O
. The tangent line at
A
A
A
meet
B
C
BC
BC
and
B
D
BD
B
D
at
E
E
E
and
F
F
F
. If
∣
A
B
∣
=
10
|AB|=10
∣
A
B
∣
=
10
, calculate
∣
A
E
∣
⋅
∣
A
F
∣
|AE|\cdot |AF|
∣
A
E
∣
⋅
∣
A
F
∣
.