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National and Regional Contests
Norway Contests
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
1996 Abels Math Contest (Norwegian MO)
1996 Abels Math Contest (Norwegian MO)
Part of
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
Subcontests
(2)
4
1
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f(f(1995)) = 95, f(xy) = f(x)f(y), f(x) <= x , f(1995) =?
Let
f
:
N
→
N
f : N \to N
f
:
N
→
N
be a function such that
f
(
f
(
1995
)
)
=
95
,
f
(
x
y
)
=
f
(
x
)
f
(
y
)
f(f(1995)) = 95, f(xy) = f(x)f(y)
f
(
f
(
1995
))
=
95
,
f
(
x
y
)
=
f
(
x
)
f
(
y
)
and
f
(
x
)
≤
x
f(x) \le x
f
(
x
)
≤
x
for all
x
,
y
x,y
x
,
y
. Find all possible values of
f
(
1995
)
f(1995)
f
(
1995
)
.
1
1
Hide problems
given circle and secant, locus as a circle wanted
Let
S
S
S
be a circle with center
C
C
C
and radius
r
r
r
, and let
P
≠
C
P \ne C
P
=
C
be an arbitrary point. A line
ℓ
\ell
ℓ
through
P
P
P
intersects the circle in
X
X
X
and
Y
Y
Y
. Let
Z
Z
Z
be the midpoint of
X
Y
XY
X
Y
. Prove that the points
Z
Z
Z
, as
ℓ
\ell
ℓ
varies, describe a circle. Find the center and radius of this circle.