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All-Russian Olympiad
1976 All Soviet Union Mathematical Olympiad
234
234
Part of
1976 All Soviet Union Mathematical Olympiad
Problems
(1)
ASU 234 All Soviet Union MO 1976 g(x_1)^2 + g(x_2)^2 + g(x_3)^2 = 1
Source:
7/6/2019
Given a sphere of unit radius with the big circle (i.e of unit radius) that will be called "equator". We shall use the words "pole", "parallel","meridian" as self-explanatory. a) Let
g
(
x
)
g(x)
g
(
x
)
, where
x
x
x
is a point on the sphere, be the distance from this point to the equator plane. Prove that
g
(
x
)
g(x)
g
(
x
)
has the property if
x
1
,
x
2
,
x
3
x_1, x_2, x_3
x
1
,
x
2
,
x
3
are the ends of the pairwise orthogonal radiuses, then
g
(
x
1
)
2
+
g
(
x
2
)
2
+
g
(
x
3
)
2
=
1
(
∗
)
g(x_1)^2 + g(x_2)^2 + g(x_3)^2 = 1 \,\,\,\, (*)
g
(
x
1
)
2
+
g
(
x
2
)
2
+
g
(
x
3
)
2
=
1
(
∗
)
Let function
f
(
x
)
f(x)
f
(
x
)
be an arbitrary nonnegative function on a sphere that satisfies (*) property. b) Let
x
1
x_1
x
1
and
x
2
x_2
x
2
points be on the same meridian between the north pole and equator, and
x
1
x_1
x
1
is closer to the pole than
x
2
x_2
x
2
. Prove that
f
(
x
1
)
>
f
(
x
2
)
f(x_1) > f(x_2)
f
(
x
1
)
>
f
(
x
2
)
. c) Let
y
1
y_1
y
1
be closer to the pole than
y
2
y_2
y
2
. Prove that
f
(
y
1
)
>
f
(
y
2
)
f(y_1) > f(y_2)
f
(
y
1
)
>
f
(
y
2
)
. d) Let
z
1
z_1
z
1
and
z
2
z_2
z
2
be on the same parallel. Prove that
f
(
z
1
)
=
f
(
z
2
)
f(z_1) = f(z_2)
f
(
z
1
)
=
f
(
z
2
)
. e) Prove that for all
x
,
f
(
x
)
=
g
(
x
)
x , f(x) = g(x)
x
,
f
(
x
)
=
g
(
x
)
.
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