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Contests
National and Regional Contests
Slovenia Contests
Slovenia Team Selection Tests
1997 Slovenia Team Selection Test
1997 Slovenia Team Selection Test
Part of
Slovenia Team Selection Tests
Subcontests
(4)
6
1
Hide problems
if 2^p +3^p = a^n for some integer n, then n = 1
Let
p
p
p
be a prime number and
a
a
a
be an integer. Prove that if
2
p
+
3
p
=
a
n
2^p +3^p = a^n
2
p
+
3
p
=
a
n
for some integer
n
n
n
, then
n
=
1
n = 1
n
=
1
.
3
1
Hide problems
coloring on n points on circle
Let
A
1
,
A
2
,
.
.
.
,
A
n
A_1,A_2,...,A_n
A
1
,
A
2
,
...
,
A
n
be
n
≥
2
n \ge 2
n
≥
2
distinct points on a circle. Find the number of colorings of these points with
p
≥
2
p \ge 2
p
≥
2
colors such that every two adjacent points receive different colors
1
1
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common tangent of externally tangent circles wanted
Circles
K
1
K_1
K
1
and
K
2
K_2
K
2
are externally tangent to each other at
A
A
A
and are internally tangent to a circle
K
K
K
at
A
1
A_1
A
1
and
A
2
A_2
A
2
respectively. The common tangent to
K
1
K_1
K
1
and
K
2
K_2
K
2
at
A
A
A
meets
K
K
K
at point
P
P
P
. Line
P
A
1
PA_1
P
A
1
meets
K
1
K_1
K
1
again at
B
1
B_1
B
1
and
P
A
2
PA_2
P
A
2
meets
K
2
K_2
K
2
again at
B
2
B_2
B
2
. Show that
B
1
B
2
B_1B_2
B
1
B
2
is a common tangent of
K
1
K_1
K
1
and
K
2
K_2
K
2
.
2
1
Hide problems
xp(x)p(1-x)+x^3 +100 \ge 0 , polynomial
Find all polynomials
p
p
p
with real coefficients such that for all real
x
x
x
,
x
p
(
x
)
p
(
1
−
x
)
+
x
3
+
100
≥
0
xp(x)p(1-x)+x^3 +100 \ge 0
x
p
(
x
)
p
(
1
−
x
)
+
x
3
+
100
≥
0
.