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National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
2009 Czech and Slovak Olympiad III A
2009 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
6
1
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czech republic,third round,2009,problem 6
Given two fixed points
O
O
O
and
G
G
G
in the plane. Find the locus of the vertices of triangles whose circumcenters and centroids are
O
O
O
and
G
G
G
respectively.
5
1
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czech republic,third round,2009,problem 5
At every vertex
A
k
(
1
≤
k
≤
n
)
A_k(1\le k\le n)
A
k
(
1
≤
k
≤
n
)
of a regular
n
n
n
-gon,
k
k
k
coins are placed. We can do the following operation: in each step, one can choose two arbitrarily coins and move them to their adjacent vertices respectively, one clockwise and one anticlockwise. Find all positive integers
n
n
n
such that after a finite number of operations, we can reach the following configuration: there are
n
+
1
−
k
n+1-k
n
+
1
−
k
coins at vertex
A
k
A_k
A
k
for all
1
≤
k
≤
n
1\le k\le n
1
≤
k
≤
n
.
4
1
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czech republic,third round,2009,problem 4
A positive integer
n
n
n
is called good if and only if there exist exactly
4
4
4
positive integers
k
1
,
k
2
,
k
3
,
k
4
k_1, k_2, k_3, k_4
k
1
,
k
2
,
k
3
,
k
4
such that
n
+
k
i
∣
n
+
k
i
2
n+k_i|n+k_i^2
n
+
k
i
∣
n
+
k
i
2
(
1
≤
k
≤
4
1 \leq k \leq 4
1
≤
k
≤
4
). Prove that: [*]
58
58
58
is good;[*]
2
p
2p
2
p
is good if and only if
p
p
p
and
2
p
+
1
2p+1
2
p
+
1
are both primes (
p
>
2
p>2
p
>
2
).
3
1
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czech republic,third round,2009,problem 3
Find the least value of
x
>
0
x>0
x
>
0
such that for all positive real numbers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
satisfying
a
b
c
d
=
1
abcd=1
ab
c
d
=
1
, the inequality
a
x
+
b
x
+
c
x
+
d
x
≥
1
a
+
1
b
+
1
c
+
1
d
a^x+b^x+c^x+d^x\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}
a
x
+
b
x
+
c
x
+
d
x
≥
a
1
+
b
1
+
c
1
+
d
1
is true.
2
1
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czech republic,third round,2009,problem 2
Rectangle
A
B
C
D
ABCD
A
BC
D
is inscribed in circle
O
O
O
. Let the projections of a point
P
P
P
on minor arc
C
D
CD
C
D
onto
A
B
,
A
C
,
B
D
AB,AC,BD
A
B
,
A
C
,
B
D
be
K
,
L
,
M
K,L,M
K
,
L
,
M
, respectively. Prove that
∠
L
K
M
=
45
\angle LKM=45
∠
L
K
M
=
45
if and only if
A
B
C
D
ABCD
A
BC
D
is a square.
1
1
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czech republic,third round,2009,problem 1
Knowing that the numbers
p
,
3
p
+
2
,
5
p
+
4
,
7
p
+
6
,
9
p
+
8
p, 3p+2, 5p+4, 7p+6, 9p+8
p
,
3
p
+
2
,
5
p
+
4
,
7
p
+
6
,
9
p
+
8
, and
11
p
+
10
11p+10
11
p
+
10
are all primes, prove that
6
p
+
11
6p+11
6
p
+
11
is a composite number.