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Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
2006 Czech and Slovak Olympiad III A
2006 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
6
1
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real solutions to a system
Find all real solutions
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of the system of equations:
{
tan
2
x
+
2
cot
2
2
y
=
1
tan
2
y
+
2
cot
2
2
z
=
1
tan
2
z
+
2
cot
2
2
x
=
1
\begin{cases} \tan ^2x+2\cot^22y=1 \\ \tan^2y+2\cot^22z=1 \\ \tan^2z+2\cot^22x=1 \\ \end{cases}
⎩
⎨
⎧
tan
2
x
+
2
cot
2
2
y
=
1
tan
2
y
+
2
cot
2
2
z
=
1
tan
2
z
+
2
cot
2
2
x
=
1
5
1
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triples of 3 primes
Find all triples
(
p
,
q
,
r
)
(p,q,r)
(
p
,
q
,
r
)
of pairwise distinct primes such that
p
∣
q
+
r
,
q
∣
r
+
2
p
,
r
∣
p
+
3
q
.
p\mid q+r, q\mid r+2p, r\mid p+3q.
p
∣
q
+
r
,
q
∣
r
+
2
p
,
r
∣
p
+
3
q
.
4
1
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find the locus of centroid of a triangle
Given a segment
A
B
AB
A
B
in the plane. Let
C
C
C
be another point in the same plane,
H
,
I
,
G
H,I,G
H
,
I
,
G
denote the orthocenter,incenter and centroid of triangle
A
B
C
ABC
A
BC
. Find the locus of
M
M
M
for which
A
,
B
,
H
,
I
A,B,H,I
A
,
B
,
H
,
I
are concyclic.
3
1
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the condition for 5 points to be concyclic
In a scalene triangle
A
B
C
ABC
A
BC
,the bisectors of angle
A
,
B
A,B
A
,
B
intersect their corresponding sides at
K
,
L
K,L
K
,
L
respectively.
I
,
O
,
H
I,O,H
I
,
O
,
H
denote respectively the incenter,circumcenter and orthocenter of triangle
A
B
C
ABC
A
BC
. Prove that
A
,
B
,
K
,
L
,
O
A,B,K,L,O
A
,
B
,
K
,
L
,
O
are concyclic iff
K
L
KL
K
L
is the common tangent line of the circumcircles of the three triangles
A
L
I
,
B
H
I
ALI,BHI
A
L
I
,
B
H
I
and
B
K
I
BKI
B
K
I
.
2
1
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quadratic equation with integer root
Let
m
,
n
m,n
m
,
n
be positive integers such that the equation (in respect of
x
x
x
)
(
x
+
m
)
(
x
+
n
)
=
x
+
m
+
n
(x+m)(x+n)=x+m+n
(
x
+
m
)
(
x
+
n
)
=
x
+
m
+
n
has at least one integer root. Prove that
1
2
n
<
m
<
2
n
\frac{1}{2}n<m<2n
2
1
n
<
m
<
2
n
.
1
1
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prove the seventh term is not prime in a recursive sequence
Define a sequence of positive integers
{
a
n
}
\{a_n\}
{
a
n
}
through the recursive formula:
a
n
+
1
=
a
n
+
b
n
(
n
≥
1
)
a_{n+1}=a_n+b_n(n\ge 1)
a
n
+
1
=
a
n
+
b
n
(
n
≥
1
)
,where
b
n
b_n
b
n
is obtained by rearranging the digits of
a
n
a_n
a
n
(in decimal representation) in reverse order (for example,if
a
1
=
250
a_1=250
a
1
=
250
,then
b
1
=
52
,
a
2
=
302
b_1=52,a_2=302
b
1
=
52
,
a
2
=
302
,and so on). Can
a
7
a_7
a
7
be a prime?