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Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
2012 Czech And Slovak Olympiad IIIA
2012 Czech And Slovak Olympiad IIIA
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
5
1
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in a group of 90 children each has at least 30 friends
In a group of
90
90
90
children each has at least
30
30
30
friends (friendship is mutual). Prove that they can be divided into three
30
30
30
-member groups so that each child has its own a group of at least one friend.
3
1
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100 |u - v| \cdot |1 - uv| \le (1 + u^2)(1 + v^2)
Prove that there are two numbers
u
u
u
and
v
v
v
, between any
101
101
101
real numbers that apply
100
∣
u
−
v
∣
⋅
∣
1
−
u
v
∣
≤
(
1
+
u
2
)
(
1
+
v
2
)
100 |u - v| \cdot |1 - uv| \le (1 + u^2)(1 + v^2)
100∣
u
−
v
∣
⋅
∣1
−
uv
∣
≤
(
1
+
u
2
)
(
1
+
v
2
)
2
1
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max triangle area with medians such that m_a \le 2, m_b \le 3, m_c \le 4
Find out the maximum possible area of the triangle
A
B
C
ABC
A
BC
whose medians have lengths satisfying inequalities
m
a
≤
2
,
m
b
≤
3
,
m
c
≤
4
m_a \le 2, m_b \le 3, m_c \le 4
m
a
≤
2
,
m
b
≤
3
,
m
c
≤
4
.
6
1
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x^4+y^2+4=5yz , y^4+z^2+4=5zx, z^4+x^2+4=5xy (3x3 system)
In the set of real numbers solve the system of equations
x
4
+
y
2
+
4
=
5
y
z
x^4+y^2+4=5yz
x
4
+
y
2
+
4
=
5
yz
y
4
+
z
2
+
4
=
5
z
x
y^4+z^2+4=5zx
y
4
+
z
2
+
4
=
5
z
x
z
4
+
x
2
+
4
=
5
x
y
z^4+x^2+4=5xy
z
4
+
x
2
+
4
=
5
x
y
4
1
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a line passes through a point inside a parallelogram with max area difference
Inside the parallelogram
A
B
C
D
ABCD
A
BC
D
is a point
X
X
X
. Make a line that passes through point
X
X
X
and divides the parallelogram into two parts whose areas differ from each other the most.
1
1
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n^4 -3n^2 + 9 is prime
Find all integers for which
n
n
n
is
n
4
−
3
n
2
+
9
n^4 -3n^2 + 9
n
4
−
3
n
2
+
9
prime