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Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
2002 Czech and Slovak Olympiad III A
2002 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
6
1
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Functional equation: f(x(f(y))=f(xy)+x
Let
R
+
\mathbb{R}^{+}
R
+
denote the set of positive real numbers. Find all functions
f
:
R
+
→
R
+
f : \mathbb{R}^{+} \to \mathbb{R}^{+}
f
:
R
+
→
R
+
satisfying for all
x
,
y
∈
R
+
x, y \in \mathbb{R}^{+}
x
,
y
∈
R
+
the equality
f
(
x
f
(
y
)
)
=
f
(
x
y
)
+
x
f(xf(y))=f(xy)+x
f
(
x
f
(
y
))
=
f
(
x
y
)
+
x
5
1
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Construct a rectangle
A triangle
K
L
M
KLM
K
L
M
is given in the plane together with a point
A
A
A
lying on the half-line opposite to
K
L
KL
K
L
. Construct a rectangle
A
B
C
D
ABCD
A
BC
D
whose vertices
B
,
C
B, C
B
,
C
and
D
D
D
lie on the lines
K
M
,
K
L
KM, KL
K
M
,
K
L
and
L
M
LM
L
M
, respectively. (We allow the rectangle to be a square.)
4
1
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Equation has two solutions and the sum of them equals 12
Find all pairs of real numbers
a
,
b
a, b
a
,
b
for which the equation in the domain of the real numbers
a
x
2
−
24
x
+
b
x
2
−
1
=
x
\frac{ax^2-24x+b}{x^2-1}=x
x
2
−
1
a
x
2
−
24
x
+
b
=
x
has two solutions and the sum of them equals
12
12
12
.
3
1
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A is the square of a natural number
Show that a given natural number
A
A
A
is the square of a natural number if and only if for any natural number
n
n
n
, at least one of the differences
(
A
+
1
)
2
−
A
,
(
A
+
2
)
2
−
A
,
(
A
+
3
)
2
−
A
,
⋯
,
(
A
+
n
)
2
−
A
(A + 1)^2 - A, (A + 2)^2 - A, (A + 3)^2 - A, \cdots , (A + n)^2 - A
(
A
+
1
)
2
−
A
,
(
A
+
2
)
2
−
A
,
(
A
+
3
)
2
−
A
,
⋯
,
(
A
+
n
)
2
−
A
is divisible by
n
n
n
.
2
1
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Find locus of the midpoints of the sides $KL$ of KLM
Consider an arbitrary equilateral triangle
K
L
M
KLM
K
L
M
, whose vertices
K
,
L
K, L
K
,
L
and
M
M
M
lie on the sides
A
B
,
B
C
AB, BC
A
B
,
BC
and
C
D
CD
C
D
, respectively, of a given square
A
B
C
D
ABCD
A
BC
D
. Find the locus of the midpoints of the sides
K
L
KL
K
L
of all such triangles
K
L
M
KLM
K
L
M
.
1
1
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Solve in integers: (multiple of 'k' closest to 'n')
Solve the system
(
4
x
)
5
+
7
y
=
14
(
2
y
)
5
−
(
3
x
)
7
=
74
(4x)_5+7y=14 \\ (2y)_5 -(3x)_7=74
(
4
x
)
5
+
7
y
=
14
(
2
y
)
5
−
(
3
x
)
7
=
74
in the domain of integers, where
(
n
)
k
(n)_k
(
n
)
k
stands for the multiple of the number
k
k
k
closest to the number
n
n
n
.