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Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1963 Czech and Slovak Olympiad III A
1963 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(4)
4
1
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Quadratic equations
Consider two quadratic equations \begin{align*}x^2+ax+b&=0, \\ x^2+cx+d&=0,\end{align*} with real coefficients. Find necessary and sufficient conditions such that the first equation has (real) roots
x
,
x
1
,
x,x_1,
x
,
x
1
,
the second
x
,
x
2
x,x_2
x
,
x
2
and
x
>
0
,
x
1
>
x
2
x>0,x_1>x_2
x
>
0
,
x
1
>
x
2
.
3
1
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Locus of midpoints
A line
M
N
MN
MN
is given in the plane. Consider circles
k
1
k_1
k
1
,
k
2
k_2
k
2
tangent to the line at points
M
M
M
,
N
N
N
, respectively, while touching each other externally. Let
X
X
X
be the midpoint of the segment
P
Q
PQ
PQ
, where
P
P
P
,
Q
Q
Q
are in this order tangent points of the second common external tangent of the circles
k
1
k_1
k
1
,
k
2
k_2
k
2
. Find the locus of the points
X
X
X
for all pairs of circles of the specified properties.
2
1
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Maximum of product
Let an even positive integer
2
k
2k
2
k
be given. Find such relatively prime positive integers
x
,
y
x, y
x
,
y
that maximize the product
x
y
xy
x
y
.
1
1
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Sides of a cuboid
Consider a cuboid
A
B
C
D
A
′
B
′
C
′
D
′
ABCDA'B'C'D'
A
BC
D
A
′
B
′
C
′
D
′
(where
A
B
C
D
ABCD
A
BC
D
is a rectangle and
A
A
′
∥
B
B
′
∥
C
C
′
∥
D
D
′
AA' \parallel BB' \parallel CC' \parallel DD'
A
A
′
∥
B
B
′
∥
C
C
′
∥
D
D
′
) with
A
A
′
=
d
AA' = d
A
A
′
=
d
,
∠
A
B
D
′
=
α
,
∠
A
′
D
′
B
=
β
\angle ABD' = \alpha, \angle A'D'B = \beta
∠
A
B
D
′
=
α
,
∠
A
′
D
′
B
=
β
. Express the lengths x =
A
B
AB
A
B
,
y
=
B
C
y = BC
y
=
BC
in terms of
d
d
d
and (acute) angles
α
,
β
\alpha, \beta
α
,
β
. Discuss condition of solvability.