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Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1978 Czech and Slovak Olympiad III A
1978 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
6
1
Hide problems
Exponentially growing sequence of integers
Show that the number
p
n
=
(
3
+
5
2
)
n
+
(
3
−
5
2
)
n
−
2
p_n=\left(\frac{3+\sqrt5}{2}\right)^n+\left(\frac{3-\sqrt5}{2}\right)^n-2
p
n
=
(
2
3
+
5
)
n
+
(
2
3
−
5
)
n
−
2
is a positive integer for any positive integer
n
.
n.
n
.
Furthermore, show that the numbers
p
2
n
−
1
p_{2n-1}
p
2
n
−
1
and
p
2
n
/
5
p_{2n}/5
p
2
n
/5
are perfect squares
(
(
(
for any positive integer
n
)
.
n).
n
)
.
5
1
Hide problems
"Incentric" rectangle in trapezoid
Let
A
B
C
S
ABCS
A
BCS
be an isosceles trapezoid. Denote
A
′
,
B
′
,
C
′
,
D
′
A',B',C',D'
A
′
,
B
′
,
C
′
,
D
′
the incenters of triangles
B
C
D
,
C
D
A
,
BCD,CDA,
BC
D
,
C
D
A
,
D
A
B
,
A
B
C
,
DAB,ABC,
D
A
B
,
A
BC
,
respectively. Show that
A
′
,
B
′
,
C
′
,
D
′
A',B',C',D'
A
′
,
B
′
,
C
′
,
D
′
are vertices of a rectangle.
4
1
Hide problems
Existence of tetrahedron
Is there a tetrahedron
A
B
C
D
ABCD
A
BC
D
such that
A
B
+
B
C
+
C
D
+
D
A
=
12
cm
AB+BC+CD+DA=12\text{ cm}
A
B
+
BC
+
C
D
+
D
A
=
12
cm
with volume
V
≥
2
3
cm
3
?
\mathrm V\ge2\sqrt3\text{ cm}^3?
V
≥
2
3
cm
3
?
3
1
Hide problems
Parametrized system of trigonometric equations
Let
α
,
β
,
γ
\alpha,\beta,\gamma
α
,
β
,
γ
be angles of a triangle. Determine all real triplets
x
,
y
,
z
x,y,z
x
,
y
,
z
satisfying the system \begin{align*} x\cos\beta+\frac1z\cos\alpha &=1, \\ y\cos\gamma+\frac1x\cos\beta &=1, \\ z\cos\alpha+\frac1y\cos\gamma &=1. \end{align*}
2
1
Hide problems
Inequality with unknown parameters
Determine (at least one) pair of real numbers
k
,
q
k,q
k
,
q
such that the inequality
2
∣
1
−
x
2
−
k
x
−
q
∣
≤
2
−
1
2\left|\sqrt{1-x^2}-kx-q\right|\le\sqrt2-1
2
1
−
x
2
−
k
x
−
q
≤
2
−
1
holds for all
x
∈
[
0
,
1
]
.
x\in[0,1].
x
∈
[
0
,
1
]
.
1
1
Hide problems
Obvious Cauchy Schwarz
Let
a
1
,
…
,
a
n
,
b
1
,
…
,
b
n
a_1,\ldots,a_n,b_1,\ldots,b_n
a
1
,
…
,
a
n
,
b
1
,
…
,
b
n
be positive numbers. Show that
(
a
1
+
⋯
+
a
n
)
(
b
1
+
⋯
+
b
n
)
≥
a
1
b
1
+
⋯
+
a
n
b
n
\sqrt{\left(a_1+\cdots+a_n\right)\left(b_1+\cdots+b_n\right)}\ge\sqrt{a_1b_1}+\cdots+\sqrt{a_nb_n}
(
a
1
+
⋯
+
a
n
)
(
b
1
+
⋯
+
b
n
)
≥
a
1
b
1
+
⋯
+
a
n
b
n
and prove that equality holds if and only if
a
1
b
1
=
⋯
=
a
n
b
n
.
\frac{a_1}{b_1}=\cdots=\frac{a_n}{b_n}.
b
1
a
1
=
⋯
=
b
n
a
n
.