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Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1994 Czech And Slovak Olympiad IIIA
1994 Czech And Slovak Olympiad IIIA
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
6
1
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\sqrt{(1-a^2)(1-b^2)} > \frac{a}{2b}+\frac{b}{2a}-ab-\frac{1}{8ab}
Show that from any four distinct numbers lying in the interval
(
0
,
1
)
(0,1)
(
0
,
1
)
one can choose two distinct numbers
a
a
a
and
b
b
b
such that
(
1
−
a
2
)
(
1
−
b
2
)
>
a
2
b
+
b
2
a
−
a
b
−
1
8
a
b
\sqrt{(1-a^2)(1-b^2)} > \frac{a}{2b}+\frac{b}{2a}-ab-\frac{1}{8ab}
(
1
−
a
2
)
(
1
−
b
2
)
>
2
b
a
+
2
a
b
−
ab
−
8
ab
1
4
1
Hide problems
\sum_{p\in P}\frac{1}{\log_p a_p}< 1$
Let
a
1
,
a
2
,
.
.
.
a_1,a_2,...
a
1
,
a
2
,
...
be a sequence of natural numbers such that for each
n
n
n
, the product
(
a
n
−
1
)
(
a
n
−
2
)
.
.
.
(
a
n
−
n
2
)
(a_n - 1)(a_n- 2)...(a_n - n^2)
(
a
n
−
1
)
(
a
n
−
2
)
...
(
a
n
−
n
2
)
is a positive integral multiple of
n
n
2
−
1
n^{n^2-1}
n
n
2
−
1
. Prove that for any finite set
P
P
P
of prime numbers the following inequality holds:
∑
p
∈
P
1
log
p
a
p
<
1
\sum_{p\in P}\frac{1}{\log_p a_p}< 1
p
∈
P
∑
lo
g
p
a
p
1
<
1
5
1
Hide problems
3 triangles with equal area, equilateral maybe
In an acute-angled triangle
A
B
C
ABC
A
BC
, the altitudes
A
A
1
,
B
B
1
,
C
C
1
AA_1,BB_1,CC_1
A
A
1
,
B
B
1
,
C
C
1
intersect at point
V
V
V
. If the triangles
A
C
1
V
,
B
A
1
V
,
C
B
1
V
AC_1V, BA_1V, CB_1V
A
C
1
V
,
B
A
1
V
,
C
B
1
V
have the same area, does it follow that the triangle
A
B
C
ABC
A
BC
is equilateral?
3
1
Hide problems
convex 1994-gon, closed polygonal line consists of 997 of its diagonals
A convex
1994
1994
1994
-gon
M
M
M
is given in the plane. A closed polygonal line consists of
997
997
997
of its diagonals. Every vertex is adjacent to exactly one diagonal. Each diagonal divides
M
M
M
into two sides, and the smaller of the numbers of edges on the two sides of
M
M
M
is defined to be the length of the diagonal. Is it posible to have (a)
991
991
991
diagonals of length
3
3
3
and
6
6
6
of length
2
2
2
? (b)
985
985
985
diagonals of length
6
,
4
6, 4
6
,
4
of length
8
8
8
, and
8
8
8
of length
3
3
3
?
2
1
Hide problems
cuboid of volume V contains a convex polyhedron M, projection covers face
A cuboid of volume
V
V
V
contains a convex polyhedron
M
M
M
. The orthogonal projection of
M
M
M
onto each face of the cuboid covers the entire face. What is the smallest possible volume of polyhedron
M
M
M
?
1
1
Hide problems
f(n)+ f(n+2) \le 2 f(n+1), exists a line with infinitely many points
Let
f
:
N
→
N
f : N \to N
f
:
N
→
N
be a function which satisfies
f
(
x
)
+
f
(
x
+
2
)
≤
2
f
(
x
+
1
)
f(x)+ f(x+2) \le 2 f(x+1)
f
(
x
)
+
f
(
x
+
2
)
≤
2
f
(
x
+
1
)
for any
x
∈
N
x \in N
x
∈
N
. Prove that there exists a line in the coordinate plane containing infinitely many points of the form
(
n
,
f
(
n
)
)
,
n
∈
N
(n, f(n)), n \in N
(
n
,
f
(
n
))
,
n
∈
N
.