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Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1992 Czech And Slovak Olympiad IIIA
1992 Czech And Slovak Olympiad IIIA
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
5
1
Hide problems
f(x) = x if x is irrational,,f(x) = (p+1)/q} if x =p/q , max f on (7/8,8/9)
The function
f
:
(
0
,
1
)
→
R
f : (0,1) \to R
f
:
(
0
,
1
)
→
R
is defined by
f
(
x
)
=
x
f(x) = x
f
(
x
)
=
x
if
x
x
x
is irrational,
f
(
x
)
=
p
+
1
q
f(x) = \frac{p+1}{q}
f
(
x
)
=
q
p
+
1
if
x
=
p
q
x =\frac{p}{q}
x
=
q
p
, where
(
p
,
q
)
=
1
(p,q) = 1
(
p
,
q
)
=
1
. Find the maximum value of
f
f
f
on the interval
(
7
/
8
,
8
/
9
)
(7/8,8/9)
(
7/8
,
8/9
)
.
1
1
Hide problems
permutation inequality a_1 +...+a_k < a_{k+1} +...+a_{17}
For a permutation
p
(
a
1
,
a
2
,
.
.
.
,
a
17
)
p(a_1,a_2,...,a_{17})
p
(
a
1
,
a
2
,
...
,
a
17
)
of
1
,
2
,
.
.
.
,
17
1,2,...,17
1
,
2
,
...
,
17
, let
k
p
k_p
k
p
denote the largest
k
k
k
for which
a
1
+
.
.
.
+
a
k
<
a
k
+
1
+
.
.
.
+
a
17
a_1 +...+a_k < a_{k+1} +...+a_{17}
a
1
+
...
+
a
k
<
a
k
+
1
+
...
+
a
17
. Find the maximum and minimum values of
k
p
k_p
k
p
and find the sum
∑
p
k
p
\sum_{p} k_p
∑
p
k
p
over all permutations
p
p
p
.
4
1
Hide problems
cos 12x = 5sin 3x+9 tan ^2x+ cot ^2x
Solve the equation
cos
12
x
=
5
sin
3
x
+
9
t
a
n
2
x
+
c
o
t
2
x
\cos 12x = 5\sin 3x+9\ tan ^2x+\ cot ^2x
cos
12
x
=
5
sin
3
x
+
9
t
a
n
2
x
+
co
t
2
x
3
1
Hide problems
S(n) = S(2n) = S(3n) = ... = S(n^2), sum of digits
Let
S
(
n
)
S(n)
S
(
n
)
denote the sum of digits of
n
∈
N
n \in N
n
∈
N
. Find all
n
n
n
such that
S
(
n
)
=
S
(
2
n
)
=
S
(
3
n
)
=
.
.
.
=
S
(
n
2
)
S(n) = S(2n) = S(3n) =... = S(n^2)
S
(
n
)
=
S
(
2
n
)
=
S
(
3
n
)
=
...
=
S
(
n
2
)
2
1
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tetrahedron area ineduality S \le \frac{\sqrt3}{6} (a^2 +b^2 +...+ f^2)
Let
S
S
S
be the total area of a tetrahedron whose edges have lengths
a
,
b
,
c
,
d
,
e
,
f
a,b,c,d, e, f
a
,
b
,
c
,
d
,
e
,
f
. Prove that
S
≤
3
6
(
a
2
+
b
2
+
.
.
.
+
f
2
)
S \le \frac{\sqrt3}{6} (a^2 +b^2 +...+ f^2)
S
≤
6
3
(
a
2
+
b
2
+
...
+
f
2
)
6
1
Hide problems
concyclic wanted, altitudes and circles related
Let
A
B
C
ABC
A
BC
be an acute triangle. The altitude from
B
B
B
meets the circle with diameter
A
C
AC
A
C
at points
P
,
Q
P,Q
P
,
Q
, and the altitude from
C
C
C
meets the circle with diameter
A
B
AB
A
B
at
M
,
N
M,N
M
,
N
. Prove that the points
M
,
N
,
P
,
Q
M,N,P,Q
M
,
N
,
P
,
Q
lie on a circle.