MathDB
Problems
Contests
International Contests
Czech-Polish-Slovak Match
1998 Czech and Slovak Match
1998 Czech and Slovak Match
Part of
Czech-Polish-Slovak Match
Subcontests
(6)
6
1
Hide problems
n girls, 2n-1 boys, dances with unknown, different ways
In a summer camp there are
n
n
n
girls
D
1
,
D
2
,
.
.
.
,
D
n
D_1,D_2, ... ,D_n
D
1
,
D
2
,
...
,
D
n
and
2
n
−
1
2n-1
2
n
−
1
boys
C
1
,
C
2
,
.
.
.
,
C
2
n
−
1
C_1,C_2, ...,C_{2n-1}
C
1
,
C
2
,
...
,
C
2
n
−
1
. The girl
D
i
,
i
=
1
,
2
,
.
.
.
,
n
,
D_i, i = 1,2,... ,n,
D
i
,
i
=
1
,
2
,
...
,
n
,
knows only the boys
C
1
,
C
2
,
.
.
.
,
C
2
i
−
1
C_1,C_2, ... ,C_{2i-1}
C
1
,
C
2
,
...
,
C
2
i
−
1
. Let
A
(
n
,
r
)
A(n, r)
A
(
n
,
r
)
be the number of different ways in which
r
r
r
girls can dance with
r
r
r
boys forming
r
r
r
pairs, each girl with a boy she knows. Prove that
A
(
n
,
r
)
=
(
n
r
)
r
!
(
n
−
r
)
!
.
A(n, r) = \binom{n}{r} \frac{r!}{(n-r)!}.
A
(
n
,
r
)
=
(
r
n
)
(
n
−
r
)!
r
!
.
5
1
Hide problems
maximum of sin CAT +sin CBT if T centroid m <TAB = <ACT
In a triangle
A
B
C
ABC
A
BC
,
T
T
T
is the centroid and
∠
T
A
B
=
∠
A
C
T
\angle TAB = \angle ACT
∠
T
A
B
=
∠
A
CT
. Find the maximum possible value of
s
i
n
∠
C
A
T
+
s
i
n
∠
C
B
T
sin \angle CAT +sin \angle CBT
s
in
∠
C
A
T
+
s
in
∠
CBT
.
4
1
Hide problems
f (n)+ f (n+1)= f (n+2) +f (n+3) -168
Find all functions
f
:
N
→
N
−
{
1
}
f : N\rightarrow N - \{1\}
f
:
N
→
N
−
{
1
}
satisfying
f
(
n
)
+
f
(
n
+
1
)
=
f
(
n
+
2
)
+
f
(
n
+
3
)
−
168
f (n)+ f (n+1)= f (n+2) +f (n+3) -168
f
(
n
)
+
f
(
n
+
1
)
=
f
(
n
+
2
)
+
f
(
n
+
3
)
−
168
for all
n
∈
N
n \in N
n
∈
N
.
3
1
Hide problems
inequality in a convex hexagon with equal sides in pairs
Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a convex hexagon such that
A
B
=
B
C
,
C
D
=
D
E
,
E
F
=
F
A
AB = BC, CD = DE, EF = FA
A
B
=
BC
,
C
D
=
D
E
,
EF
=
F
A
. Prove that
B
C
B
E
+
D
E
D
A
+
F
A
F
C
≥
3
2
\frac{BC}{BE} +\frac{DE}{DA} +\frac{FA}{FC} \ge \frac{3}{2}
BE
BC
+
D
A
D
E
+
FC
F
A
≥
2
3
. When does equality occur?
2
1
Hide problems
integer roots of P(P(x)) when P(x) has n roots included 0
A polynomial
P
(
x
)
P(x)
P
(
x
)
of degree
n
≥
5
n \ge 5
n
≥
5
with integer coefficients has
n
n
n
distinct integer roots, one of which is
0
0
0
. Find all integer roots of the polynomial
P
(
P
(
x
)
)
P(P(x))
P
(
P
(
x
))
.
1
1
Hide problems
equal angles from an interior point of parallelogram
Let
P
P
P
be an interior point of the parallelogram
A
B
C
D
ABCD
A
BC
D
. Prove that
∠
A
P
B
+
∠
C
P
D
=
18
0
∘
\angle APB+ \angle CPD = 180^\circ
∠
A
PB
+
∠
CP
D
=
18
0
∘
if and only if
∠
P
D
C
=
∠
P
B
C
\angle PDC = \angle PBC
∠
P
D
C
=
∠
PBC
.