MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1995 Poland - Second Round
1995 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
4
1
Hide problems
\sum x_i^t \le \sum x_i ^{t+1} if \sum x_i \le \sum x_i ^2
Positive real numbers
x
1
,
x
2
,
.
.
.
,
x
n
x_1,x_2,...,x_n
x
1
,
x
2
,
...
,
x
n
satisfy the condition
∑
i
=
1
n
x
i
≤
∑
i
=
1
n
x
i
2
\sum_{i=1}^n x_i \le \sum_{i=1}^n x_i ^2
∑
i
=
1
n
x
i
≤
∑
i
=
1
n
x
i
2
. Prove the inequality
∑
i
=
1
n
x
i
t
≤
∑
i
=
1
n
x
i
t
+
1
\sum_{i=1}^n x_i^t \le \sum_{i=1}^n x_i ^{t+1}
∑
i
=
1
n
x
i
t
≤
∑
i
=
1
n
x
i
t
+
1
for all real numbers
t
>
1
t > 1
t
>
1
.
6
1
Hide problems
n x n can be cut intosquares 2 x 2 and 3 x 3
Determine all positive integers
n
n
n
for which the square
n
×
n
n \times n
n
×
n
can be cut into squares
2
×
2
2\times 2
2
×
2
and
3
×
3
3\times3
3
×
3
(with the sides parallel to the sides of the big square).
5
1
Hide problems
points of tangency of incircles to 4 edges in tetrahedron are concyclic
The incircles of the faces
A
B
C
ABC
A
BC
and
A
B
D
ABD
A
B
D
of a tetrahedron
A
B
C
D
ABCD
A
BC
D
are tangent to the edge
A
B
AB
A
B
in the same point. Prove that the points of tangency of these incircles to the edges
A
C
,
B
C
,
A
D
,
B
D
AC,BC,AD,BD
A
C
,
BC
,
A
D
,
B
D
are concyclic.
3
1
Hide problems
a+b = 1. Show that c+d = 1 iff [na]+[nb] = [nc]+[nd] for all n
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be positive irrational numbers with
a
+
b
=
1
a+b = 1
a
+
b
=
1
. Show that
c
+
d
=
1
c+d = 1
c
+
d
=
1
if and only if
[
n
a
]
+
[
n
b
]
=
[
n
c
]
+
[
n
d
]
[na]+[nb] = [nc]+[nd]
[
na
]
+
[
nb
]
=
[
n
c
]
+
[
n
d
]
for all positive integers
n
n
n
.
2
1
Hide problems
perpendicular lines from 3 vertices of convex hexagon are concurrent
Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a convex hexagon with
A
B
=
B
C
,
C
D
=
D
E
AB = BC, CD = DE
A
B
=
BC
,
C
D
=
D
E
and
E
F
=
F
A
EF = FA
EF
=
F
A
. Prove that the lines through
C
,
E
,
A
C,E,A
C
,
E
,
A
perpendicular to
B
D
,
D
F
,
F
B
BD,DF,FB
B
D
,
D
F
,
FB
are concurrent.
1
1
Hide problems
P(5) is divisible by 2,P(2) is divisible by 5 then P(7) is divisible by 10
For a polynomial
P
P
P
with integer coefficients,
P
(
5
)
P(5)
P
(
5
)
is divisible by
2
2
2
and
P
(
2
)
P(2)
P
(
2
)
is divisible by
5
5
5
. Prove that
P
(
7
)
P(7)
P
(
7
)
is divisible by
10
10
10
.