MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1990 Poland - Second Round
1990 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
3
1
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players in a In a chess tournament
In a chess tournament, each player played at most one game against each other, and the number of games played by each player is not less than the set natural number
n
n
n
. Prove that it is possible to divide players into two groups
A
A
A
and
B
B
B
in such a way that the number of games played by each player of group
A
A
A
with players of group
B
B
B
is not less than
n
/
2
n/2
n
/2
and at the same time the number of games played by each player of the
B
B
B
group with players of the
A
A
A
group was not less than
n
/
2
n/2
n
/2
.
6
1
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numbers of areas of convex polygons
For any convex polygon
W
W
W
with area 1, let us denote by
f
(
W
)
f(W)
f
(
W
)
the area of the convex polygon whose vertices are the centers of all sides of the polygon
W
W
W
. For each natural number
n
≥
3
n \geq 3
n
≥
3
, determine the lower limit and the upper limit of the set of numbers
f
(
W
)
f(W)
f
(
W
)
when
W
W
W
runs through the set of all
n
n
n
convex angles with area 1.
5
1
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n naturals whose sum is equal to their product
There are
n
n
n
natural numbers (
n
≥
2
n\geq 2
n
≥
2
) whose sum is equal to their product. Prove that this common value does not exceed
2
n
2n
2
n
.
4
1
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(sin x)^k + (cos x)^{-m} = (cos x)^k + (sin x)^{-m}
For each pair of even natural numbers
k
k
k
,
m
m
m
determine all real numbers
x
x
x
that satisfy the equation
(
sin
x
)
k
+
(
cos
x
)
−
m
=
(
cos
x
)
k
+
(
sin
x
)
−
m
(\sin x)^k + (\cos x)^{-m} = (\cos x)^k + (\sin x)^{-m}
(
sin
x
)
k
+
(
cos
x
)
−
m
=
(
cos
x
)
k
+
(
sin
x
)
−
m
2
1
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can this set of points be a tetrahedron ?
In space, a point
O
O
O
and a finite set of vectors
v
1
→
,
…
,
v
n
→
\overrightarrow{v_1},\ldots,\overrightarrow{v_n}
v
1
,
…
,
v
n
are given . We consider the set of points
P
P
P
for which the vector
O
P
→
\overrightarrow{OP}
OP
can be represented as a sum
a
1
v
1
→
+
…
+
a
n
v
n
→
a_1 \overrightarrow{v_1} + \ldots + a_n\overrightarrow{v_n}
a
1
v
1
+
…
+
a
n
v
n
with coefficients satisfying the inequalities
0
≤
a
i
≤
1
0 \leq a_i \leq 1
0
≤
a
i
≤
1
(
i
=
1
,
2
,
…
,
n
( i = 1, 2, \ldots, n
(
i
=
1
,
2
,
…
,
n
). Decide whether this set can be a tetrahedron.
1
1
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(xy-1)^2 = (x +1)^2 + (y +1)^2, NT
Find all pairs of integers
x
x
x
,
y
y
y
satisfying the equation
(
x
y
−
1
)
2
=
(
x
+
1
)
2
+
(
y
+
1
)
2
.
(xy-1)^2 = (x +1)^2 + (y +1)^2.
(
x
y
−
1
)
2
=
(
x
+
1
)
2
+
(
y
+
1
)
2
.