MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1970 Poland - Second Round
1970 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
2
1
Hide problems
n + 1 points on sides of regular n-gon, min max area
On the sides of the regular
n
n
n
-gon,
n
+
1
n + 1
n
+
1
points are taken dividing the perimeter into equal parts. At what position of the selected points is the area of the convex polygon with these
n
+
1
n + 1
n
+
1
vertices a) the largest, b) the smallest?
1
1
Hide problems
|cos nb- cos na| <= n^2 |cos b- cos a|,
Prove that
∣
cos
n
β
−
cos
n
α
∣
≤
n
2
∣
cos
β
−
cos
α
∣
,
|\cos n\beta - \cos n\alpha| \leq n^2 |\cos \beta - \cos\alpha|,
∣
cos
n
β
−
cos
n
α
∣
≤
n
2
∣
cos
β
−
cos
α
∣
,
where
n
n
n
is a natural number . Check for what values of
n
n
n
,
α
\alpha
α
,
β
\beta
β
equality holds.
6
1
Hide problems
maximum number of mutually independent subsets of a $2^n $-element set
If
A
A
A
is a subset of
X
X
X
, then we take
A
1
=
A
A^1 = A
A
1
=
A
,
A
−
1
=
X
−
A
A^{-1} = X - A
A
−
1
=
X
−
A
. The subsets
A
1
,
A
2
,
…
,
A
k
A_1, A_2, \ldots, A_k
A
1
,
A
2
,
…
,
A
k
are called mutually independent if the product
A
1
ε
1
∩
A
2
ε
2
…
A
k
ε
k
A_1^{\varepsilon_1} \cap A_2^{\varepsilon_2} \ldots A_k^{\varepsilon_k}
A
1
ε
1
∩
A
2
ε
2
…
A
k
ε
k
is nonempty for every system of numbers
ε
1
,
ε
2
,
…
,
ε
k
\varepsilon_1 , \varepsilon_2, \ldots, \varepsilon_k
ε
1
,
ε
2
,
…
,
ε
k
, such that
∣
ε
2
∣
=
|\varepsilon_2| =
∣
ε
2
∣
=
1 for
i
=
1
,
2
,
…
,
k
i = 1, 2, \ldots, k
i
=
1
,
2
,
…
,
k
. What is the maximum number of mutually independent subsets of a
2
n
2^n
2
n
-element set?
5
1
Hide problems
Q(x) = P(x) P(x^3) P(x^9) P(x^{27}) P(x^{81})
Given the polynomial
P
(
x
)
=
1
2
−
1
3
x
+
1
6
x
2
P(x) = \frac{1}{2} - \frac{1}{3}x + \frac{1}{6}x^2
P
(
x
)
=
2
1
−
3
1
x
+
6
1
x
2
. Let
Q
(
x
)
=
∑
k
=
0
m
b
k
x
k
Q(x) = \sum_{k=0}^{m} b_k x^k
Q
(
x
)
=
∑
k
=
0
m
b
k
x
k
be a polynomial given by
Q
(
x
)
=
P
(
x
)
⋅
P
(
x
3
)
⋅
P
(
x
9
)
⋅
P
(
x
27
)
⋅
P
(
x
81
)
.
Q(x) = P(x) \cdot P(x^3) \cdot P(x^9) \cdot P(x^{27}) \cdot P(x^{81}).
Q
(
x
)
=
P
(
x
)
⋅
P
(
x
3
)
⋅
P
(
x
9
)
⋅
P
(
x
27
)
⋅
P
(
x
81
)
.
Calculate
∑
k
=
0
m
∣
b
k
∣
\sum_{k=0}^m |b_k|
∑
k
=
0
m
∣
b
k
∣
.
4
1
Hide problems
perimeter of inscribed triangle is less
Prove that if triangle
T
1
T_1
T
1
contains triangle
T
2
T_2
T
2
, then the perimeter of triangle
T
1
T_1
T
1
is not less than the perimeter of triangle
T
2
T_2
T
2
.
3
1
Hide problems
2^n - 1 is never divisible by n
Prove the theorem: There is no natural number
n
>
1
n > 1
n
>
1
such that the number
2
n
−
1
2^n - 1
2
n
−
1
is divisible by
n
n
n
.