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Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1977 Poland - Second Round
1977 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
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max no of parts n squares cut a plane
What is the greatest number of parts into which the plane can be cut by the edges of
n
n
n
squares?
5
1
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w_1(x) = x^2 - 1, w_{n+1}(x) = w_n(x)^2 - 1,
Let the polynomials
w
n
w_n
w
n
be given by the formulas: w_1(x) = x^2 - 1, w_{n+1}(x) = w_n(x)^2 - 1, (n = 1, 2, \ldots) and let
a
a
a
be a real number. How many different real solutions does the equation
w
n
(
x
)
=
a
w_n(x) = a
w
n
(
x
)
=
a
have?
4
1
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touchpoints of these circles with base of pyramid are concyclic
A pyramid with a quadrangular base is given such that each pair of circles inscribed in adjacent faces has a common point. Prove that the touchpoints of these circles with the base of the pyramid lie on one circle.
3
1
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7 pieces of paper in the hat.
There are 7 pieces of paper in the hat. On the
n
n
n
th piece of paper there is written the number
2
n
−
1
2^n-1
2
n
−
1
(
n
=
1
,
2
,
…
,
7
n = 1, 2, \ldots, 7
n
=
1
,
2
,
…
,
7
). We draw cards randomly until the sum exceeds 124. What is the most probable value of this sum?
2
1
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XA *XB * XC >=8 d(X, AB) * d(X, BC) * d(X, AC)
Let
X
X
X
be the interior point of triangle
A
B
C
ABC
A
BC
. prove that the product of the distances of point
X
X
X
from the vertices
A
,
B
,
C
A, B, C
A
,
B
,
C
is at least eight times greater than the product of the distances of this point from the lines
A
B
,
B
C
,
C
A
AB, BC, CA
A
B
,
BC
,
C
A
.
1
1
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|x_1c_1 + x_2c_2 +... + x_nc_n| >= |b-a|/2(|c_1|+|c_2|+\ldots+|c_n|)
Let
a
a
a
and
b
b
b
be different real numbers. Prove that for any real numbers
c
1
,
c
2
,
…
,
c
n
c_1, c_2, \ldots,c_n
c
1
,
c
2
,
…
,
c
n
there exists a sequence of
n
n
n
-elements
(
x
i
)
(x_i)
(
x
i
)
, each term of which is equal to one of the numbers
a
a
a
or
b
b
b
such that
∣
x
1
c
1
+
x
2
c
2
+
…
+
x
n
c
n
∣
≥
∣
b
−
a
∣
2
(
∣
c
1
∣
+
∣
c
2
∣
+
…
+
∣
c
n
∣
)
.
|x_1c_1 + x_2c_2 + \ldots + x_nc_n| \geq \frac{|b-a|}{2}(|c_1|+|c_2|+\ldots+|c_n|).
∣
x
1
c
1
+
x
2
c
2
+
…
+
x
n
c
n
∣
≥
2
∣
b
−
a
∣
(
∣
c
1
∣
+
∣
c
2
∣
+
…
+
∣
c
n
∣
)
.