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Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1983 Poland - Second Round
1983 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
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probability randomly selecting a pair of integers
For a given number
n
n
n
, let us denote by
p
n
p_n
p
n
the probability that when randomly selecting a pair of integers
k
,
m
k, m
k
,
m
satisfying the conditions
0
≤
k
≤
m
≤
2
n
0 \leq k \leq m \leq 2^n
0
≤
k
≤
m
≤
2
n
(the selection of each pair is equally probable) the number
(
m
k
)
\binom{m}{k}
(
k
m
)
will be even. Calculate
lim
n
→
∞
p
n
\lim_{n\to \infty} p_n
lim
n
→
∞
p
n
.
5
1
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AK+BL+CM > AB+BC+CA
The bisectors of the angles
C
A
B
,
A
B
C
,
B
C
A
CAB, ABC, BCA
C
A
B
,
A
BC
,
BC
A
of the triangle
A
B
C
ABC
A
BC
intersect the circle circumcribed around this triangle at points
K
,
L
,
M
K, L, M
K
,
L
,
M
, respectively. Prove that
A
K
+
B
L
+
C
M
>
A
B
+
B
C
+
C
A
.
AK+BL+CM > AB+BC+CA.
A
K
+
B
L
+
CM
>
A
B
+
BC
+
C
A
.
4
1
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largest odd number by which k is divisible, sum
Let
a
(
k
)
a(k)
a
(
k
)
be the largest odd number by which
k
k
k
is divisible. Prove that
∑
k
=
1
2
n
a
(
k
)
=
1
3
(
4
n
+
2
)
.
\sum_{k=1}^{2^n} a(k) = \frac{1}{3}(4^n+2).
k
=
1
∑
2
n
a
(
k
)
=
3
1
(
4
n
+
2
)
.
3
1
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DL = DM if < PAC = \< PBC
The point
P
P
P
lies inside the triangle
A
B
C
ABC
A
BC
, with
∡
P
A
C
=
∡
P
B
C
\measuredangle PAC = \measuredangle PBC
∡
P
A
C
=
∡
PBC
. The points
L
L
L
and
M
M
M
are the projections
P
P
P
onto the lines
B
C
BC
BC
and
C
A
CA
C
A
, respectively,
D
D
D
is the midpoint of the segment
A
B
AB
A
B
. Prove that
D
L
=
D
M
DL = DM
D
L
=
D
M
.
2
1
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\sqrt{a+b-c}+\sqrt{a-b+c}+\sqrt{-a+b+c} \leq \sqrt{a}+\sqrt{b} + \sqrt{c}
There are three non-negative numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
such that the sum of each two is not less than the remaining one. Prove that
a
+
b
−
c
+
a
−
b
+
c
+
−
a
+
b
+
c
≤
a
+
b
+
c
.
\sqrt{a+b-c} + \sqrt{a-b+c} + \sqrt{-a+b+c} \leq \sqrt{a} + \sqrt{b} + \sqrt{c}.
a
+
b
−
c
+
a
−
b
+
c
+
−
a
+
b
+
c
≤
a
+
b
+
c
.
1
1
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twice the area of this polygon with latttice points is an integer.
On a plane with a fixed coordinate system, there is a convex polygon whose all vertices have integer coordinates. Prove that twice the area of this polygon is an integer.