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Problems
Contests
National and Regional Contests
Poland Contests
Polish MO Finals
1979 Polish MO Finals
1979 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(6)
4
1
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x_m / y_m <= B x_n / y_n
Let
A
>
1
A > 1
A
>
1
and
B
>
1
B > 1
B
>
1
be real numbers and (xn) be a sequence of numbers in the interval
[
1
,
A
B
]
[1,AB]
[
1
,
A
B
]
. Prove that there exists a sequence
(
y
n
)
(y_n)
(
y
n
)
of numbers in the interval
[
1
,
A
]
[1,A]
[
1
,
A
]
such that
x
m
x
n
≤
B
y
m
y
n
f
o
r
a
l
l
m
,
n
=
1
,
2
,
.
.
.
\frac{x_m}{x_n}\le B\frac{y_m}{y_n} \,\,\, for \,\,\, all \,\,\, m,n = 1,2,...
x
n
x
m
≤
B
y
n
y
m
f
or
a
ll
m
,
n
=
1
,
2
,
...
6
1
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sum 1 / w′(x_i)=0 for polynomial with x_i dinstinct roots
A polynomial
w
w
w
of degree
n
>
1
n > 1
n
>
1
has
n
n
n
distinct zeros
x
1
,
x
2
,
.
.
.
,
x
n
x_1,x_2,...,x_n
x
1
,
x
2
,
...
,
x
n
. Prove that:
1
w
′
(
x
1
)
+
1
w
′
(
x
2
)
+
.
.
.
⋅
⋅
⋅
+
1
w
′
(
x
n
)
=
0.
\frac{1}{w'(x_1)}+\frac{1}{w'(x_2)}+...···+\frac{1}{w'(x_n)}= 0.
w
′
(
x
1
)
1
+
w
′
(
x
2
)
1
+
...
⋅⋅⋅
+
w
′
(
x
n
)
1
=
0.
3
1
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experiment consists of performing n independent tests
An experiment consists of performing
n
n
n
independent tests. The
i
i
i
-th test is successful with the probability equal to
p
i
p_i
p
i
. Let
r
k
r_k
r
k
be the probability that exactly
k
k
k
tests succeed. Prove that
∑
i
=
1
n
p
i
=
∑
k
=
0
n
k
r
k
.
\sum_{i=1}^n p_i =\sum_{k=0}^n kr_k.
i
=
1
∑
n
p
i
=
k
=
0
∑
n
k
r
k
.
1
1
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r_i +r_j -n is divisible by m
Let be given a set
{
r
1
,
r
2
,
.
.
.
,
r
k
}
\{r_1,r_2,...,r_k\}
{
r
1
,
r
2
,
...
,
r
k
}
of natural numbers that give distinct remainders when divided by a natural number
m
m
m
. Prove that if
k
>
m
/
2
k > m/2
k
>
m
/2
, then for every integer
n
n
n
there exist indices
i
i
i
and
j
j
j
(not necessarily distinct) such that
r
i
+
r
j
−
n
r_i +r_j -n
r
i
+
r
j
−
n
is divisible by
m
m
m
.
5
1
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product of the sides of cyclic quadrilateral
Prove that the product of the sides of a quadrilateral inscribed in a circle with radius
1
1
1
does not exceed
4
4
4
.
2
1
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concurrency of lines joining vertices with incenters of opp. faces
Prove that the four lines, joining the vertices of a tetrahedron with the incenters of the opposite faces, have a common point if and only if the three products of the lengths of opposite sides are equal.