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Problems
Contests
National and Regional Contests
Poland Contests
Polish MO Finals
1972 Polish MO Finals
1972 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(6)
3
1
Hide problems
|P(x) -1/2 | <1/1000, integer
Prove that there is a polynomial
P
(
x
)
P(x)
P
(
x
)
with integer coefficients such that for all
x
x
x
in the interval
[
1
10
,
9
10
]
\left[ \frac{1}{10} , \frac{9}{10}\right]
[
10
1
,
10
9
]
we have
∣
P
(
x
)
−
1
2
∣
<
1
1000
.
\left|P(x) -\frac12 \right| < \frac{ 1}{1000 }.
P
(
x
)
−
2
1
<
1000
1
.
1
1
Hide problems
u_1(x)^n +u_2(x)^n = u_3(x)^n, ui(x) = a_ix+b_i
Polynomials
u
i
(
x
)
=
a
i
x
+
b
i
u_i(x) = a_ix+b_i
u
i
(
x
)
=
a
i
x
+
b
i
(
a
i
,
b
i
∈
R
a_i,b_i \in R
a
i
,
b
i
∈
R
,
i
=
1
,
2
,
3
i = 1,2,3
i
=
1
,
2
,
3
) satisfy
u
1
(
x
)
n
+
u
2
(
x
)
n
=
u
3
(
x
)
n
u_1(x)^n +u_2(x)^n = u_3(x)^n
u
1
(
x
)
n
+
u
2
(
x
)
n
=
u
3
(
x
)
n
for some integer
n
≥
2.
n \ge 2.
n
≥
2.
Prove that there exist real numbers
A
A
A
,
B
B
B
,
c
1
c_1
c
1
,
c
2
c_2
c
2
,
c
3
c_3
c
3
such that
u
i
(
x
)
=
c
i
(
A
x
+
B
)
u_i(x) = c_i(Ax+B)
u
i
(
x
)
=
c
i
(
A
x
+
B
)
for
i
=
1
,
2
,
3
i = 1,2,3
i
=
1
,
2
,
3
.
6
1
Hide problems
sum of digits of number 1972^n
Prove that the sum of digits of the number
197
2
n
1972^n
197
2
n
is not bounded from above when
n
n
n
tends to infinity.
5
1
Hide problems
all subsets of a finite set can be arranged in a sequence
Prove that all subsets of a finite set can be arranged in a sequence in which every two successive subsets differ in exactly one element.
4
1
Hide problems
triangle with max perimeter also has a max area, sphere
Points
A
A
A
and
B
B
B
are given on a line having no common points with a sphere
K
K
K
. The feet
P
P
P
of the perpendicular from the center of
K
K
K
to the line
A
B
AB
A
B
is positioned between
A
A
A
and
B
B
B
, and the lengths of segments
A
P
AP
A
P
and
B
P
BP
BP
both exceed the radius of
K
K
K
. Consider the set
Z
Z
Z
of all triangles
A
B
C
ABC
A
BC
whose sides
A
C
AC
A
C
and
B
C
BC
BC
are tangent to
K
K
K
. Prove that among all triangles in
Z
Z
Z
, a triangle
T
T
T
with a maximum perimeter also has a maximum area.
2
1
Hide problems
minimum length among all closed polygonal lines
On the plane are given
n
>
2
n > 2
n
>
2
points, no three of which are collinear. Prove that among all closed polygonal lines passing through these points, any one with the minimum length is non-selfintersecting.