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Contests
National and Regional Contests
Poland Contests
Polish MO Finals
1981 Polish MO Finals
1981 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(6)
3
1
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\prod (x-a^k )/(x+a^k ineq
Prove that for any natural number
n
n
n
and real numbers
a
a
a
and
x
x
x
satisfying
a
n
+
1
≤
x
≤
1
a^{n+1} \le x \le 1
a
n
+
1
≤
x
≤
1
and
0
<
a
<
1
0 < a < 1
0
<
a
<
1
it holds that
∏
k
=
1
n
∣
x
−
a
k
x
+
a
k
∣
≤
∏
k
=
1
n
1
−
a
k
1
+
a
k
\prod_{k=1}^n \left|\frac{x-a^k}{x+a^k}\right| \le \prod_{k=1}^n \frac{1-a^k}{1+a^k}
k
=
1
∏
n
x
+
a
k
x
−
a
k
≤
k
=
1
∏
n
1
+
a
k
1
−
a
k
6
1
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volume of tetrahedron ineq
In a tetrahedron of volume
V
V
V
the sum of the squares of the lengths of its edges equals
S
S
S
. Prove that
V
≤
S
S
72
3
V \le \frac{S\sqrt{S}}{72\sqrt{3}}
V
≤
72
3
S
S
5
1
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x^3 +x^2y+xy^2 +y^3 = 8(x^2 +xy+y^2 +1) NT
Determine all pairs of integers
(
x
,
y
)
(x,y)
(
x
,
y
)
satisfying the equation
x
3
+
x
2
y
+
x
y
2
+
y
3
=
8
(
x
2
+
x
y
+
y
2
+
1
)
.
x^3 +x^2y+xy^2 +y^3 = 8(x^2 +xy+y^2 +1).
x
3
+
x
2
y
+
x
y
2
+
y
3
=
8
(
x
2
+
x
y
+
y
2
+
1
)
.
4
1
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n markers on a table, each of which is denoted by an integer
On a table are given
n
n
n
markers, each of which is denoted by an integer. At any time, if some two markers are denoted with the same number, say
k
k
k
, we can redenote one of them with
k
+
1
k +1
k
+
1
and the other one with
k
−
1
k -1
k
−
1
. Prove that after a finite number of moves all the markers will be denoted with different numbers.
2
1
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BC = XY iff tanBtanC = 3 or tanBtanC = -1.
In a triangle
A
B
C
ABC
A
BC
, the perpendicular bisectors of sides
A
B
AB
A
B
and
A
C
AC
A
C
intersect
B
C
BC
BC
at
X
X
X
and
Y
Y
Y
. Prove that
B
C
=
X
Y
BC = XY
BC
=
X
Y
if and only if
tan
B
tan
C
=
3
\tan B\tan C = 3
tan
B
tan
C
=
3
or
tan
B
tan
C
=
−
1
\tan B\tan C = -1
tan
B
tan
C
=
−
1
.
1
1
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circle with every point on intersection of 2 planes
Two intersecting lines
a
a
a
and
b
b
b
are given in a plane. Consider all pairs of orthogonal planes
α
\alpha
α
,
β
\beta
β
such that
a
⊂
α
a \subset \alpha
a
⊂
α
and
b
⊂
β
b\subset \beta
b
⊂
β
. Prove that there is a circle such that every its point lies on the line
α
∩
β
\alpha \cap \beta
α
∩
β
for some
α
\alpha
α
and
β
\beta
β
.