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Contests
National and Regional Contests
Poland Contests
Polish MO Finals
1963 Polish MO Finals
1963 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(6)
6
1
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coplanar lines
Through the vertex of a trihedral angle in which no edge is perpendicular to the opposite face, a straight line is drawn in the plane of each face perpendicular to the opposite edge. Prove that the three straight lines obtained lie in one plane.
5
1
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P(x) = x^5 - 3x^4 + 6x^3 - 3x^2 + 9x - 6
Prove that a fifth-degree polynomial
P
(
x
)
=
x
5
−
3
x
4
+
6
x
3
−
3
x
2
+
9
x
−
6
P(x) = x^5 - 3x^4 + 6x^3 - 3x^2 + 9x - 6
P
(
x
)
=
x
5
−
3
x
4
+
6
x
3
−
3
x
2
+
9
x
−
6
is not the product of two lower-degree polynomials with integer coefficients.
4
1
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1 +1/\sqrt{2} + 1/\sqrt{3} + ... +1/ {\sqrt{n}} > \sqrt{n-1}.
Prove that for every natural number
n
n
n
the inequality holds
1
+
1
2
+
1
3
+
…
+
1
n
>
n
−
1
.
1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \ldots + \frac{1}{\sqrt{n}} > \sqrt{n-1}.
1
+
2
1
+
3
1
+
…
+
n
1
>
n
−
1
.
3
1
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rectangle with max area from a triangle
From a given triangle, cut out the rectangle with the largest area.
2
1
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common midpoints for 6 segments
In space there are given four distinct points
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
. Prove that the three segments connecting the midpoints of the segments
A
B
AB
A
B
and
C
D
CD
C
D
,
A
C
AC
A
C
and
B
D
BD
B
D
,
A
D
AD
A
D
and
B
C
BC
BC
have a common midpoint.
1
1
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relative prime naturals with all digits 1
Prove that two natural numbers whose digits are all ones are relatively prime if and only if the numbers of their digits are relatively prime.