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National and Regional Contests
Poland Contests
Polish MO Finals
1967 Polish MO Finals
1967 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(6)
1
1
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L_n = (n+1)(n+2) ... 2n
Find the highest power of 2 that is a factor of the number
L
n
=
(
n
+
1
)
(
n
+
2
)
.
.
.
2
n
,
L_n = (n+1)(n+2)... 2n,
L
n
=
(
n
+
1
)
(
n
+
2
)
...2
n
,
where
n
n
n
is a natural number.
2
1
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AC_1^2 + BA_1^2 + CB_1^2 = AB_1^2 + BC_1^2 + CA_1^2
Prove that if points
A
1
,
B
1
,
C
1
A_1, B_1, C_1
A
1
,
B
1
,
C
1
lying on the sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
of a triangle
A
B
C
ABC
A
BC
are the orthogonal projections of a point
P
P
P
of the triangle onto these sides, then
A
C
1
2
+
B
A
1
2
+
C
B
1
2
=
A
B
1
2
+
B
C
1
2
+
C
A
1
2
.
AC_1^2 + BA_1^2 + CB_1^2 = AB_1^2 + BC_1^2 + CA_1^2.
A
C
1
2
+
B
A
1
2
+
C
B
1
2
=
A
B
1
2
+
B
C
1
2
+
C
A
1
2
.
4
1
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x^3 + x +1 is factor of P_n(x) = x^{n + 2} + (x+1)^{2n+1}
Prove that the polynomial
x
3
+
x
+
1
x^3 + x + 1
x
3
+
x
+
1
is a factor of the polynomial
P
n
(
x
)
=
x
n
+
2
+
(
x
+
1
)
2
n
+
1
P_n(x) = x^{n + 2} + (x+1)^{2n+1}
P
n
(
x
)
=
x
n
+
2
+
(
x
+
1
)
2
n
+
1
for every integer
n
≥
0
n \geq 0
n
≥
0
.
6
1
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3d locus, cones circumcribed on a sphere
Given a sphere and a plane that has no common points with the sphere. Find the geometric locus of the centers of the circles of tangency with the sphere of those cones circumcribed on the sphere whose vertices lie on the given plane.
5
1
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cyclic polygon with an odd number of sides has all angles equal
Prove that if a cyclic polygon with an odd number of sides has all angles equal, then this polygon is regular.
3
1
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100 persons, everyone knows at least 67
There are 100 persons in a hall, everyone knowing at least 67 of the others. Prove that there always exist four of them who know each other