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Problems
Contests
National and Regional Contests
Poland Contests
Polish MO Finals
1964 Polish MO Finals
1964 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(6)
6
1
Hide problems
concyclic projections, pyramid related
Given is a pyramid
S
A
B
C
D
SABCD
S
A
BC
D
whose base is a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
with perpendicular diagonals
A
C
AC
A
C
and
B
D
BD
B
D
, and the orthogonal projection of vertex
S
S
S
onto the base is the point
0
0
0
of the intersection of the diagonals of the base. Prove that the orthogonal projections of point
O
O
O
onto the lateral faces of the pyramid lie on the circle.
5
1
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min sum of distances from sides of angle
Given an acute angle and a circle inside the angle. Find a point
M
M
M
on the circle such that the sum of the distances of the point
M
M
M
from the sides of the angle is a minimum.
4
1
Hide problems
x^3 + ax^2 + bx + c = 0 has real roots
Prove that if the roots of the equation
x
3
+
a
x
2
+
b
x
+
c
=
0
x^3 + ax^2 + bx + c = 0
x
3
+
a
x
2
+
b
x
+
c
=
0
, with real coefficients, are real, then the roots of the equation
3
x
2
+
2
a
x
+
b
=
0
3x^2 + 2ax + b = 0
3
x
2
+
2
a
x
+
b
=
0
are also real.
3
1
Hide problems
coplanar touchpoints of tetrahedron and sphere
Given a tetrahedron
A
B
C
D
ABCD
A
BC
D
whose edges
A
B
,
B
C
,
C
D
,
D
A
AB, BC, CD, DA
A
B
,
BC
,
C
D
,
D
A
are tangent to a certain sphere. Prove that the points of tangency lie in the same plane.
2
1
Hide problems
sum a_i * sum b_y < n sum a_ib_i
Prove that if
a
1
<
a
2
<
…
<
a
n
a_1 < a_2 < \ldots < a_n
a
1
<
a
2
<
…
<
a
n
and
b
1
<
b
2
<
…
<
b
n
b_1 < b_2 < \ldots < b_n
b
1
<
b
2
<
…
<
b
n
, where
n
≥
2
n \geq 2
n
≥
2
, then
(
a
1
+
a
2
+
…
+
a
n
)
(
b
1
+
b
2
+
…
+
b
n
)
<
n
(
a
1
b
1
+
a
2
b
2
+
…
+
a
n
b
n
)
.
\qquad (a_1 + a_2 + \ldots + a_n)(b_1 + b_2 + \ldots + b_n) < n(a_1b_1 + a_2b_2 + \ldots + a_nb_n).
(
a
1
+
a
2
+
…
+
a
n
)
(
b
1
+
b
2
+
…
+
b
n
)
<
n
(
a
1
b
1
+
a
2
b
2
+
…
+
a
n
b
n
)
.
1
1
Hide problems
1/3 <= tan 3a / tan a <= 3 not true
Prove that the inequality
1
3
≤
tan
3
α
tan
α
≤
3
\frac{1}{3} \leq \frac{\tan 3\alpha}{\tan \alpha} \leq 3
3
1
≤
tan
α
tan
3
α
≤
3
is not true for any value of
α
\alpha
α
.