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Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1967 Poland - Second Round
1967 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
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concyclic points on a sphere
Prove that the points
A
1
,
A
2
,
…
,
A
n
A_1, A_2, \ldots, A_n
A
1
,
A
2
,
…
,
A
n
(
n
≥
7
n \geq 7
n
≥
7
) located on the surface of the sphere lie on a circle if and only if the planes tangent to the surface of the sphere at these points have a common point or are parallel to one straight line.
4
1
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xy+yz+zx = xyz + 2 diophantine
Solve the equation in natural numbers
x
y
+
y
z
+
z
x
=
x
y
z
+
2.
xy+yz+zx = xyz + 2.
x
y
+
yz
+
z
x
=
x
yz
+
2.
3
1
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angle bisector wanted
Two circles touch internally at point
A
A
A
. A chord
B
C
BC
BC
of the larger circle is drawn tangent to the smaller one at point
D
D
D
. Prove that
A
D
AD
A
D
is the bisector of angle
B
A
C
BAC
B
A
C
.
1
1
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a_{k-1}+a_{k+1} >= 2a_k
Real numbers
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
(
n
≥
3
n \ge 3
n
≥
3
) satisfy the conditions
a
1
=
a
n
=
0
a_1 = a_n = 0
a
1
=
a
n
=
0
and
a
k
−
1
+
a
k
+
1
≥
2
a
k
a_{k-1}+a_{k+1} \ge 2a_k
a
k
−
1
+
a
k
+
1
≥
2
a
k
for
k
=
2
k = 2
k
=
2
,
3
3
3
,
.
.
.
,
,...,
,
...
,
n
−
1
n -1
n
−
1
. Prove that none of the numbers
a
1
a_1
a
1
,
.
.
.
...
...
,
a
n
a_n
a
n
is positive.
5
1
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2 triangles and 1 line
On the plane are placed two triangles exterior to each other. Show that there always exists a line passing through two vertices of one triangle and separating the third vertex from all vertices of the other triangle.
2
1
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100 persons , everyone knows at least 66
There are 100 persons in a hall, everyone knowing at least 66 of the others. Prove that there is a case in which among any four some two don’t know each other.