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Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1962 Poland - Second Round
1962 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
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3-digit number wanted
Find a three-digit number with the property that the number represented by these digits and in the same order, but with a numbering base different than
10
10
10
, is twice as large as the given number.
5
1
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locus of vertex of equilateral
In the plane there is a square
Q
Q
Q
and a point
P
P
P
. The point
K
K
K
runs through the perimeter of the square
Q
Q
Q
. Find the locus of the vertex
M
M
M
of the equilateral triangle
K
P
M
KPM
K
PM
.
4
1
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a < b < c => d_a > d_b > d_c.
Prove that if the sides
a
a
a
,
b
b
b
,
c
c
c
of a triangle satisfy the inequality
a
<
b
<
c
a < b < c
a
<
b
<
c
then the angle bisectors
d
a
d_a
d
a
,
d
b
d_b
d
b
,
d
c
d_c
d
c
of opposite angles satisfy the inequality
d
a
>
d
b
>
d
c
.
d_a > d_b > d_c.
d
a
>
d
b
>
d
c
.
3
1
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concurrent of segments connecting vertices of tetrahedron with centroids
Prove that the four segments connecting the vertices of the tetrahedron with the centers of gravity of the opposite faces have a common point.
2
1
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ax^2 + 2bxy + cy^2 + 2dx + 2ey + f = P(x) Q(x)
What conditions should real numbers
a
a
a
,
b
b
b
,
c
c
c
,
d
d
d
,
e
e
e
,
f
f
f
meet in order for a polynomial of second degree
a
x
2
+
2
b
x
y
+
c
y
2
+
2
d
x
+
2
e
y
+
f
ax^2 + 2bxy + cy^2 + 2dx + 2ey + f
a
x
2
+
2
b
x
y
+
c
y
2
+
2
d
x
+
2
ey
+
f
was the product of two first degree polynomials with real coefficients ?
1
1
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x + y + z = a, 1/x +1/y +1/ z = 1/a
Prove that if the numbers
x
x
x
,
y
y
y
,
z
z
z
satisfy the equationw
x
+
y
+
z
=
a
,
x + y + z = a,
x
+
y
+
z
=
a
,
1
x
+
1
y
+
1
z
=
1
a
,
\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{a},
x
1
+
y
1
+
z
1
=
a
1
,
then at least one of them is equal to
a
a
a
.