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Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1963 Poland - Second Round
1963 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
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3 planes intersect along one line
From the point
S
S
S
of space arise
3
3
3
half-lines:
S
A
SA
S
A
,
S
B
SB
SB
and
S
C
SC
SC
, none of which is perpendicular to both others. Through each of these rays, a plane is drawn perpendicular to the plane containing the other two rays. Prove that the drawn planes intersect along one line
d
d
d
.
5
1
Hide problems
P(x) = nx^{n+2} -(n + 2)x^{n+1} + (n + 2)x-n is divisible (x - 1)^3
Prove that the polynomial
P
(
x
)
=
n
x
n
+
2
−
(
n
+
2
)
x
n
+
1
+
(
n
+
2
)
x
−
n
P(x) = nx^{n+2} -(n + 2)x^{n+1} + (n + 2)x-n
P
(
x
)
=
n
x
n
+
2
−
(
n
+
2
)
x
n
+
1
+
(
n
+
2
)
x
−
n
is divisible by the polynomial
(
x
−
1
)
3
(x - 1)^3
(
x
−
1
)
3
.
4
1
Hide problems
4 projections are collinear
In the triangle
A
B
C
ABC
A
BC
, the bisectors of the internal and external angles are drawn at the vertices
A
A
A
and
B
B
B
. Prove that the orthogonal projections of the point
C
C
C
on these bisectors lie on one straight line.
3
1
Hide problems
NT system, x + y + z = 3, x^3 + y^3 + z^3 = 3
Solve the system of equations in integers
x
+
y
+
z
=
3
x + y + z = 3
x
+
y
+
z
=
3
x
3
+
y
3
+
z
3
=
3
x^3 + y^3 + z^3 = 3
x
3
+
y
3
+
z
3
=
3
2
1
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parallelogram construction
In the plane there is a quadrilateral
A
B
C
D
ABCD
A
BC
D
and a point
M
M
M
. Construct a parallelogram with center
M
M
M
and its vertices lying on the lines
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
,
D
A
DA
D
A
.
1
1
Hide problems
a^2 + b^2 + c^2 = (pa + qb + rc)^2 + (qa + rb + pc)^2 + (ra + pb + qc)^2
Prove that if the numbers
p
p
p
,
q
q
q
,
r
r
r
satisfy the equality
p
+
q
+
r
=
1
p+q + r=1
p
+
q
+
r
=
1
1
p
+
1
q
+
1
r
=
0
\frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 0
p
1
+
q
1
+
r
1
=
0
then for any numbers
a
a
a
,
b
b
b
,
c
c
c
equality holds
a
2
+
b
2
+
c
2
=
(
p
a
+
q
b
+
r
c
)
2
+
(
q
a
+
r
b
+
p
c
)
2
+
(
r
a
+
p
b
+
q
c
)
2
.
a^2 + b^2 + c^2 = (pa + qb + rc)^2 + (qa + rb + pc)^2 + (ra + pb + qc)^2.
a
2
+
b
2
+
c
2
=
(
p
a
+
q
b
+
rc
)
2
+
(
q
a
+
r
b
+
p
c
)
2
+
(
r
a
+
p
b
+
q
c
)
2
.