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Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1950 Poland - Second Round
1950 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
5
1
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2 concentric circles trisect segment, construction
Given two concentric circles and a point
A
A
A
. Through point
A
A
A
, draw a secant such that its segment contained by the larger circle is divided by the smaller circle into three equal parts.
6
1
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y^3-x^3=91 diophantine
Solve the equation in integer numbers
y
3
−
x
3
=
91
y^3-x^3=91
y
3
−
x
3
=
91
4
1
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P interior of ABC with <PAB=<PBC =<PCA
Inside the triangle
A
B
C
ABC
A
BC
there is a point
P
P
P
such that
∠
P
A
B
=
∠
P
B
C
=
∠
P
C
A
=
ϕ
.
\angle PAB=\angle PBC =\angle PCA = \phi.
∠
P
A
B
=
∠
PBC
=
∠
PC
A
=
ϕ
.
Prove that
1
sin
2
ϕ
=
1
sin
2
A
+
1
sin
2
B
+
1
sin
2
C
\frac{1}{\sin^2 \phi}=\frac{1}{\sin^2 A} +\frac{1}{\sin^2 B} +\frac{1}{\sin^2 C}
sin
2
ϕ
1
=
sin
2
A
1
+
sin
2
B
1
+
sin
2
C
1
3
1
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lines are angle bisectors of cyclic quad
The diagonals of a quadrangle inscribed in a circle intersect at point
K
K
K
. The projections of the point
K
K
K
onto the subsequent sides of this quadrangle are points
M
,
N
,
P
,
Q
M, N, P, Q
M
,
N
,
P
,
Q
. Prove that these lines
K
M
KM
K
M
,
K
N
KN
K
N
,
K
P
KP
K
P
,
K
Q
KQ
K
Q
are the angle bisectors of the quadrangle
M
N
P
Q
MNPQ
MNPQ
.
2
1
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a+b+c >=3 if a ,b > 0, abc=1 - Polish MO Second Round 1950 p2
Prove that if
a
>
0
a > 0
a
>
0
,
b
>
0
b > 0
b
>
0
,
a
b
c
=
1
abc=1
ab
c
=
1
, then
a
+
b
+
c
≥
3
a+b+c \ge 3
a
+
b
+
c
≥
3
1
1
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x^2+x+y=8, y^2+2xy+z=168, z^2+2yz+2xz=12480
Solve the system of equations
{
x
2
+
x
+
y
=
8
y
2
+
2
x
y
+
z
=
168
z
2
+
2
y
z
+
2
x
z
=
12480
\begin{cases} x^2+x+y=8\\ y^2+2xy+z=168\\ z^2+2yz+2xz=12480 \end{cases}
⎩
⎨
⎧
x
2
+
x
+
y
=
8
y
2
+
2
x
y
+
z
=
168
z
2
+
2
yz
+
2
x
z
=
12480