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Problems
Contests
International Contests
Czech-Polish-Slovak Match
2005 Czech-Polish-Slovak Match
2005 Czech-Polish-Slovak Match
Part of
Czech-Polish-Slovak Match
Subcontests
(6)
5
1
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Locus of points for which products of areas are equal
Given a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
, find the locus of the points
P
P
P
inside the quadrilateral such that
S
P
A
B
⋅
S
P
C
D
=
S
P
B
C
⋅
S
P
D
A
S_{PAB}\cdot S_{PCD} = S_{PBC}\cdot S_{PDA}
S
P
A
B
⋅
S
PC
D
=
S
PBC
⋅
S
P
D
A
(where
S
X
S_X
S
X
denotes the area of triangle
X
X
X
).
6
1
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y(x+y)=x^3-7x^2+11x-3
Determine all pairs of integers
(
x
,
y
)
(x, y)
(
x
,
y
)
satisfying the equation
y
(
x
+
y
)
=
x
3
−
7
x
2
+
11
x
−
3.
y(x + y) = x^3- 7x^2 + 11x - 3.
y
(
x
+
y
)
=
x
3
−
7
x
2
+
11
x
−
3.
4
1
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Arrange balls so that A gets as many as 4 others
We distribute
n
≥
1
n\ge1
n
≥
1
labelled balls among nine persons
A
,
B
,
C
,
…
,
I
A,B,C, \dots , I
A
,
B
,
C
,
…
,
I
. How many ways are there to do this so that
A
A
A
gets the same number of balls as
B
,
C
,
D
B,C,D
B
,
C
,
D
and
E
E
E
together?
3
1
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For which n is the polynomial reducible?
Find all integers
n
≥
3
n \ge 3
n
≥
3
for which the polynomial
W
(
x
)
=
x
n
−
3
x
n
−
1
+
2
x
n
−
2
+
6
W(x) = x^n - 3x^{n-1} + 2x^{n-2} + 6
W
(
x
)
=
x
n
−
3
x
n
−
1
+
2
x
n
−
2
+
6
can be written as a product of two non-constant polynomials with integer coefficients.
2
1
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Circumcentre, Incentre, Intersection of Diagonals Collinear
A convex quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in a circle with center
O
O
O
and circumscribed to a circle with center
I
I
I
. Its diagonals meet at
P
P
P
. Prove that points
O
,
I
O, I
O
,
I
and
P
P
P
lie on a line.
1
1
Hide problems
Simultaneous equations in n variables
Let
n
n
n
be a given positive integer. Solve the system
x
1
+
x
2
2
+
x
3
3
+
⋯
+
x
n
n
=
n
,
x_1 + x_2^2 + x_3^3 + \cdots + x_n^n = n,
x
1
+
x
2
2
+
x
3
3
+
⋯
+
x
n
n
=
n
,
x
1
+
2
x
2
+
3
x
3
+
⋯
+
n
x
n
=
n
(
n
+
1
)
2
x_1 + 2x_2 + 3x_3 + \cdots + nx_n = \frac{n(n+1)}{2}
x
1
+
2
x
2
+
3
x
3
+
⋯
+
n
x
n
=
2
n
(
n
+
1
)
in the set of nonnegative real numbers.