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Problems
Contests
International Contests
Czech-Polish-Slovak Match
2010 Czech-Polish-Slovak Match
2010 Czech-Polish-Slovak Match
Part of
Czech-Polish-Slovak Match
Subcontests
(3)
2
2
Hide problems
Sum of the distances to 60 points on a unit circle ≤ 80
Given any
60
60
60
points on a circle of radius
1
1
1
, prove that there is a point on the circle the sum of whose distances to these
60
60
60
points is at most
80
80
80
.
Minimum of cyclic Σ ( x^2 + x/(y^2+z+1) )
Let
x
x
x
,
y
y
y
,
z
z
z
be positive real numbers satisfying
x
+
y
+
z
≥
6
x+y+z\ge 6
x
+
y
+
z
≥
6
. Find, with proof, the minimum value of
x
2
+
y
2
+
z
2
+
x
y
2
+
z
+
1
+
y
z
2
+
x
+
1
+
z
x
2
+
y
+
1
.
x^2+y^2+z^2+\frac{x}{y^2+z+1}+\frac{y}{z^2+x+1}+\frac{z}{x^2+y+1}.
x
2
+
y
2
+
z
2
+
y
2
+
z
+
1
x
+
z
2
+
x
+
1
y
+
x
2
+
y
+
1
z
.
1
2
Hide problems
System of equations: a√b - c = a; b√c - a = b; c√a - b = c
Find all triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of positive real numbers satisfying the system of equations
a
b
−
c
&
=
a
,
b
c
−
a
&
=
b
,
c
a
−
b
&
=
c
.
a\sqrt{b}-c \&= a,\qquad b\sqrt{c}-a \&= b,\qquad c\sqrt{a}-b \&= c.
a
b
−
c
&
=
a
,
b
c
−
a
&
=
b
,
c
a
−
b
&
=
c
.
Rearranging sides of triangles to make other triangles
Given any collection of
2010
2010
2010
nondegenerate triangles, their sides are painted so that each triangle has one red side, one blue side, and one white side. For each color, arrange the side lengths in order: [*]let
b
1
≤
b
2
≤
⋯
≤
b
2011
b_1\le b_2\le\cdots\le b_{2011}
b
1
≤
b
2
≤
⋯
≤
b
2011
denote the lengths of the blue sides; [*]let
r
1
≤
r
2
≤
⋯
≤
r
2011
r_1\le r_2\le\cdots\le r_{2011}
r
1
≤
r
2
≤
⋯
≤
r
2011
denote the lengths of the red sides; and [*]let
w
1
≤
w
2
≤
⋯
≤
w
2011
w_1\le w_2\le\cdots\le w_{2011}
w
1
≤
w
2
≤
⋯
≤
w
2011
denote the lengths of the white sides. Find the largest integer
k
k
k
for which there necessarily exists at least
k
k
k
indices
j
j
j
such that
b
j
b_j
b
j
,
r
j
r_j
r
j
,
w
j
w_j
w
j
are the side lengths of a nondegenerate triangle.
3
2
Hide problems
Choosing squares in an array that do not form a rectangle
Let
p
p
p
be a prime number. Prove that from a
p
2
×
p
2
p^2\times p^2
p
2
×
p
2
array of squares, we can select
p
3
p^3
p
3
of the squares such that the centers of any four of the selected squares are not the vertices of a rectangle with sides parallel to the edges of the array.
Conditions on sides & diagonals => ABCD is a parallelogram
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral for which
A
B
+
C
D
=
2
⋅
A
C
and
B
C
+
D
A
=
2
⋅
B
D
.
AB+CD=\sqrt{2}\cdot AC\qquad\text{and}\qquad BC+DA=\sqrt{2}\cdot BD.
A
B
+
C
D
=
2
⋅
A
C
and
BC
+
D
A
=
2
⋅
B
D
.
Prove that
A
B
C
D
ABCD
A
BC
D
is a parallelogram.