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Problems
Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine Team Selection Test
2013 Ukraine Team Selection Test
2013 Ukraine Team Selection Test
Part of
Ukraine Team Selection Test
Subcontests
(8)
12
1
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coloring 2013 interior points and the vertices of a convex 2013-gon
4026
4026
4026
points were noted on the plane, not three of which lie on a straight line. The
2013
2013
2013
points are the vertices of a convex polygon, and the other
2013
2013
2013
vertices are inside this polygon. It is allowed to paint each point in one of two colors. Coloring will be good if some pairs of dots can be combined segments with the following conditions:
∙
\bullet
∙
Each segment connects dots of the same color.
∙
\bullet
∙
No two drawn segments intersect at their inner points.
∙
\bullet
∙
For an arbitrary pair of dots of the same color, there is a path along the lines from one point to another. Please note that the sides of the convex
2013
2013
2013
rectangle are not automatically drawn segments, although some (or all) can be drawn as needed. Prove that the total number of good colors does not depend on the specific locations of the points and find that number.
11
1
Hide problems
infinite primes p divide 2^{2^n} + a
Specified natural number
a
a
a
. Prove that there are an infinite number of prime numbers
p
p
p
such that for some natural
n
n
n
the number
2
2
n
+
a
2^{2^n} + a
2
2
n
+
a
is divisible by
p
p
p
.
7
1
Hide problems
2013 users on social network, max friendship under conditions
2013
2013
2013
users have registered on the social network "Graph". Some users are friends, and friendship in "Graph" is mutual. It is known that among network users there are no three, each of whom would be friends. Find the biggest one possible number of pairs of friends in "Graph".
2
1
Hide problems
stars in a 2013 x 2013 table, no more than m stars in each row or column
The teacher reported to Peter an odd integer
m
≤
2013
m \le 2013
m
≤
2013
and gave the guy a homework. Petrick should star the cells in the
2013
×
2013
2013 \times 2013
2013
×
2013
table so to make the condition true: if there is an asterisk in some cell in the table, then or in row or column containing this cell should be no more than
m
m
m
stars (including this one). Thus in each cell of the table the guy can put at most one star. The teacher promised Peter that his assessment would be just the number of stars that the guy will be able to place. What is the greatest number will the stars be able to place in the table Petrick?
6
1
Hide problems
PP', QQ' , RR' parallel or intersect, cyclic hexagon and perp. bisectors related
Six different points
A
,
B
,
C
,
D
,
E
,
F
A, B, C, D, E, F
A
,
B
,
C
,
D
,
E
,
F
are marked on the plane, no four of them lie on one circle and no two segments with ends at these points lie on parallel lines. Let
P
,
Q
,
R
P, Q,R
P
,
Q
,
R
be the points of intersection of the perpendicular bisectors to pairs of segments
(
A
D
,
B
E
)
(AD, BE)
(
A
D
,
BE
)
,
(
B
E
,
C
F
)
(BE, CF)
(
BE
,
CF
)
,
(
C
F
,
D
A
)
(CF, DA)
(
CF
,
D
A
)
respectively, and
P
′
,
Q
′
,
R
′
P', Q' ,R'
P
′
,
Q
′
,
R
′
are points the intersection of the perpendicular bisectors to the pairs of segments
(
A
E
,
B
D
)
(AE, BD)
(
A
E
,
B
D
)
,
(
B
F
,
C
E
)
(BF, CE)
(
BF
,
CE
)
,
(
C
A
,
D
F
)
(CA, DF)
(
C
A
,
D
F
)
respectively. Show that
P
≠
P
′
,
Q
≠
Q
′
,
R
≠
R
′
P \ne P', Q \ne Q', R \ne R'
P
=
P
′
,
Q
=
Q
′
,
R
=
R
′
, and prove that the lines
P
P
′
,
Q
Q
′
PP', QQ'
P
P
′
,
Q
Q
′
and
R
R
′
RR'
R
R
′
intersect at one point or are parallel.
1
1
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midpoint of MN lies on the circumcircle of PQR, starting with isosceles
Let
A
B
C
ABC
A
BC
be an isosceles triangle
A
B
C
ABC
A
BC
with base
B
C
BC
BC
insribed in a circle. The segment
A
D
AD
A
D
is the diameter of the circle, and point
P
P
P
lies on the smaller arc
B
D
BD
B
D
. Line
D
P
DP
D
P
intersects rays
A
B
AB
A
B
and
A
C
AC
A
C
at points
M
M
M
and
N
N
N
, and the lines
B
P
BP
BP
and
C
P
CP
CP
intersects the line
A
D
AD
A
D
at points
Q
Q
Q
and
R
R
R
. Prove that the midpoint of the segment
M
N
MN
MN
lies on the circumscribed circle of triangle
P
Q
R
PQR
PQR
.
5
1
Hide problems
\sqrt[3]{(x+y)/2z}+\sqrt[3]{(y+z)/2y}+\sqrt[3]{(z+x)/2y} <= [5(x+y+z)+9]/8
For positive
x
,
y
x, y
x
,
y
, and
z
z
z
that satisfy the condition
x
y
z
=
1
xyz = 1
x
yz
=
1
, prove the inequality
x
+
y
2
z
3
+
y
+
z
2
x
3
+
z
+
x
2
y
3
≤
5
(
x
+
y
+
z
)
+
9
8
\sqrt[3]{\frac{x+y}{2z}}+\sqrt[3]{\frac{y+z}{2x}}+\sqrt[3]{\frac{z+x}{2y}}\le \frac{5(x+y+z)+9}{8}
3
2
z
x
+
y
+
3
2
x
y
+
z
+
3
2
y
z
+
x
≤
8
5
(
x
+
y
+
z
)
+
9
9
1
Hide problems
Functional equation
Determine all functions
f
:
R
→
R
f:\Bbb{R}\to\Bbb{R}
f
:
R
→
R
such that
f
2
(
x
+
y
)
=
f
2
(
x
)
+
2
f
(
x
y
)
+
f
2
(
y
)
,
f^2(x+y)=f^2(x)+2f(xy)+f^2(y),
f
2
(
x
+
y
)
=
f
2
(
x
)
+
2
f
(
x
y
)
+
f
2
(
y
)
,
for all
x
,
y
∈
R
.
x,y\in \Bbb{R}.
x
,
y
∈
R
.