MathDB
Problems
Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine Team Selection Test
2015 Ukraine Team Selection Test
2015 Ukraine Team Selection Test
Part of
Ukraine Team Selection Test
Subcontests
(7)
12
1
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set A, subset of {1,2,...,n}, contains a three-element arithmetic sequence
For a given natural
n
n
n
, we consider the set
A
⊂
{
1
,
2
,
.
.
.
,
n
}
A\subset \{1,2, ..., n\}
A
⊂
{
1
,
2
,
...
,
n
}
, which consists of at least
[
n
+
1
2
]
\left[\frac{n+1}{2}\right]
[
2
n
+
1
]
items. Prove that for
n
≥
2015
n \ge 2015
n
≥
2015
the set
A
A
A
contains a three-element arithmetic sequence.
7
1
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elements of AB form finite arithmetic progression, A or B has at most 3 elements
Let
A
A
A
and
B
B
B
be two sets of real numbers. Suppose that the elements of the set
A
B
=
{
a
b
:
a
∈
A
,
b
∈
B
}
AB = \{ab: a\in A, b\in B\}
A
B
=
{
ab
:
a
∈
A
,
b
∈
B
}
form a finite arithmetic progression. Prove that one of these sets contains no more than three elements
9
1
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n points in plane, convex heptagon and pentagon conditions
The set
M
M
M
consists of
n
n
n
points on the plane and satisfies the conditions:
∙
\bullet
∙
there are
7
7
7
points in the set
M
M
M
, which are vertices of a convex heptagon,
∙
\bullet
∙
for arbitrary five points with
M
M
M
, which are vertices of a convex pentagon, there is a point that also belongs to
M
M
M
and lies inside this pentagon. Find the smallest possible value that
n
n
n
can take .
8
1
Hide problems
f(x)f(yf(x)-1)=x^2f(y)-f(x)
Find all functions
f
:
R
→
R
f: R \to R
f
:
R
→
R
such that
f
(
x
)
f
(
y
f
(
x
)
−
1
)
=
x
2
f
(
y
)
−
f
(
x
)
f(x)f(yf(x)-1)=x^2f(y)-f(x)
f
(
x
)
f
(
y
f
(
x
)
−
1
)
=
x
2
f
(
y
)
−
f
(
x
)
for all real
x
,
y
x ,y
x
,
y
4
1
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sum of integers in one set equals product of integers in other set, mod p
A prime number
p
>
3
p> 3
p
>
3
is given. Prove that integers less than
p
p
p
, it is possible to divide them into two non-empty sets such that the sum of the numbers in the first set will be congruent modulo p to the product of the numbers in the second set.
2
1
Hide problems
n teams in a football tournament
In a football tournament,
n
n
n
teams play one round (
n
⋮
2
n \vdots 2
n
⋮
2
). In each round should play
n
/
2
n / 2
n
/2
pairs of teams that have not yet played. Schedule of each round takes place before its holding. For which smallest natural
k
k
k
such that the following situation is possible: after
k
k
k
tours, making a schedule of
k
+
1
k + 1
k
+
1
rounds already is not possible, i.e. these
n
n
n
teams cannot be divided into
n
/
2
n / 2
n
/2
pairs, in each of which there are teams that have not played in the previous
k
k
k
rounds.PS. The 3 vertical dots notation in the first row, I do not know what it means.
1
1
Hide problems
equal angles wanted, circumcircle and angle bisector related
Let
O
O
O
be the circumcenter of the triangle
A
B
C
,
A
′
ABC, A'
A
BC
,
A
′
be a point symmetric of
A
A
A
wrt line
B
C
,
X
BC, X
BC
,
X
is an arbitrary point on the ray
A
A
′
AA'
A
A
′
(
X
≠
A
X \ne A
X
=
A
). Angle bisector of angle
B
A
C
BAC
B
A
C
intersects the circumcircle of triangle
A
B
C
ABC
A
BC
at point
D
D
D
(
D
≠
A
D \ne A
D
=
A
). Let
M
M
M
be the midpoint of the segment
D
X
DX
D
X
. A line passing through point
O
O
O
parallel to
A
D
AD
A
D
, intersects
D
X
DX
D
X
at point
N
N
N
. Prove that angles
B
A
M
BAM
B
A
M
and
C
A
N
CAN
C
A
N
angles are equal.