MathDB
Problems
Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine Team Selection Test
2018 Ukraine Team Selection Test
2018 Ukraine Team Selection Test
Part of
Ukraine Team Selection Test
Subcontests
(8)
11
1
Hide problems
2n students in a math competition, no of ways competition is honest
2
n
2n
2
n
students take part in a math competition. First, each of the students sends its task to the members of the jury, after which each of the students receives from the jury one of proposed tasks (all received tasks are different). Let's call the competition honest, if there are
n
n
n
students who were given the tasks suggested by the remaining
n
n
n
participants. Prove that the number of task distributions in which the competition is honest is a square of natural numbers.
10
1
Hide problems
equal angles wanted, AH =3KH on altitude AH, circumcircle related
Let
A
B
C
ABC
A
BC
be a triangle with
A
H
AH
A
H
altitude. The point
K
K
K
is chosen on the segment
A
H
AH
A
H
as follows such that
A
H
=
3
K
H
AH =3KH
A
H
=
3
KH
. Let
O
O
O
be the center of the circle circumscribed around by triangle
A
B
C
,
M
ABC, M
A
BC
,
M
and
N
N
N
be the midpoints of
A
C
AC
A
C
and AB respectively. Lines
K
O
KO
K
O
and
M
N
MN
MN
intersect at the point
Z
Z
Z
, a perpendicular to
O
K
OK
O
K
passing through point
Z
Z
Z
intersects lines
A
C
AC
A
C
and
A
B
AB
A
B
at points
X
X
X
and
Y
Y
Y
respectively. Prove that
∠
X
K
Y
=
∠
C
K
B
\angle XKY =\angle CKB
∠
X
K
Y
=
∠
C
K
B
.
9
1
Hide problems
concyclic wanted, altitudes, orthocenter, circumcircle and tangents related
Let
A
A
1
,
B
B
1
,
C
C
1
AA_1, BB_1, CC_1
A
A
1
,
B
B
1
,
C
C
1
be the heights of triangle
A
B
C
ABC
A
BC
and
H
H
H
be its orthocenter. Liune
ℓ
\ell
ℓ
parallel to
A
C
AC
A
C
, intersects straight lines
A
A
1
AA_1
A
A
1
and
C
C
1
CC_1
C
C
1
at points
A
2
A_2
A
2
and
C
2
C_2
C
2
, respectively. Suppose that point
B
1
B_1
B
1
lies outside the circumscribed circle of triangle
A
2
H
C
2
A_2 HC_2
A
2
H
C
2
. Let
B
1
P
B_1P
B
1
P
and
B
1
T
B_1T
B
1
T
be tangent to of this circle. Prove that points
A
1
,
C
1
,
P
A_1, C_1, P
A
1
,
C
1
,
P
, and
T
T
T
are cyclic.
7
1
Hide problems
pn^2 has no more than one divisor d such that n^2+d is perfect square
The prime number
p
>
2
p > 2
p
>
2
and the integer
n
n
n
are given. Prove that the number
p
n
2
pn^2
p
n
2
has no more than one divisor
d
d
d
for which
n
2
+
d
n^2+d
n
2
+
d
is the square of the natural number. .
5
1
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min of sum of square of distances of 12 points with max distance 1
Find the smallest positive number
λ
\lambda
λ
such that for an arbitrary
12
12
12
points on the plane
P
1
,
P
2
,
.
.
.
P
12
P_1,P_2,...P_{12}
P
1
,
P
2
,
...
P
12
(points may coincide), with distance between arbitrary two of them does not exceeds
1
1
1
, holds the inequality
∑
1
≤
i
≤
j
≤
12
P
i
P
j
2
≤
λ
\sum_{1\le i\le j\le 12} P_iP_j^2 \le \lambda
∑
1
≤
i
≤
j
≤
12
P
i
P
j
2
≤
λ
4
1
Hide problems
red or blue edges in a square nxn lattice
Let
n
n
n
be an odd integer. Consider a square lattice of size
n
×
n
n \times n
n
×
n
, consisting of
n
2
n^2
n
2
unit squares and
2
n
(
n
+
1
)
2n(n +1)
2
n
(
n
+
1
)
edges. All edges are painted in red or blue so that the number of red edges does not exceed
n
2
n^2
n
2
. Prove that there is a cell that has at least three blue edges.
3
1
Hide problems
lattice points game in 3D, n consecutive points on a line // to axes wanted
Consider the set of all integer points in
Z
3
Z^3
Z
3
. Sasha and Masha play such a game. At first, Masha marks an arbitrary point. After that, Sasha marks all the points on some a plane perpendicular to one of the coordinate axes and at no point, which Masha noted. Next, they continue to take turns (Masha can't to select previously marked points, Sasha cannot choose the planes on which there are points said Masha). Masha wants to mark
n
n
n
consecutive points on some line that parallel to one of the coordinate axes, and Sasha seeks to interfere with it. Find all
n
n
n
, in which Masha can achieve the desired result.
12
1
Hide problems
Sum of f on a template is zero
Let
n
n
n
be a positive integer and
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\dots,a_n
a
1
,
a
2
,
…
,
a
n
be integers. Function
f
:
Z
→
R
f: \mathbb{Z} \rightarrow \mathbb{R}
f
:
Z
→
R
is such that for all integers
k
k
k
and
l
l
l
,
l
≠
0
l \neq 0
l
=
0
,
∑
i
=
1
n
f
(
k
+
a
i
l
)
=
0.
\sum_{i=1}^n f(k+a_il)=0.
i
=
1
∑
n
f
(
k
+
a
i
l
)
=
0.
Prove that
f
≡
0
f \equiv 0
f
≡
0
.