MathDB
Problems
Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine Team Selection Test
2014 Ukraine Team Selection Test
2014 Ukraine Team Selection Test
Part of
Ukraine Team Selection Test
Subcontests
(11)
7
1
Hide problems
k chips of two colors on cells of a nxn board
For each natural
n
≥
4
n \ge 4
n
≥
4
, find the smallest natural number
k
k
k
that satisfies following condition: For an arbitrary arrangement of
k
k
k
chips of two colors on
n
×
n
n\times n
n
×
n
board, there exists a non-empty set such that all columns and rows contain even number (
0
0
0
is also possible) of chips each color.
8
1
Hide problems
collinear wanted, inscribed ABCD and perpendiculars related
The quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in the circle
ω
\omega
ω
with the center
O
O
O
. Suppose that the angles
B
B
B
and
C
C
C
are obtuse and lines
A
D
AD
A
D
and
B
C
BC
BC
are not parallel. Lines
A
B
AB
A
B
and
C
D
CD
C
D
intersect at point
E
E
E
. Let
P
P
P
and
R
R
R
be the feet of the perpendiculars from the point
E
E
E
on the lines
B
C
BC
BC
and
A
D
AD
A
D
respectively.
Q
Q
Q
is the intersection point of
E
P
EP
EP
and
A
D
,
S
AD, S
A
D
,
S
is the intersection point of
E
R
ER
ER
and
B
C
BC
BC
. Let K be the midpoint of the segment
Q
S
QS
QS
. Prove that the points
E
,
K
E, K
E
,
K
, and
O
O
O
are collinear.
6
1
Hide problems
blue and yellow paths in a n x n board, one path starts or end at center
Let
n
≥
3
n \ge 3
n
≥
3
be an odd integer. Each cell is a
n
×
n
n \times n
n
×
n
board painted in yellow or blue. Let's call the sequence of cells
S
1
,
S
2
,
.
.
.
,
S
m
S_1, S_2,...,S_m
S
1
,
S
2
,
...
,
S
m
path if they are all the same color and the cells
S
i
S_i
S
i
and
S
j
S_j
S
j
have one in common an edge if and only if
∣
i
−
j
∣
=
1
|i - j| = 1
∣
i
−
j
∣
=
1
. Suppose that all yellow cells form a path and all the blue cells form a path. Prove that one of the two paths begins or ends at the center of the board.
4
1
Hide problems
circumcenter of XYM lies on BC, excircle and circumcircles related
The
A
A
A
-excircle of the triangle
A
B
C
ABC
A
BC
touches the side
B
C
BC
BC
at point
K
K
K
. The circumcircles of triangles
A
K
B
AKB
A
K
B
and
A
K
C
AKC
A
K
C
intersect for the second time with the bisector of angle
A
A
A
at points
X
X
X
and
Y
Y
Y
respectively. Let
M
M
M
be the midpoint of
B
C
BC
BC
. Prove that the circumcenter of triangle
X
Y
M
XYM
X
Y
M
lies on
B
C
BC
BC
.
10
1
Hide problems
n - 3 points inside a triangle with vertices on convex hull of n points
Find all positive integers
n
≥
4
n \ge 4
n
≥
4
for which there are
n
n
n
points in general position on the plane such that an arbitrary triangle with vertices belonging to the convex hull of these
n
n
n
points, containing exactly one of
n
−
3
n - 3
n
−
3
points inside remained.
12
1
Hide problems
no of pos. integers n suh that p | n! +1 is <= cp^{2/3}
Prove that for an arbitrary prime
p
≥
3
p \ge 3
p
≥
3
the number of positive integers
n
n
n
, for which
p
∣
n
!
+
1
p | n! +1
p
∣
n
!
+
1
does not exceed
c
p
2
/
3
cp^{2/3}
c
p
2/3
, where c is a constant that does not depend on
p
p
p
.
11
1
Hide problems
(f (x) - f (y)) (u - v) = (f (u) - f (v)) (x - y) when x + y = u + v
Find all functions
f
:
R
→
R
f: R \to R
f
:
R
→
R
that satisfy the condition
(
f
(
x
)
−
f
(
y
)
)
(
u
−
v
)
=
(
f
(
u
)
−
f
(
v
)
)
(
x
−
y
)
(f (x) - f (y)) (u - v) = (f (u) - f (v)) (x -y)
(
f
(
x
)
−
f
(
y
))
(
u
−
v
)
=
(
f
(
u
)
−
f
(
v
))
(
x
−
y
)
for arbitrary real
x
,
y
,
u
,
v
x, y, u, v
x
,
y
,
u
,
v
such that
x
+
y
=
u
+
v
x + y = u + v
x
+
y
=
u
+
v
.
9
1
Hide problems
x^m+y^m+z^m=a, x^n+y^n+z^n=b , 3parameter diophantine system
Let
m
,
n
m, n
m
,
n
be odd prime numbers. Find all pairs of integers numbers
a
,
b
a, b
a
,
b
for which the system of equations:
x
m
+
y
m
+
z
m
=
a
x^m+y^m+z^m=a
x
m
+
y
m
+
z
m
=
a
,
x
n
+
y
n
+
z
n
=
b
x^n+y^n+z^n=b
x
n
+
y
n
+
z
n
=
b
has many solutions in integers
x
,
y
,
z
x, y, z
x
,
y
,
z
.
5
1
Hide problems
i+j = C_{n}^{i} + C_{n}^{j} (mod 2)
Find all positive integers
n
≥
2
n \ge 2
n
≥
2
such that equality
i
+
j
≡
C
n
i
+
C
n
j
i+j \equiv C_{n}^{i} + C_{n}^{j}
i
+
j
≡
C
n
i
+
C
n
j
(mod
2
2
2
) is true for arbitrary
0
≤
i
≤
j
≤
n
0 \le i \le j \le n
0
≤
i
≤
j
≤
n
.
1
1
Hide problems
k times, ants jump from one vertex to a neighbour one, on 2n-gon
Given an integer
n
≥
2
n \ge 2
n
≥
2
and a regular
2
n
2n
2
n
-polygon at each vertex of which sitting on an ant. At some points in time, each ant creeps into one of two adjacent peaks (some peaks may have several ants at a time). Through
k
k
k
such operations, it turned out to be an arbitrary line connecting two different ones the vertices of a polygon with ants do not pass through its center. For given
n
n
n
find the lowest possible value of
k
k
k
.
2
1
Hide problems
Find the maximum
Let
x
1
,
x
2
,
⋯
,
x
n
x_1,x_2,\cdots,x_n
x
1
,
x
2
,
⋯
,
x
n
be postive real numbers such that
x
1
x
2
⋯
x
n
=
1
x_1x_2\cdots x_n=1
x
1
x
2
⋯
x
n
=
1
,
S
=
x
1
3
+
x
2
3
+
⋯
+
x
n
3
S=x^3_1+x^3_2+\cdots+x^3_n
S
=
x
1
3
+
x
2
3
+
⋯
+
x
n
3
.Find the maximum of
x
1
S
−
x
1
3
+
x
1
2
+
x
2
S
−
x
2
3
+
x
2
2
+
⋯
+
x
n
S
−
x
n
3
+
x
n
2
\frac{x_1}{S-x^3_1+x^2_1}+\frac{x_2}{S-x^3_2+x^2_2}+\cdots+\frac{x_n}{S-x^3_n+x^2_n}
S
−
x
1
3
+
x
1
2
x
1
+
S
−
x
2
3
+
x
2
2
x
2
+
⋯
+
S
−
x
n
3
+
x
n
2
x
n