MathDB
Problems
Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine Team Selection Test
2011 Ukraine Team Selection Test
2011 Ukraine Team Selection Test
Part of
Ukraine Team Selection Test
Subcontests
(9)
12
1
Hide problems
| a_ {i +1} - a_ i |= 1given, wanted a _ 1 - a _ {2n} = n iff 1<= a _ {2k} <= n
Let
n
n
n
be a natural number. Consider all permutations
(
a
1
,
…
,
a
2
n
)
({{a} _ {1}}, \ \ldots, \ {{a} _ {2n}})
(
a
1
,
…
,
a
2
n
)
of the first
2
n
2n
2
n
natural numbers such that the numbers
∣
a
i
+
1
−
a
i
∣
,
i
=
1
,
…
,
2
n
−
1
,
| {{a} _ {i +1}} - {{a} _ {i}} |, \ i = 1, \ \ldots, \ 2n-1,
∣
a
i
+
1
−
a
i
∣
,
i
=
1
,
…
,
2
n
−
1
,
are pairwise different. Prove that
a
1
−
a
2
n
=
n
{{a} _ {1}} - {{a} _ {2n}} = n
a
1
−
a
2
n
=
n
if and only if
1
≤
a
2
k
≤
n
1 \le {{a} _ {2k}} \le n
1
≤
a
2
k
≤
n
for all
k
=
1
,
…
,
n
k = 1, \ \ldots, \ n
k
=
1
,
…
,
n
.
11
1
Hide problems
S (x^ 3) - S^2(x) <1/4 (P ^2 (x^3) +Q(x^3) , where S (x) = P (x) Q (x)
Let
P
(
x
)
P (x)
P
(
x
)
and
Q
(
x
)
Q (x)
Q
(
x
)
be polynomials with real coefficients such that
P
(
0
)
>
0
P (0)> 0
P
(
0
)
>
0
and all coefficients of the polynomial
S
(
x
)
=
P
(
x
)
⋅
Q
(
x
)
S (x) = P (x) \cdot Q (x)
S
(
x
)
=
P
(
x
)
⋅
Q
(
x
)
are integers. Prove that for any positive
x
x
x
the inequality holds:
S
(
x
2
)
−
S
2
(
x
)
≤
1
4
(
P
2
(
x
3
)
+
Q
(
x
3
)
)
.
S ({{x} ^ {2}}) - {{S} ^ {2}} (x) \le \frac {1} {4} ({{P} ^ {2}} ({{ x} ^ {3}}) + Q ({{x} ^ {3}})).
S
(
x
2
)
−
S
2
(
x
)
≤
4
1
(
P
2
(
x
3
)
+
Q
(
x
3
))
.
8
1
Hide problems
any natural number can be given as a_i+a_j, a _n>n^2 /16, increasing
Is there an increasing sequence of integers
0
=
a
0
<
a
1
<
a
2
<
…
0 = {{a} _{0}} <{{a} _{1}} <{{a} _{2}} <\ldots
0
=
a
0
<
a
1
<
a
2
<
…
for which the following two conditions are satisfied simultaneously: 1) any natural number can be given as
a
i
+
a
j
{{a} _{i}} + {{a} _{j}}
a
i
+
a
j
for some (possibly equal)
i
≥
0
i \ge 0
i
≥
0
,
j
≥
0
j \ge 0
j
≥
0
, 2)
a
n
>
n
2
16
{{a} _ {n}}> \tfrac {{{n} ^ {2}}} {16}
a
n
>
16
n
2
for all natural
n
n
n
?
4
1
Hide problems
a_i+ a_i* =a _ {i-1} +a_ {i + 1}
Suppose an ordered set of
(
a
1
,
a
2
,
…
,
a
n
)
({{a} _{1}}, \ {{a} _{2}},\ \ldots,\ {{a} _{n}})
(
a
1
,
a
2
,
…
,
a
n
)
real numbers,
n
≥
3
n \ge 3
n
≥
3
. It is possible to replace the number
a
i
{{a} _ {i}}
a
i
,
i
=
2
,
n
−
1
‾
i = \overline {2, \ n-1}
i
=
2
,
n
−
1
by the number
a
i
∗
a_ {i} ^ {*}
a
i
∗
that
a
i
+
a
i
∗
=
a
i
−
1
+
a
i
+
1
{{a} _ {i}} + a_ {i} ^ {*} = {{a} _ {i-1}} + {{a} _ {i + 1}}
a
i
+
a
i
∗
=
a
i
−
1
+
a
i
+
1
. Let
(
b
1
,
b
2
,
…
,
b
n
)
({{b} _ {1}},\ {{b} _ {2}}, \ \ldots, \ {{b} _ {n}})
(
b
1
,
b
2
,
…
,
b
n
)
be the set with the largest sum of numbers that can be obtained from this, and
(
c
1
,
c
2
,
…
,
c
n
)
({{c} _ {1}},\ {{c} _ {2}}, \ \ldots, \ {{c} _ {n}})
(
c
1
,
c
2
,
…
,
c
n
)
is a similar set with the least amount. For the odd
n
≥
3
n \ge 3
n
≥
3
and set
(
1
,
3
,
…
,
n
,
2
,
4
,
…
,
n
−
1
)
(1,\ 3, \ \ldots, \ n, \ 2, \ 4, \ \ldots,\ n-1)
(
1
,
3
,
…
,
n
,
2
,
4
,
…
,
n
−
1
)
find the values of the expressions
b
1
+
b
2
+
…
+
b
n
{{b} _ {1}} + {{b} _ {2}} + \ldots + {{b} _ {n}}
b
1
+
b
2
+
…
+
b
n
and
c
1
+
c
2
+
…
+
c
n
{{c} _ {1}} + {{c} _ {2}} + \ldots + {{c} _ {n}}
c
1
+
c
2
+
…
+
c
n
.
3
1
Hide problems
n different integers in (k^n,(k+1)^n), product of which is n-th power of integer
Given a positive integer
n
>
2
n> 2
n
>
2
. Prove that there exists a natural
K
K
K
such that for all integers
k
≥
K
k \ge K
k
≥
K
on the open interval
(
k
n
,
(
k
+
1
)
n
)
({{k} ^{n}}, \ {{(k + 1)} ^{n}})
(
k
n
,
(
k
+
1
)
n
)
there are
n
n
n
different integers, the product of which is the
n
n
n
-th power of an integer.
1
1
Hide problems
sum \sin (<A _i MA_i)> sin 2 \pi/n + (n-2) sin \pi /n
Given a right
n
n
n
-angle
A
1
A
2
…
A
n
{{A} _ {1}} {{A} _ {2}} \ldots {{A} _ {n}}
A
1
A
2
…
A
n
,
n
≥
4
n \ge 4
n
≥
4
, and a point
M
M
M
inside it. Prove the inequality
sin
(
∠
A
1
M
A
2
)
+
sin
(
∠
A
2
M
A
3
)
+
…
+
sin
(
∠
A
n
M
A
1
)
>
sin
2
π
n
+
(
n
−
2
)
s
i
n
π
n
\sin (\angle {{A} _ {1}} M {{A} _ {2}}) + \sin (\angle {{A} _ {2}} M {{A} _ {3}} ) + \ldots + \sin (\angle {{A} _ {n}} M {{A} _ {1}})> \sin \frac{2 \pi}{n} + (n-2) sin \frac{\pi}{n}
sin
(
∠
A
1
M
A
2
)
+
sin
(
∠
A
2
M
A
3
)
+
…
+
sin
(
∠
A
n
M
A
1
)
>
sin
n
2
π
+
(
n
−
2
)
s
in
n
π
10
1
Hide problems
concurrency wanted, <APE =<BAC, <CQF = < BCA, orthocenter,
Let
H
H
H
be the point of intersection of the altitudes
A
P
AP
A
P
and
C
Q
CQ
CQ
of the acute-angled triangle
A
B
C
ABC
A
BC
. The points
E
E
E
and
F
F
F
are marked on the median
B
M
BM
BM
such that
∠
A
P
E
=
∠
B
A
C
\angle APE = \angle BAC
∠
A
PE
=
∠
B
A
C
,
∠
C
Q
F
=
∠
B
C
A
\angle CQF = \angle BCA
∠
CQF
=
∠
BC
A
, with point
E
E
E
lying inside the triangle
A
P
B
APB
A
PB
and point
F
F
F
is inside the triangle
C
Q
B
CQB
CQB
. Prove that the lines
A
E
,
C
F
AE, CF
A
E
,
CF
, and
B
H
BH
B
H
intersect at one point.
9
1
Hide problems
circle tangent to 2 lines and 2 circumcircles, cyclic, <PBC = <PDA, <PCB =<PAD
Inside the inscribed quadrilateral
A
B
C
D
ABCD
A
BC
D
, a point
P
P
P
is marked such that
∠
P
B
C
=
∠
P
D
A
\angle PBC = \angle PDA
∠
PBC
=
∠
P
D
A
,
∠
P
C
B
=
∠
P
A
D
\angle PCB = \angle PAD
∠
PCB
=
∠
P
A
D
. Prove that there exists a circle that touches the straight lines
A
B
AB
A
B
and
C
D
CD
C
D
, as well as the circles circumscribed by the triangles
A
B
P
ABP
A
BP
and
C
D
P
CDP
C
D
P
.
6
1
Hide problems
3 collinear leads to 4 collinear , incircle related
The circle
ω
\omega
ω
inscribed in triangle
A
B
C
ABC
A
BC
touches its sides
A
B
,
B
C
,
C
A
AB, BC, CA
A
B
,
BC
,
C
A
at points
K
,
L
,
M
K, L, M
K
,
L
,
M
respectively. In the arc
K
L
KL
K
L
of the circle
ω
\omega
ω
that does not contain the point
M
M
M
, we select point
S
S
S
. Denote by
P
,
Q
,
R
,
T
P, Q, R, T
P
,
Q
,
R
,
T
the intersection points of straight
A
S
AS
A
S
and
K
M
,
M
L
KM, ML
K
M
,
M
L
and
S
C
,
L
P
SC, LP
SC
,
L
P
and
K
Q
,
A
Q
KQ, AQ
K
Q
,
A
Q
and
P
C
PC
PC
respectively. It turned out that the points
R
,
S
R, S
R
,
S
and
M
M
M
are collinear. Prove that the point
T
T
T
also lies on the line
S
M
SM
SM
.