MathDB
Problems
Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine Team Selection Test
2010 Ukraine Team Selection Test
2010 Ukraine Team Selection Test
Part of
Ukraine Team Selection Test
Subcontests
(8)
12
1
Hide problems
r=b+\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}
Is there a positive integer
n
n
n
for which the following holds: for an arbitrary rational
r
r
r
there exists an integer
b
b
b
and non-zero integers
a
1
,
a
2
,
.
.
.
,
a
n
a _1, a_2, ..., a_n
a
1
,
a
2
,
...
,
a
n
such that
r
=
b
+
1
a
1
+
1
a
2
+
.
.
.
+
1
a
n
r=b+\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}
r
=
b
+
a
1
1
+
a
2
1
+
...
+
a
n
1
?
11
1
Hide problems
FN > ND, excircle of ABC and circumcircle of CDM related
Let
A
B
C
ABC
A
BC
be the triangle in which
A
B
>
A
C
AB> AC
A
B
>
A
C
. Circle
ω
a
\omega_a
ω
a
touches the segment of the
B
C
BC
BC
at point
D
D
D
, the extension of the segment
A
B
AB
A
B
towards point
B
B
B
at the point
F
F
F
, and the extension of the segment
A
C
AC
A
C
towards point
C
C
C
at the point
E
E
E
. The ray
A
D
AD
A
D
intersects circle
ω
a
\omega_a
ω
a
for second time at point
M
M
M
. Denote the circle circumscribed around the triangle
C
D
M
CDM
C
D
M
by
ω
\omega
ω
. Circle
ω
\omega
ω
intersects the segment
D
F
DF
D
F
at N. Prove that
F
N
>
N
D
FN > ND
FN
>
N
D
.
8
1
Hide problems
if a_n <a_m then a_{kn} <a _{km}, consistent sequences
Consider an infinite sequence of positive integers in which each positive integer occurs exactly once. Let
{
a
n
}
,
n
≥
1
\{a_n\}, n\ge 1
{
a
n
}
,
n
≥
1
be such a sequence. We call it consistent if, for an arbitrary natural
k
k
k
and every natural
n
,
m
n ,m
n
,
m
such that
a
n
<
a
m
a_n <a_m
a
n
<
a
m
, the inequality
a
k
n
<
a
k
m
a_{kn} <a _{km}
a
kn
<
a
km
also holds. For example, the sequence
a
n
=
n
a_n = n
a
n
=
n
is consistent . a) Prove that there are consistent sequences other than
a
n
=
n
a_n = n
a
n
=
n
. b) Are there consistent sequences for which
a
n
≠
n
,
n
≥
2
a_n \ne n, n\ge 2
a
n
=
n
,
n
≥
2
? c) Are there consistent sequences for which
a
n
≠
n
,
n
≥
1
a n \ne n, n\ge 1
an
=
n
,
n
≥
1
?
7
1
Hide problems
AB + AC >= BC cos A + 2h sin A
Denote in the triangle
A
B
C
ABC
A
BC
by
h
h
h
the length of the height drawn from vertex
A
A
A
, and by
α
=
∠
B
A
C
\alpha = \angle BAC
α
=
∠
B
A
C
. Prove that the inequality
A
B
+
A
C
≥
B
C
⋅
cos
α
+
2
h
⋅
sin
α
AB + AC \ge BC \cdot \cos \alpha + 2h \cdot \sin \alpha
A
B
+
A
C
≥
BC
⋅
cos
α
+
2
h
⋅
sin
α
. Are there triangles for which this inequality turns into equality?
2
1
Hide problems
collinear wanted, starting with a cyclic ABCD, incircle , excircle related
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral inscribled in a circle with the center
O
,
P
O, P
O
,
P
be the point of intersection of the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
,
B
C
∦
A
D
BC\nparallel AD
BC
∦
A
D
. Rays
A
B
AB
A
B
and
D
C
DC
D
C
intersect at the point
E
E
E
. The circle with center
I
I
I
inscribed in the triangle
E
B
C
EBC
EBC
touches
B
C
BC
BC
at point
T
1
T_1
T
1
. The
E
E
E
-excircle with center
J
J
J
in the triangle
E
A
D
EAD
E
A
D
touches the side
A
D
AD
A
D
at the point T
2
_2
2
. Line
I
T
1
IT_1
I
T
1
and
J
T
2
JT_2
J
T
2
intersect at
Q
Q
Q
. Prove that the points
O
,
P
O, P
O
,
P
, and
Q
Q
Q
lie on a straight line.
4
1
Hide problems
a/(b+c)+b/(c+a)+c/(a+b)+\sqrt{(ab+bc+ca)/(a^2+b^2+c^2)}>= 5/2
For the nonnegative numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
prove the inequality:
a
b
+
c
+
b
c
+
a
+
c
a
+
b
+
a
b
+
b
c
+
c
a
a
2
+
b
2
+
c
2
≥
5
2
\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\sqrt{\frac{ab+bc+ca}{a^2+b^2+c^2}}\ge \frac52
b
+
c
a
+
c
+
a
b
+
a
+
b
c
+
a
2
+
b
2
+
c
2
ab
+
b
c
+
c
a
≥
2
5
6
1
Hide problems
(c^n+1)/(2^na+b) is an integer for all n
Find all pairs of odd integers
a
a
a
and
b
b
b
for which there exists a natural number
c
c
c
such that the number
c
n
+
1
2
n
a
+
b
\frac{c^n+1}{2^na+b}
2
n
a
+
b
c
n
+
1
is integer for all natural
n
n
n
.
1
1
Hide problems
game with 2010 red and 2010 white cards
There are
2010
2010
2010
red cards and
2010
2010
2010
white cards. All of these
4020
4020
4020
cards are shuffled and dealt in two randomly to each of the
2010
2010
2010
round table players. The game consists of several rounds, each of which players simultaneously hand over cards to each other according to the following rules. If a player holds at least one red card, he passes one red card to the player sitting to his left, otherwise he transfers one white card to the left. The game ends after the round when each player has one red card and one white card. Determine as many rounds as possible.