MathDB
Problems
Contests
National and Regional Contests
Lithuania Contests
Grand Duchy of Lithuania
2013 Grand Duchy of Lithuania
2013 Grand Duchy of Lithuania
Part of
Grand Duchy of Lithuania
Subcontests
(4)
4
1
Hide problems
number n \choose i + n \choose j -i -j is even
A positive integer
n
≥
2
n \ge 2
n
≥
2
is called peculiar if the number
(
n
i
)
n \choose i
(
i
n
)
+
(
n
j
)
n \choose j
(
j
n
)
−
i
−
j
-i-j
−
i
−
j
is even for all integers
i
i
i
and
j
j
j
such that
0
≤
i
≤
j
≤
n
0 \le i \le j \le n
0
≤
i
≤
j
≤
n
. Determine all peculiar numbers.
3
1
Hide problems
2player game with no 1234567890 written on blackboard
The number
1234567890
1234567890
1234567890
is written on the blackboard. Two players
A
A
A
and
B
B
B
play the following game taking alternate moves. In one move, a player erases the number which is written on the blackboard, say,
m
m
m
, subtracts from
m
m
m
any positive integer not exceeding the sum of the digits of
m
m
m
and writes the obtained result instead of
m
m
m
. The first player who reduces the number written on the blackboard to
0
0
0
wins. Determine which of the players has the winning strategy if the player
A
A
A
makes the first move.
1
1
Hide problems
f(x) - g(x) is an integer , f(x) is an integer iff g(x) is an integer
Let
f
:
R
→
R
f : R \to R
f
:
R
→
R
and
g
:
R
→
R
g : R \to R
g
:
R
→
R
be strictly increasing linear functions such that
f
(
x
)
f(x)
f
(
x
)
is an integer if and only if
g
(
x
)
g(x)
g
(
x
)
is an integer. Prove that
f
(
x
)
−
g
(
x
)
f(x) - g(x)
f
(
x
)
−
g
(
x
)
is an integer for any
x
∈
R
x \in R
x
∈
R
.
2
1
Hide problems
BC is tangent to circumcircle of DEF iff D is midpoint of BC, AB=AC, <FDE=<ABC
Let
A
B
C
ABC
A
BC
be an isosceles triangle with
A
B
=
A
C
AB = AC
A
B
=
A
C
. The points
D
,
E
D, E
D
,
E
and
F
F
F
are taken on the sides
B
C
,
C
A
BC, CA
BC
,
C
A
and
A
B
AB
A
B
, respectively, so that
∠
F
D
E
=
∠
A
B
C
\angle F DE = \angle ABC
∠
F
D
E
=
∠
A
BC
and
F
E
FE
FE
is not parallel to
B
C
BC
BC
. Prove that the line
B
C
BC
BC
is tangent to the circumcircle of
△
D
E
F
\vartriangle DEF
△
D
EF
if and only if
D
D
D
is the midpoint of the side
B
C
BC
BC
.