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Problems
Contests
National and Regional Contests
Lithuania Contests
Lithuania National Olympiad
2010 Lithuania National Olympiad
2010 Lithuania National Olympiad
Part of
Lithuania National Olympiad
Subcontests
(4)
4
2
Hide problems
divisible by 37
Decimal digits
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfy
37
∣
(
a
0
a
0
…
a
0
b
0
c
0
c
…
0
c
)
10
37\mid (a0a0\ldots a0b0c0c\ldots 0c)_{10}
37
∣
(
a
0
a
0
…
a
0
b
0
c
0
c
…
0
c
)
10
where there are
1001
1001
1001
a's and
1001
1001
1001
c's. Prove that
b
=
a
+
c
b=a+c
b
=
a
+
c
.
residues modulo 5
Arrange arbitrarily
1
,
2
,
…
,
25
1,2,\ldots ,25
1
,
2
,
…
,
25
on a circumference. We consider all
25
25
25
sums obtained by adding
5
5
5
consecutive numbers. If the number of distinct residues of those sums modulo
5
5
5
is
d
d
d
(
0
≤
d
≤
5
)
(0\le d\le 5)
(
0
≤
d
≤
5
)
,find all possible values of
d
d
d
.
3
1
Hide problems
move a stone in a chessboard
In an
m
×
n
m\times n
m
×
n
rectangular chessboard,there is a stone in the lower leftmost square. Two persons A,B move the stone alternately. In each step one can move the stone upward or rightward any number of squares. The one who moves it into the upper rightmost square wins. Find all
(
m
,
n
)
(m,n)
(
m
,
n
)
such that the first person has a winning strategy.
2
2
Hide problems
bisector of A
In trapezoid
A
B
C
D
ABCD
A
BC
D
,
A
D
AD
A
D
is parallel to
B
C
BC
BC
. Knowing that
A
B
=
A
D
+
B
C
AB=AD+BC
A
B
=
A
D
+
BC
, prove that the bisector of
∠
A
\angle A
∠
A
also bisects
C
D
CD
C
D
.
find angle B
Let
I
I
I
be the incenter of a triangle
A
B
C
ABC
A
BC
.
D
,
E
,
F
D,E,F
D
,
E
,
F
are the symmetric points of
I
I
I
with respect to
B
C
,
A
C
,
A
B
BC,AC,AB
BC
,
A
C
,
A
B
respectively. Knowing that
D
,
E
,
F
,
B
D,E,F,B
D
,
E
,
F
,
B
are concyclic,find all possible values of
∠
B
\angle B
∠
B
.
1
2
Hide problems
an ineq with 2 variables
Let
a
,
b
a,b
a
,
b
be real numbers. Prove the inequality
2
(
a
4
+
a
2
b
2
+
b
4
)
≥
3
(
a
3
b
+
a
b
3
)
.
2(a^4+a^2b^2+b^4)\ge 3(a^3b+ab^3).
2
(
a
4
+
a
2
b
2
+
b
4
)
≥
3
(
a
3
b
+
a
b
3
)
.
the range of a+b
a
,
b
a,b
a
,
b
are real numbers such that:
a
3
+
b
3
=
8
−
6
a
b
.
a^3+b^3=8-6ab.
a
3
+
b
3
=
8
−
6
ab
.
Find the maximal and minimal value of
a
+
b
a+b
a
+
b
.