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Contests
National and Regional Contests
Lithuania Contests
Grand Duchy of Lithuania
2016 Grand Duchy of Lithuania
2016 Grand Duchy of Lithuania
Part of
Grand Duchy of Lithuania
Subcontests
(4)
1
1
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sum a/(a+b^2) <= 1/4 (sum 1/a) when a + b + c = 1, a,b,c>0
Let
a
,
b
a, b
a
,
b
and
c
c
c
be positive real numbers such that
a
+
b
+
c
=
1
a + b + c = 1
a
+
b
+
c
=
1
. Prove that
a
a
+
b
2
+
b
b
+
c
2
+
c
c
+
a
2
≤
1
4
(
1
a
+
1
b
+
1
c
)
\frac{a}{a+b^2}+\frac{b}{b+c^2}+\frac{c}{c+a^2} \le \frac{1}{4} \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right)
a
+
b
2
a
+
b
+
c
2
b
+
c
+
a
2
c
≤
4
1
(
a
1
+
b
1
+
c
1
)
4
1
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7^n -1 is divisible by 6^n -1
Determine all positive integers
n
n
n
such that
7
n
−
1
7^n -1
7
n
−
1
is divisible by
6
n
−
1
6^n -1
6
n
−
1
.
2
1
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during a school year 44 competitions among students were held
During a school year
44
44
44
competitions were held. Exactly
7
7
7
students won in each of the competitions. For any two competitions, there exists exactly
1
1
1
student who won in both competitions. Is it true that there exists a student who won all of the competitions?
3
1
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BD = EF iff AF = EC, ABC isosceles, BF = BE,ED is angle bisector of <BEC
Let
A
B
C
ABC
A
BC
be an isosceles triangle with
A
B
=
A
C
AB = AC
A
B
=
A
C
. Let
D
,
E
D, E
D
,
E
and
F
F
F
be points on line segments
B
C
,
C
A
BC, CA
BC
,
C
A
and
A
B
AB
A
B
, respectively, such that
B
F
=
B
E
BF = BE
BF
=
BE
and such that
E
D
ED
E
D
is the angle bisector of
∠
B
E
C
\angle BEC
∠
BEC
. Prove that
B
D
=
E
F
BD = EF
B
D
=
EF
if and only if
A
F
=
E
C
AF = EC
A
F
=
EC
.