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Problems
Contests
National and Regional Contests
Lithuania Contests
Grand Duchy of Lithuania
2015 Grand Duchy of Lithuania
2015 Grand Duchy of Lithuania
Part of
Grand Duchy of Lithuania
Subcontests
(4)
4
1
Hide problems
max a+b+c given gcd(a, b, c)=1, gcd(a, b+c)>1, gcd(b, c+a)>1, gcd(c, a+b)>1
We denote by gcd (...) the greatest common divisor of the numbers in (...). (For example, gcd
(
4
,
6
,
8
)
=
2
(4, 6, 8)=2
(
4
,
6
,
8
)
=
2
and gcd
(
12
,
15
)
=
3
(12, 15)=3
(
12
,
15
)
=
3
.) Suppose that positive integers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfy the following four conditions:
∙
\bullet
∙
gcd
(
a
,
b
,
c
)
=
1
(a, b, c)=1
(
a
,
b
,
c
)
=
1
,
∙
\bullet
∙
gcd
(
a
,
b
+
c
)
>
1
(a, b + c)>1
(
a
,
b
+
c
)
>
1
,
∙
\bullet
∙
gcd
(
b
,
c
+
a
)
>
1
(b, c + a)>1
(
b
,
c
+
a
)
>
1
,
∙
\bullet
∙
gcd
(
c
,
a
+
b
)
>
1
(c, a + b)>1
(
c
,
a
+
b
)
>
1
. a) Is it possible that
a
+
b
+
c
=
2015
a + b + c = 2015
a
+
b
+
c
=
2015
? b) Determine the minimum possible value that the sum
a
+
b
+
c
a+ b+ c
a
+
b
+
c
can take.
3
1
Hide problems
numbers 1 to 17 in 17x17 square table
A table consists of
17
×
17
17 \times 17
17
×
17
squares. In each square one positive integer from
1
1
1
to
17
17
17
is written, every such number is written in exactly
17
17
17
squares. Prove that there is a row or a column of the table that contains at least
5
5
5
different numbers.
1
1
Hide problems
y^2 + y + \sqrt{y - x^2 -xy} <= 3xy
Find all pairs of real numbers
(
x
,
y
)
(x, y)
(
x
,
y
)
for which the inequality
y
2
+
y
+
y
−
x
2
−
x
y
≤
3
x
y
y^2 + y + \sqrt{y - x^2 -xy} \le 3xy
y
2
+
y
+
y
−
x
2
−
x
y
≤
3
x
y
holds.
2
1
Hide problems
perpendicular wanted, intersecting circles related
Let
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
be two circles , with respective centres
O
1
O_1
O
1
and
O
2
O_2
O
2
, that intersect each other in
A
A
A
and
B
B
B
. The line
O
1
A
O_1A
O
1
A
intersects
ω
2
\omega_2
ω
2
in
A
A
A
and
C
C
C
and the line
O
2
A
O_2A
O
2
A
inetersects
ω
1
\omega_1
ω
1
in
A
A
A
and
D
D
D
. The line through
B
B
B
parallel to
A
D
AD
A
D
intersects
ω
1
\omega_1
ω
1
in
B
B
B
and
E
E
E
. Suppose that
O
1
A
O_1A
O
1
A
is parallel to
D
E
DE
D
E
. Show that
C
D
CD
C
D
is perpendicular to
O
2
C
O_2C
O
2
C
.