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National and Regional Contests
Lithuania Contests
Grand Duchy of Lithuania
2022 Grand Duchy of Lithuania
2022 Grand Duchy of Lithuania
Part of
Grand Duchy of Lithuania
Subcontests
(4)
1
1
Hide problems
max no of roots of x^{20} + a_{19}x^{19} + ... + a_1x + a_0 in (99, 100)
Given a polynomial with integer coefficients
P
(
x
)
=
x
20
+
a
19
x
19
+
.
.
.
+
a
1
x
+
a
0
,
P(x) = x^{20} + a_{19}x^{19} +... + a_1x + a_0,
P
(
x
)
=
x
20
+
a
19
x
19
+
...
+
a
1
x
+
a
0
,
having
20
20
20
different real roots. Determine the maximum number of roots such a polynomial
P
P
P
can have in the interval
(
99
,
100
)
(99, 100)
(
99
,
100
)
.
4
1
Hide problems
2^a + 2^b + 2^c + 3 is perfect square
Find all triples of natural numbers
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
for which the number
2
a
+
2
b
+
2
c
+
3
2^a + 2^b + 2^c + 3
2
a
+
2
b
+
2
c
+
3
is the square of an integer.
3
1
Hide problems
circumcenter of BDF lies on circumcircle of AEO, ABC isosceles
The center
O
O
O
of the circle
ω
\omega
ω
passing through the vertex
C
C
C
of the isosceles triangle
A
B
C
ABC
A
BC
(
A
B
=
A
C
AB = AC
A
B
=
A
C
) is the interior point of the triangle
A
B
C
ABC
A
BC
. This circle intersects segments
B
C
BC
BC
and
A
C
AC
A
C
at points
D
≠
C
D \ne C
D
=
C
and
E
≠
C
E \ne C
E
=
C
, respectively, and the circumscribed circle
Ω
\Omega
Ω
of the triangle
A
E
O
AEO
A
EO
at the point
F
≠
E
F \ne E
F
=
E
. Prove that the center of the circumcircle of the triangle
B
D
F
BDF
B
D
F
lies on the circle
Ω
\Omega
Ω
.
2
1
Hide problems
max no of students that could participate in the Olympics
During the mathematics Olympiad, students solved three problems. Each task was evaluated with an integer number of points from
0
0
0
to
7
7
7
. There is at most one problem for each pair of students, for which they got after the same number of points. Determine the maximum number of students could participate in the Olympics.