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Contests
National and Regional Contests
Lithuania Contests
Grand Duchy of Lithuania
2021 Grand Duchy of Lithuania
2021 Grand Duchy of Lithuania
Part of
Grand Duchy of Lithuania
Subcontests
(4)
4
1
Hide problems
diophantine x^3 - yz^3 = 2021 with at least 2 primes
A triplet of positive integers
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
satisfying
x
,
y
,
z
>
1
x, y, z > 1
x
,
y
,
z
>
1
and
x
3
−
y
z
3
=
2021
x^3 - yz^3 = 2021
x
3
−
y
z
3
=
2021
is called primary if at least two of the integers
x
,
y
,
z
x, y, z
x
,
y
,
z
are prime numbers. a) Find at least one primary triplet. b) Show that there are infinitely many primary triplets.
3
1
Hide problems
right triangle from AC,BD,CD if <ADB + <ACB = <CAB + <DBA = 30^o, AD = BC.
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral satisfying
∠
A
D
B
+
∠
A
C
B
=
∠
C
A
B
+
∠
D
B
A
=
3
0
o
\angle ADB + \angle ACB = \angle CAB + \angle DBA = 30^o
∠
A
D
B
+
∠
A
CB
=
∠
C
A
B
+
∠
D
B
A
=
3
0
o
,
A
D
=
B
C
AD = BC
A
D
=
BC
. Prove that there exists a right-angled triangle with side lengths
A
C
AC
A
C
,
B
D
BD
B
D
,
C
D
CD
C
D
.
2
1
Hide problems
bw colorings of 1, 2, ... , 2021
Every number in the sequence
1
,
2
,
.
.
.
,
2021
1, 2, ... , 2021
1
,
2
,
...
,
2021
is either white or black. At one step Alice can choose three numbers of the sequence and change the color of each of them (white to black and black to white) if one of those three numbers is the arithmetic mean of the other two. Alice wants to perform several steps so that at the end all the numbers in the sequence are black. For which initial colorings of numbers can Alice achieve this?
1
1
Hide problems
f(x) = g(h(x)) - h(g(x)).
Prove that for any polynomial
f
(
x
)
f(x)
f
(
x
)
(with real coefficients) there exist polynomials
g
(
x
)
g(x)
g
(
x
)
and
h
(
x
)
h(x)
h
(
x
)
(with real coefficients) such that
f
(
x
)
=
g
(
h
(
x
)
)
−
h
(
g
(
x
)
)
f(x) = g(h(x)) - h(g(x))
f
(
x
)
=
g
(
h
(
x
))
−
h
(
g
(
x
))
.