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Problems
Contests
National and Regional Contests
Lithuania Contests
Grand Duchy of Lithuania
2018 Grand Duchy of Lithuania
2018 Grand Duchy of Lithuania
Part of
Grand Duchy of Lithuania
Subcontests
(4)
4
1
Hide problems
d - k and d are divisors of n
Find all positive integers
n
n
n
for which there exists a positive integer
k
k
k
such that for every positive divisor
d
d
d
of
n
n
n
, the number
d
−
k
d - k
d
−
k
is also a (not necessarily positive) divisor of
n
n
n
.
1
1
Hide problems
x^2 + y^2 > 1 if (x^2 + y^2 -1)(z^2 + t^2 - 1) > (xz + yt -1)^2
Let
x
,
y
,
z
,
t
x, y, z, t
x
,
y
,
z
,
t
be real numbers such that
(
x
2
+
y
2
−
1
)
(
z
2
+
t
2
−
1
)
>
(
x
z
+
y
t
−
1
)
2
(x^2 + y^2 -1)(z^2 + t^2 - 1) > (xz + yt -1)^2
(
x
2
+
y
2
−
1
)
(
z
2
+
t
2
−
1
)
>
(
x
z
+
y
t
−
1
)
2
. Prove that
x
2
+
y
2
>
1
x^2 + y^2 > 1
x
2
+
y
2
>
1
.
2
1
Hide problems
10 distinct numbers from {1, 2, 3, ... , 37 }
Ten distinct numbers are chosen at random from the set
{
1
,
2
,
3
,
.
.
.
,
37
}
\{1, 2, 3, ... , 37\}
{
1
,
2
,
3
,
...
,
37
}
. Show that one can select four distinct numbers out of those ten so that the sum of two of them is equal to the sum of the other two.
3
1
Hide problems
right angle wanted, orthocenter, circumcenter, midpoint, parallel related
The altitudes
A
D
AD
A
D
and
B
E
BE
BE
of an acute triangle
A
B
C
ABC
A
BC
intersect at point
H
H
H
. Let
F
F
F
be the intersection of the line
A
B
AB
A
B
and the line that is parallel to the side BC and goes through the circumcenter of
A
B
C
ABC
A
BC
. Let
M
M
M
be the midpoint of the segment
A
H
AH
A
H
. Prove that
∠
C
M
F
=
9
0
o
\angle CMF = 90^o
∠
CMF
=
9
0
o