MathDB
Problems
Contests
National and Regional Contests
Bosnia Herzegovina Contests
Bosnia and Herzegovina EGMO Team Selection Test
2019 Bosnia and Herzegovina EGMO TST
2019 Bosnia and Herzegovina EGMO TST
Part of
Bosnia and Herzegovina EGMO Team Selection Test
Subcontests
(4)
4
1
Hide problems
n blue, n red and 1 green point on a circle
Let
n
n
n
be a natural number. There are
n
n
n
blue points ,
n
n
n
red points and one green point on the circle . Prove that it is possible to draw
n
n
n
lengths whose ends are in the given points, so that a maximum of one segment emerges from each point, no more than two segments intersect and the endpoints of none of the segments are blue and red points.[hide=original wording]Нека je ? природан број. На кружници се налази ? плавих, ? црвених и једна зелена тачка. Доказати да је могуће повући ? дужи чији су крајеви у датим тачкама, тако да из сваке тачке излази максимално једна дуж, никоје две дужи се не сијеку и крајње тачке ниједне од дужи нису плава и црвена тачка.
2
1
Hide problems
1 = d_1 < d_2 < ...< d_k = n=d_2d_3 + d_2d_5+d_3d_5
Let
1
=
d
1
<
d
2
<
.
.
.
<
d
k
=
n
1 = d_1 < d_2 < ...< d_k = n
1
=
d
1
<
d
2
<
...
<
d
k
=
n
be all natural divisors of the natural number
n
n
n
. Find all possible values of the number
k
k
k
if
n
=
d
2
d
3
+
d
2
d
5
+
d
3
d
5
n=d_2d_3 + d_2d_5+d_3d_5
n
=
d
2
d
3
+
d
2
d
5
+
d
3
d
5
.
1
1
Hide problems
x_k+x_{k+1}=x^2_{k+2} , n x n system
Let
x
1
,
x
2
,
.
.
.
,
x
n
x_1,x_2, ..., x_n
x
1
,
x
2
,
...
,
x
n
be non-negative real numbers. Solve the system of equations:
x
k
+
x
k
+
1
=
x
k
+
2
2
,
(
k
=
1
,
2
,
.
.
.
,
n
)
,
x_k+x_{k+1}=x^2_{k+2}\,\,,\,\,\, (k =1,2,...,n),
x
k
+
x
k
+
1
=
x
k
+
2
2
,
(
k
=
1
,
2
,
...
,
n
)
,
where
x
n
+
1
=
x
1
x_{n+1} = x_1
x
n
+
1
=
x
1
,
x
n
+
2
=
x
2
x_{n+2} = x_2
x
n
+
2
=
x
2
.
3
1
Hide problems
intersection of circumcircles lies on a line
The circle inscribed in the triangle
A
B
C
ABC
A
BC
touches the sides
A
B
AB
A
B
and
A
C
AC
A
C
at the points
K
K
K
and
L
L
L
, respectively. The angle bisectors from
B
B
B
and
C
C
C
intersect the altitude of the triangle from the vertex
A
A
A
at the points
Q
Q
Q
and
R
R
R
, respectively. Prove that one of the points of intersection of the circles circumscribed around the triangles
B
K
Q
BKQ
B
K
Q
and
C
P
L
CPL
CP
L
lies on
B
C
BC
BC
.