MathDB
Problems
Contests
National and Regional Contests
Serbia Contests
Serbia JBMO TST
2019 Serbia JBMO TST
2019 Serbia JBMO TST
Part of
Serbia JBMO TST
Subcontests
(4)
1
1
Hide problems
number of divisors of n! is divisible by 2019
Does there exist a positive integer
n
n
n
, such that the number of divisors of
n
!
n!
n
!
is divisible by
2019
2019
2019
?
4
1
Hide problems
JBMO TST Serbia
4.
4.
4.
On a table there are notes of values:
1
1
1
,
2
2
2
,
5
5
5
,
10
10
10
,
20
20
20
,
50
50
50
,
100
100
100
,
200
200
200
,
500
500
500
,
1000
1000
1000
,
2000
2000
2000
and
5000
5000
5000
(the number of any of these notes can be any non-negative integer). Two players , First and Second play a game in turns (First plays first). With one move a player can take any one note of value higher than
1
1
1
, and replace it with notes of less value. The value of the chosen note is equal to the sum of the values of the replaced notes. The loser is the player which can not play any more moves. Which player has the winning strategy?
3
1
Hide problems
JBMO TST Serbia Problem 3
3.
3.
3.
Congruent circles
k
1
k_{1}
k
1
and
k
2
k_{2}
k
2
intersect in the points
A
A
A
and
B
B
B
. Let
P
P
P
be a variable point of arc
A
B
AB
A
B
of circle
k
2
k_{2}
k
2
which is inside
k
1
k_{1}
k
1
and let
A
P
AP
A
P
intersect
k
1
k_{1}
k
1
once more in point
C
C
C
, and the ray
C
B
CB
CB
intersects
k
2
k_{2}
k
2
once more in
D
D
D
. Let the angle bisector of
∠
C
A
D
\angle CAD
∠
C
A
D
intersect
k
1
k_{1}
k
1
in
E
E
E
, and the circle
k
2
k_{2}
k
2
in
F
F
F
. Ray
F
B
FB
FB
intersects
k
1
k_{1}
k
1
in
Q
Q
Q
. If
X
X
X
is one of the intersection points of circumscribed circles of triangles
C
D
P
CDP
C
D
P
and
E
Q
F
EQF
EQF
, prove that the triangle
C
F
X
CFX
CFX
is equilateral.
2
1
Hide problems
Weird inequality
If a b c positive reals smaller than 1, prove: a+b+c+2abc>ab+bc+ca+2(abc)^(1/2)