MathDB
Problems
Contests
National and Regional Contests
Greece Contests
Greece JBMO TST
2023 Greece JBMO TST
2023 Greece JBMO TST
Part of
Greece JBMO TST
Subcontests
(3)
1
1
Hide problems
student that has at least 10 friends
A class has
24
24
24
students. Each group consisting of three of the students meet, and choose one of the other
21
21
21
students, A, to make him a gift. In this case, A considers each member of the group that offered him a gift as being his friend. Prove that there is a student that has at least
10
10
10
friends.
3
1
Hide problems
sum (a^2 + b^2)/2ab + 2(ab + bc + ca)/3 >=5
Let
a
,
b
,
a, b,
a
,
b
,
and
c
c
c
be positive real numbers such that
a
2
+
b
2
+
c
2
=
3
a^2 + b^2 + c^2 = 3
a
2
+
b
2
+
c
2
=
3
. Prove that
a
2
+
b
2
2
a
b
+
b
2
+
c
2
2
b
c
+
c
2
+
a
2
2
c
a
+
2
(
a
b
+
b
c
+
c
a
)
3
≥
5
\frac{a^2 + b^2}{2ab} + \frac{b^2 + c^2}{2bc} + \frac{c^2 + a^2}{2ca} + \frac{2(ab + bc + ca)}{3} \ge 5
2
ab
a
2
+
b
2
+
2
b
c
b
2
+
c
2
+
2
c
a
c
2
+
a
2
+
3
2
(
ab
+
b
c
+
c
a
)
≥
5
When equality holds?
2
1
Hide problems
Equal segments in a cyclic quadrilateral
Consider a cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
in which
B
C
=
C
D
BC = CD
BC
=
C
D
and
A
B
<
A
D
AB < AD
A
B
<
A
D
. Let
E
E
E
be a point on the side
A
D
AD
A
D
and
F
F
F
a point on the line
B
C
BC
BC
such that
A
E
=
A
B
=
A
F
AE = AB = AF
A
E
=
A
B
=
A
F
. Prove that
E
F
∥
B
D
EF \parallel BD
EF
∥
B
D
.