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Problems
Contests
National and Regional Contests
Estonia Contests
SAFEST (South Africa - Estonia) Olympiad
2019 SAFEST Olympiad
2019 SAFEST Olympiad
Part of
SAFEST (South Africa - Estonia) Olympiad
Subcontests
(3)
4
1
Hide problems
sum 1/(a_i +2019)=1/2019, min of product a_1a_2... a_{2019}, a_i>0
Let
a
1
,
a
2
,
.
.
.
,
a
2019
a_1, a_2, . . . , a_{2019}
a
1
,
a
2
,
...
,
a
2019
be any positive real numbers such that
1
a
1
+
2019
+
1
a
2
+
2019
+
.
.
.
+
1
a
2019
+
2019
=
1
2019
\frac{1}{a_1 + 2019}+\frac{1}{a_2 + 2019}+ ... +\frac{1}{a_{2019} + 2019}=\frac{1}{2019}
a
1
+
2019
1
+
a
2
+
2019
1
+
...
+
a
2019
+
2019
1
=
2019
1
. Find the minimum value of
a
1
a
2
.
.
.
a
2019
a_1a_2... a_{2019}
a
1
a
2
...
a
2019
and determine for which values of
a
1
,
a
2
,
.
.
.
,
a
2019
a_1, a_2, . . . , a_{2019}
a
1
,
a
2
,
...
,
a
2019
this minimum occurs
5
1
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25 imo IMO participants at a party, same language to at least 10 others
There are
25
25
25
IMO participants attending a party. Every two of them speak to each other in some language, and they use only one language even if they both know some other language as well. Among every three participants there is a person who uses the same language to speak to the other two (in that group of three). Prove that there is an IMO participant who speaks the same language to at least
10
10
10
other participants
1
1
Hide problems
midpoint of MN lies on the circumcircle of PQR, isosceles related
Let
A
B
C
ABC
A
BC
be an isosceles triangle with
A
B
=
A
C
AB = AC
A
B
=
A
C
. Let
A
D
AD
A
D
be the diameter of the circumcircle of
A
B
C
ABC
A
BC
and let
P
P
P
be a point on the smaller arc
B
D
BD
B
D
. The line
D
P
DP
D
P
intersects the rays
A
B
AB
A
B
and
A
C
AC
A
C
at points
M
M
M
and
N
N
N
, respectively. The line
A
D
AD
A
D
intersects the lines
B
P
BP
BP
and
C
P
CP
CP
at points
Q
Q
Q
and
R
R
R
, respectively. Prove that the midpoint of
M
N
MN
MN
lies on the circumcircle of
P
Q
R
PQR
PQR