MathDB
Problems
Contests
National and Regional Contests
Saudi Arabia Contests
Saudi Arabia IMO TST
2021 Saudi Arabia IMO TST
2021 Saudi Arabia IMO TST
Part of
Saudi Arabia IMO TST
Subcontests
(3)
1
1
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P(T)-a is an odd integer , where p(T) is product of all elements of T
For a non-empty set
T
T
T
denote by
p
(
T
)
p(T)
p
(
T
)
the product of all elements of
T
T
T
. Does there exist a set
T
T
T
of
2021
2021
2021
elements such that for any
a
∈
T
a\in T
a
∈
T
one has that
P
(
T
)
−
a
P(T)-a
P
(
T
)
−
a
is an odd integer? Consider two cases: 1) All elements of
T
T
T
are irrational numbers. 2) At least one element of
T
T
T
is a rational number.
5
1
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OK=3R wanted, incenter, external bisector, perp. bisector, IL _|_IK
Let
A
B
C
ABC
A
BC
be a non isosceles triangle with incenter
I
I
I
. The circumcircle of the triangle
A
B
C
ABC
A
BC
has radius
R
R
R
. Let
A
L
AL
A
L
be the external angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
with
L
∈
B
C
L \in BC
L
∈
BC
. Let
K
K
K
be the point on perpendicular bisector of
B
C
BC
BC
such that
I
L
⊥
I
K
IL \perp IK
I
L
⊥
I
K
.Prove that
O
K
=
3
R
OK=3R
O
K
=
3
R
.
2
1
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n =perfect number, \phi (n)= 2^k
Find all positive integers
n
n
n
, such that
n
n
n
is a perfect number and
φ
(
n
)
\varphi (n)
φ
(
n
)
is power of
2
2
2
.Note:a positive integer
n
n
n
, is called perfect if the sum of all its positive divisors is equal to
2
n
2n
2
n
.