MathDB
Problems
Contests
National and Regional Contests
Saudi Arabia Contests
Saudi Arabia JBMO TST
2021 Saudi Arabia JBMO TST
2021 Saudi Arabia JBMO TST
Part of
Saudi Arabia JBMO TST
Subcontests
(4)
4
2
Hide problems
Sum a_ib_i =0
Let
F
F
F
is the set of all sequences
{
(
a
1
,
a
2
,
.
.
.
,
a
2020
)
}
\{(a_1, a_2, . . . , a_{2020})\}
{(
a
1
,
a
2
,
...
,
a
2020
)}
with
a
i
∈
{
−
1
,
1
}
a_i \in \{-1, 1\}
a
i
∈
{
−
1
,
1
}
for all
i
=
1
,
2
,
.
.
.
,
2020
i = 1,2,...,2020
i
=
1
,
2
,
...
,
2020
. Prove that there exists a set
S
S
S
, such that
S
⊂
F
S \subset F
S
⊂
F
,
∣
S
∣
=
2020
|S| = 2020
∣
S
∣
=
2020
and for any
(
a
1
,
a
2
,
.
.
.
,
a
2020
)
∈
F
(a_1,a_2,...,a_{2020}) \in F
(
a
1
,
a
2
,
...
,
a
2020
)
∈
F
there exists
(
b
1
,
b
2
,
.
.
.
,
b
2020
)
∈
S
(b_1,b_2,...,b_{2020}) \in S
(
b
1
,
b
2
,
...
,
b
2020
)
∈
S
, such that
∑
i
=
1
2020
a
i
b
i
=
0
\sum_{i=1}^{2020} a_ib_i = 0
∑
i
=
1
2020
a
i
b
i
=
0
.
every perfect square is amazing
Let us call a set of positive integers nice if the number of its elements equals to the average of its numbers. Call a positive integer
n
n
n
an amazing number if the set
{
1
,
2
,
.
.
.
,
n
}
\{1, 2 , . . . , n\}
{
1
,
2
,
...
,
n
}
can be partitioned into nice subsets. a) Prove that every perfect square is amazing. b) Show that there are infinitely many positive integers which are not amazing.
1
1
Hide problems
a_n = a^2_{n-1} + 15a_{n-1} has no perfect squares
Let
(
a
n
)
n
≥
1
(a_n)_{n\ge 1}
(
a
n
)
n
≥
1
be a sequence given by
a
1
=
45
a_1 = 45
a
1
=
45
and
a
n
=
a
n
−
1
2
+
15
a
n
−
1
a_n = a^2_{n-1} + 15a_{n-1}
a
n
=
a
n
−
1
2
+
15
a
n
−
1
for
n
>
1
n > 1
n
>
1
. Prove that the sequence contains no perfect squares.
3
1
Hide problems
Sum of n numbers is 0 when each is divisible by other’s sum
We have
n
>
2
n > 2
n
>
2
nonzero integers such that everyone of them is divisible by the sum of the other
n
−
1
n - 1
n
−
1
numbers, Show that the sum of the
n
n
n
numbers is precisely
0
0
0
.
2
2
Hide problems
BC=KM if CM=CK and <CBM = 2 <ABM and K lies on median BM
In a triangle
A
B
C
ABC
A
BC
, let
K
K
K
be a point on the median
B
M
BM
BM
such that
C
M
=
C
K
CM = CK
CM
=
C
K
. It turned out that
∠
C
B
M
=
2
∠
A
B
M
\angle CBM = 2\angle ABM
∠
CBM
=
2∠
A
BM
. Show that
B
C
=
K
M
BC = KM
BC
=
K
M
.
AE // MN wanted, starting with cyclic hexagon
In a circle
O
O
O
, there are six points,
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
,
E
E
E
,
F
F
F
in a counterclockwise order such that
B
D
⊥
C
F
BD \perp CF
B
D
⊥
CF
, and
C
F
CF
CF
,
B
E
BE
BE
,
A
D
AD
A
D
are concurrent. Let the perpendicular from
B
B
B
to
A
C
AC
A
C
be
M
M
M
, and the perpendicular from
D
D
D
to
C
E
CE
CE
be
N
N
N
. Prove that
A
E
∥
M
N
AE \parallel MN
A
E
∥
MN
.