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Problems
Contests
National and Regional Contests
Saudi Arabia Contests
Saudi Arabia IMO TST
2022 Saudi Arabia IMO TST
2022 Saudi Arabia IMO TST
Part of
Saudi Arabia IMO TST
Subcontests
(2)
1
2
Hide problems
exists index i, such prime p|a_i, a_{n+1}=a_n^2 + n a_n .
Let
(
a
n
)
(a_n)
(
a
n
)
be the integer sequence which is defined by
a
1
=
1
a_1= 1
a
1
=
1
and
a
n
+
1
=
a
n
2
+
n
⋅
a
n
,
∀
n
≥
1.
a_{n+1}=a_n^2 + n \cdot a_n \,\, , \,\, \forall n \ge 1.
a
n
+
1
=
a
n
2
+
n
⋅
a
n
,
∀
n
≥
1.
Let
S
S
S
be the set of all primes
p
p
p
such that there exists an index
i
i
i
such that
p
∣
a
i
p|a_i
p
∣
a
i
. Prove that the set
S
S
S
is an infinite set and it is not equal to the set of all primes.
2022 / 2023 plates around a round table, 2 player game
There are a)
2022
2022
2022
, b)
2023
2023
2023
plates placed around a round table and on each of them there is one coin. Alice and Bob are playing a game that proceeds in rounds indefinitely as follows. In each round, Alice first chooses a plate on which there is at least one coin. Then Bob moves one coin from this plate to one of the two adjacent plates, chosen by him. Determine whether it is possible for Bob to select his moves so that, no matter how Alice selects her moves, there are never more than two coins on any plate.
3
2
Hide problems
f(ab + bc + ca) =f(a)f(b) +f(b)f(c)+f(c)f(a) in Q^+
Find all non-constant functions
f
:
Q
+
→
Q
+
f : Q^+ \to Q^+
f
:
Q
+
→
Q
+
satisfying the equation
f
(
a
b
+
b
c
+
c
a
)
=
f
(
a
)
f
(
b
)
+
f
(
b
)
f
(
c
)
+
f
(
c
)
f
(
a
)
f(ab + bc + ca) =f(a)f(b) +f(b)f(c)+f(c)f(a)
f
(
ab
+
b
c
+
c
a
)
=
f
(
a
)
f
(
b
)
+
f
(
b
)
f
(
c
)
+
f
(
c
)
f
(
a
)
for all
a
,
b
,
c
∈
Q
+
a, b,c \in Q^+
a
,
b
,
c
∈
Q
+
.
concyclic wanted, common external tangents of 2 circles meet on circle
Let
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
be points on the line
d
d
d
in that order and
A
B
=
C
D
AB = CD
A
B
=
C
D
. Denote
(
P
)
(P)
(
P
)
as some circle that passes through
A
,
B
A, B
A
,
B
with its tangent lines at
A
,
B
A, B
A
,
B
are
a
,
b
a,b
a
,
b
. Denote
(
Q
)
(Q)
(
Q
)
as some circle that passes through
C
,
D
C, D
C
,
D
with its tangent lines at
C
,
D
C, D
C
,
D
are
c
,
d
c,d
c
,
d
. Suppose that
a
a
a
cuts
c
,
d
c, d
c
,
d
at
K
,
L
K, L
K
,
L
respectively and
b
b
b
cuts
c
,
d
c, d
c
,
d
at
M
,
N
M, N
M
,
N
respectively. Prove that four points
K
,
L
,
M
,
N
K, L, M,N
K
,
L
,
M
,
N
belong to a same circle
(
ω
)
(\omega)
(
ω
)
and the common external tangent lines of circles
(
P
)
(P)
(
P
)
,
(
Q
)
(Q)
(
Q
)
meet on
(
ω
)
(\omega)
(
ω
)
.