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Problems
Contests
National and Regional Contests
Saudi Arabia Contests
Saudi Arabia BMO TST
2022 Saudi Arabia BMO + EGMO TST
2022 Saudi Arabia BMO + EGMO TST
Part of
Saudi Arabia BMO TST
Subcontests
(11)
p1
1
Hide problems
a_{n+1} = a_n+rad(a_n), product of all distinct prime factors
By
r
a
d
(
x
)
rad(x)
r
a
d
(
x
)
we denote the product of all distinct prime factors of a positive integer
n
n
n
. Given
a
∈
N
a \in N
a
∈
N
, a sequence
(
a
n
)
(a_n)
(
a
n
)
is defined by
a
0
=
a
a_0 = a
a
0
=
a
and
a
n
+
1
=
a
n
+
r
a
d
(
a
n
)
a_{n+1} = a_n+rad(a_n)
a
n
+
1
=
a
n
+
r
a
d
(
a
n
)
for all
n
≥
0
n \ge 0
n
≥
0
. Prove that there exists an index
n
n
n
for which
a
n
r
a
d
(
a
n
)
=
2022
\frac{a_n}{rad(a_n)} = 2022
r
a
d
(
a
n
)
a
n
=
2022
p3
1
Hide problems
numbering partitions of a positive integer into sum of exponents of 2
We consider all partitions of a positive integer n into a sum of (nonnegative integer) exponents of
2
2
2
(i.e.
1
1
1
,
2
2
2
,
4
4
4
,
8
8
8
,
.
.
.
. . .
...
). A number in the sum is allowed to repeat an arbitrary number of times (e.g.
7
=
2
+
2
+
1
+
1
+
1
7 = 2 + 2 + 1 + 1 + 1
7
=
2
+
2
+
1
+
1
+
1
) and two partitions differing only in the order of summands are considered to be equal (e.g.
8
=
4
+
2
+
1
+
1
8 = 4 + 2 + 1 + 1
8
=
4
+
2
+
1
+
1
and
8
=
1
+
2
+
1
+
4
8 = 1 + 2 + 1 + 4
8
=
1
+
2
+
1
+
4
are regarded to be the same partition). Let
E
(
n
)
E(n)
E
(
n
)
be the number of partitions in which an even number of exponents appear an odd number of times and
O
(
n
)
O(n)
O
(
n
)
the number of partitions in which an odd number of exponents appear an odd number of times. For example, for
n
=
5
n = 5
n
=
5
partitions counted in
E
(
n
)
E(n)
E
(
n
)
are
5
=
4
+
1
5 = 4 + 1
5
=
4
+
1
and
5
=
2
+
1
+
1
+
1
5 = 2 + 1 + 1 + 1
5
=
2
+
1
+
1
+
1
, whereas partitions counted in O(n) are
5
=
2
+
2
+
1
5 = 2 + 2 + 1
5
=
2
+
2
+
1
and
5
=
1
+
1
+
1
+
1
+
1
5 = 1 + 1 + 1 + 1 + 1
5
=
1
+
1
+
1
+
1
+
1
, hence
E
(
5
)
=
O
(
5
)
=
2
E(5) = O(5) = 2
E
(
5
)
=
O
(
5
)
=
2
. Find
E
(
n
)
−
O
(
n
)
E(n) - O(n)
E
(
n
)
−
O
(
n
)
as a function of
n
n
n
.
p2
1
Hide problems
f(g(x)) = x^3 and g(f(x)) = x^2.
Determine if there exist functions
f
,
g
:
R
→
R
f, g : R \to R
f
,
g
:
R
→
R
satisfying for every
x
∈
R
x \in R
x
∈
R
the following equations
f
(
g
(
x
)
)
=
x
3
f(g(x)) = x^3
f
(
g
(
x
))
=
x
3
and
g
(
f
(
x
)
)
=
x
2
g(f(x)) = x^2
g
(
f
(
x
))
=
x
2
.
2.4
2
Hide problems
2f(x)f(x + y)- f(x^2) = x}/2 (f(2x) + 4f(f(y)))
Find all functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
such that
2
f
(
x
)
f
(
x
+
y
)
−
f
(
x
2
)
=
x
2
(
f
(
2
x
)
+
4
f
(
f
(
y
)
)
)
2f(x)f(x + y) -f(x^2) =\frac{x}{2}(f(2x) + 4f(f(y)))
2
f
(
x
)
f
(
x
+
y
)
−
f
(
x
2
)
=
2
x
(
f
(
2
x
)
+
4
f
(
f
(
y
)))
for all
x
,
y
∈
R
x, y \in R
x
,
y
∈
R
.
f(x + 2y + f(x + y)) = f(2x) + f(3y),
Consider the function
f
:
R
+
→
R
+
f : R^+ \to R^+
f
:
R
+
→
R
+
and satisfying
f
(
x
+
2
y
+
f
(
x
+
y
)
)
=
f
(
2
x
)
+
f
(
3
y
)
,
∀
x
,
y
>
0.
f(x + 2y + f(x + y)) = f(2x) + f(3y), \,\, \forall \,\, x, y > 0.
f
(
x
+
2
y
+
f
(
x
+
y
))
=
f
(
2
x
)
+
f
(
3
y
)
,
∀
x
,
y
>
0.
1. Find all functions
f
(
x
)
f(x)
f
(
x
)
that satisfy the given condition.2. Suppose that
f
(
4
sin
4
x
)
f
(
4
cos
4
x
)
≥
f
2
(
1
)
f(4\sin^4x)f(4\cos^4x) \ge f^2(1)
f
(
4
sin
4
x
)
f
(
4
cos
4
x
)
≥
f
2
(
1
)
for all
x
∈
(
0
π
2
)
x \in \left(0\frac{\pi}{2}\right)
x
∈
(
0
2
π
)
. Find the minimum value of
f
(
2022
)
f(2022)
f
(
2022
)
.
2.3
2
Hide problems
m + i = r + 3, rectangle partitioned in rectangles
A rectangle
R
R
R
is partitioned into smaller rectangles whose sides are parallel with the sides of
R
R
R
. Let
B
B
B
be the set of all boundary points of all the rectangles in the partition, including the boundary of
R
R
R
. Let S be the set of all (closed) segments whose points belong to
B
B
B
. Let a maximal segment be a segment in
S
S
S
which is not a proper subset of any other segment in
S
S
S
. Let an intersection point be a point in which
4
4
4
rectangles of the partition meet. Let
m
m
m
be the number of maximal segments,
i
i
i
the number of intersection points and
r
r
r
the number of rectangles. Prove that
m
+
i
=
r
+
3
m + i = r + 3
m
+
i
=
r
+
3
.
n real numbers on a board, (a,b) -> B
Let
n
n
n
be an even positive integer. On a board n real numbers are written. In a single move we can erase any two numbers from the board and replace each of them with their product. Prove that for every
n
n
n
initial numbers one can in finite number of moves obtain
n
n
n
equal numbers on the board.
1.4
2
Hide problems
12n + 6 chairs around a round table, 6n + 3 married couples
At a gala banquet,
12
n
+
6
12n + 6
12
n
+
6
chairs, where
n
∈
N
n \in N
n
∈
N
, are equally arranged around a large round table. A seating will be called a proper seating of rank
n
n
n
if a gathering of
6
n
+
3
6n + 3
6
n
+
3
married couples sit around this table such that each seated person also has exactly one sibling (brother/sister) of the opposite gender present (siblings cannot be married to each other) and each man is seated closer to his wife than his sister. Among all proper seats of rank n find the maximum possible number of women seated closer to their brother than their husband. (The maximum is taken not only across all possible seating arrangements for a given gathering, but also across all possible gatherings.)
max no of swords (6 unit square tile) cut from 6x11
The sword is a figure consisting of
6
6
6
unit squares presented in the picture below (and any other figure obtained from it by rotation). https://cdn.artofproblemsolving.com/attachments/4/3/08494627d043ea575703564e9e6b5ba63dc2ef.png Determine the largest number of swords that can be cut from a
6
×
11
6\times 11
6
×
11
piece of paper divided into unit squares (each sword should consist of six such squares).
1.1
2
Hide problems
product of first k primes + 1= perfect power
Find all positive integers
k
k
k
such that the product of the first
k
k
k
primes increased by
1
1
1
is a power of an integer (with an exponent greater than
1
1
1
).
M_{P(x)} = x\in [0,2021] \max}{|P(x)|.
For each non-constant integer polynomial
P
(
x
)
P(x)
P
(
x
)
, let’s define
M
P
(
x
)
=
max
x
∈
[
0
,
2021
]
∣
P
(
x
)
∣
.
M_{P(x)} = \underset{x\in [0,2021]}{\max} |P(x)|.
M
P
(
x
)
=
x
∈
[
0
,
2021
]
max
∣
P
(
x
)
∣.
1. Find the minimum value of
M
P
(
x
)
M_{P(x)}
M
P
(
x
)
when deg
P
(
x
)
=
1
P(x) = 1
P
(
x
)
=
1
.2. Suppose that
P
(
x
)
∈
Z
[
x
]
P(x) \in Z[x]
P
(
x
)
∈
Z
[
x
]
when deg
P
(
x
)
=
n
P(x) = n
P
(
x
)
=
n
and
2
≤
n
≤
2022
2 \le n \le 2022
2
≤
n
≤
2022
. Prove that
M
P
(
x
)
≥
1011
M_{P(x)} \ge 1011
M
P
(
x
)
≥
1011
.
2.1
2
Hide problems
a_n is coprime to 2n + 1 , if a_{n+1} = a^2_n + a_n -1
Define
a
0
=
2
a_0 = 2
a
0
=
2
and
a
n
+
1
=
a
n
2
+
a
n
−
1
a_{n+1} = a^2_n + a_n -1
a
n
+
1
=
a
n
2
+
a
n
−
1
for
n
≥
0
n \ge 0
n
≥
0
. Prove that
a
n
a_n
a
n
is coprime to
2
n
+
1
2n + 1
2
n
+
1
for all
n
∈
N
n \in N
n
∈
N
.
circumcenter of ABC lies on (PXQ), AP = AB, PB//AC, AQ = AC, CQ // AB
Let
A
B
C
ABC
A
BC
be an acute-angled triangle. Point
P
P
P
is such that
A
P
=
A
B
AP = AB
A
P
=
A
B
and
P
B
∥
A
C
PB \parallel AC
PB
∥
A
C
. Point
Q
Q
Q
is such that
A
Q
=
A
C
AQ = AC
A
Q
=
A
C
and
C
Q
∥
A
B
CQ \parallel AB
CQ
∥
A
B
. Segments
C
P
CP
CP
and
B
Q
BQ
BQ
meet at point
X
X
X
. Prove that the circumcenter of triangle
A
B
C
ABC
A
BC
lies on the circumcircle of triangle
P
X
Q
PXQ
PXQ
.
1.2
2
Hide problems
ABCD cyclic wanted if AMCD, BMDC are tangentials
Point
M
M
M
on side
A
B
AB
A
B
of quadrilateral
A
B
C
D
ABCD
A
BC
D
is such that quadrilaterals
A
M
C
D
AMCD
A
MC
D
and
B
M
D
C
BMDC
BM
D
C
are circumscribed around circles centered at
O
1
O_1
O
1
and
O
2
O_2
O
2
respectively. Line
O
1
O
2
O_1O_2
O
1
O
2
cuts an isosceles triangle with vertex
M
M
M
from angle
C
M
D
CMD
CM
D
. prove that
A
B
C
D
ABCD
A
BC
D
is a cyclc quadrilateral.
c>=1, if f(x) = cx(x - 2)
Consider the polynomial f(x) = cx(x - 2) where
c
c
c
is a positive real number. For any
n
∈
Z
+
n \in Z^+
n
∈
Z
+
, the notation
g
n
(
x
)
g_n(x)
g
n
(
x
)
is a composite function
n
n
n
times of
f
f
f
and assume that the equation
g
n
(
x
)
=
0
g_n(x) = 0
g
n
(
x
)
=
0
has all of the
2
n
2^n
2
n
solutions are real numbers. 1. For
c
=
5
c = 5
c
=
5
, find in terms of
n
n
n
, the sum of all the solutions of
g
n
(
x
)
g_n(x)
g
n
(
x
)
, of which each multiple (if any) is counted only once. 2. Prove that
c
≥
1
c\ge 1
c
≥
1
.
2.2
2
Hide problems
CK + KM = BL + LM if AK = AL = BC
Given is an acute triangle
A
B
C
ABC
A
BC
with
B
C
<
C
A
<
A
B
BC < CA < AB
BC
<
C
A
<
A
B
. Points
K
K
K
and
L
L
L
lie on segments
A
C
AC
A
C
and
A
B
AB
A
B
and satisfy
A
K
=
A
L
=
B
C
AK = AL = BC
A
K
=
A
L
=
BC
. Perpendicular bisectors of segments
C
K
CK
C
K
and
B
L
BL
B
L
intersect line
B
C
BC
BC
at points
P
P
P
and
Q
Q
Q
, respectively. Segments
K
P
KP
K
P
and
L
Q
LQ
L
Q
intersect at
M
M
M
. Prove that
C
K
+
K
M
=
B
L
+
L
M
CK + KM = BL + LM
C
K
+
K
M
=
B
L
+
L
M
.
positive integers n that have precisely \sqrt{n + 1} natural divisors.
Find all positive integers
n
n
n
that have precisely
n
+
1
\sqrt{n + 1}
n
+
1
natural divisors.
1.3
2
Hide problems
BP = CM wanted , 2 circles inscribed in <BAC each tangent to a side
Given is triangle
A
B
C
ABC
A
BC
with
A
B
>
A
C
AB > AC
A
B
>
A
C
. Circles
O
B
O_B
O
B
,
O
C
O_C
O
C
are inscribed in angle
B
A
C
BAC
B
A
C
with
O
B
O_B
O
B
tangent to
A
B
AB
A
B
at
B
B
B
and
O
C
O_C
O
C
tangent to
A
C
AC
A
C
at
C
C
C
. Tangent to
O
B
O_B
O
B
from
C
C
C
different than
A
C
AC
A
C
intersects
A
B
AB
A
B
at
K
K
K
and tangent to
O
C
O_C
O
C
from
B
B
B
different than
A
B
AB
A
B
intersects
A
C
AC
A
C
at
L
L
L
. Line
K
L
KL
K
L
and the angle bisector of
B
A
C
BAC
B
A
C
intersect
B
C
BC
BC
at points
P
P
P
and
M
M
M
, respectively. Prove that
B
P
=
C
M
BP = CM
BP
=
CM
.
gcd(m^{n-1} - 1, n) > 1 if n | m^{p(n-1)} - 1
Let
p
p
p
be a prime number and let
m
,
n
m, n
m
,
n
be integers greater than
1
1
1
such that
n
∣
m
p
(
n
−
1
)
−
1
n | m^{p(n-1)} - 1
n
∣
m
p
(
n
−
1
)
−
1
. Prove that
g
c
d
(
m
n
−
1
−
1
,
n
)
>
1
gcd(m^{n-1} - 1, n) > 1
g
c
d
(
m
n
−
1
−
1
,
n
)
>
1
.