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Problems
Contests
National and Regional Contests
Saudi Arabia Contests
Saudi Arabia BMO TST
2014 Saudi Arabia BMO TST
2014 Saudi Arabia BMO TST
Part of
Saudi Arabia BMO TST
Subcontests
(5)
5
3
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Find all n
Find all positive integers
n
n
n
such that
3
n
+
4
n
+
⋯
+
(
n
+
2
)
n
=
(
n
+
3
)
n
.
3^n+4^n+\cdots+(n+2)^n=(n+3)^n.
3
n
+
4
n
+
⋯
+
(
n
+
2
)
n
=
(
n
+
3
)
n
.
Prove that lines PQ,BC, and MT are concurrent
Let
A
B
C
ABC
A
BC
be a triangle. Circle
Ω
\Omega
Ω
passes through points
B
B
B
and
C
C
C
. Circle
ω
\omega
ω
is tangent internally to
Ω
\Omega
Ω
and also to sides
A
B
AB
A
B
and
A
C
AC
A
C
at
T
,
P
,
T,~ P,
T
,
P
,
and
Q
Q
Q
, respectively. Let
M
M
M
be midpoint of arc
B
C
^
\widehat{BC}
BC
(containing T) of
Ω
\Omega
Ω
. Prove that lines
P
Q
,
B
C
,
P Q,~ BC,
PQ
,
BC
,
and
M
T
MT
MT
are concurrent.
Forming an n by n array of numbers with conditions
Let
n
>
3
n > 3
n
>
3
be an odd positive integer not divisible by
3
3
3
. Determine if it is possible to form an
n
×
n
n \times n
n
×
n
array of numbers such that[*] (a) the set of the numbers in each row is a permutation of
0
,
1
,
…
,
n
−
1
0, 1, \dots , n - 1
0
,
1
,
…
,
n
−
1
; the set of the numbers in each column is a permutation of
0
,
1
,
…
,
n
−
1
0, 1, \dots , n-1
0
,
1
,
…
,
n
−
1
;[*] (b) the board is totally non-symmetric: for
1
≤
i
<
j
≤
n
1 \le i < j \le n
1
≤
i
<
j
≤
n
and
1
≤
i
′
<
j
′
≤
n
1 \le i' < j' \le n
1
≤
i
′
<
j
′
≤
n
, if
(
i
,
j
)
≠
(
i
′
,
j
′
)
(i, j) \neq (i', j')
(
i
,
j
)
=
(
i
′
,
j
′
)
then
(
a
i
,
j
,
a
j
,
i
)
≠
(
a
i
′
,
j
′
,
a
j
′
,
i
′
)
(a_{i,j} , a_{j,i}) \neq (a_{i',j'} , a_{j',i'})
(
a
i
,
j
,
a
j
,
i
)
=
(
a
i
′
,
j
′
,
a
j
′
,
i
′
)
where
a
i
,
j
a_{i,j}
a
i
,
j
denotes the entry in the
i
th
i^\text{th}
i
th
row and
j
th
j^\text{th}
j
th
column.
4
3
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Prove that DIMP is cyclic
Let
A
B
C
ABC
A
BC
be a triangle with
∠
B
≤
∠
C
\angle B \le \angle C
∠
B
≤
∠
C
,
I
I
I
its incenter and
D
D
D
the intersection point of line
A
I
AI
A
I
with side
B
C
BC
BC
. Let
M
M
M
and
N
N
N
be points on sides
B
A
BA
B
A
and
C
A
CA
C
A
, respectively, such that
B
M
=
B
D
BM = BD
BM
=
B
D
and
C
N
=
C
D
CN = CD
CN
=
C
D
. The circumcircle of triangle
C
M
N
CMN
CMN
intersects again line
B
C
BC
BC
at
P
P
P
. Prove that quadrilateral
D
I
M
P
DIMP
D
I
MP
is cyclic.
Labeling a self intersecting polygon
Let
n
n
n
be an integer greater than
2
2
2
. Consider a set of
n
n
n
different points, with no three collinear, in the plane. Prove that we can label the points
P
1
,
P
2
,
…
,
P
n
P_1,~ P_2, \dots , P_n
P
1
,
P
2
,
…
,
P
n
such that
P
1
P
2
…
P
n
P_1P_2 \dots P_n
P
1
P
2
…
P
n
is not a self-intersecting polygon. (A polygon is self-intersecting if one of its side intersects the interior of another side. The polygon is not necessarily convex )
Injective function f
Let
f
:
N
→
N
f :\mathbb{N} \rightarrow\mathbb{N}
f
:
N
→
N
be an injective function such that
f
(
1
)
=
2
,
f
(
2
)
=
4
f(1) = 2,~ f(2) = 4
f
(
1
)
=
2
,
f
(
2
)
=
4
and
f
(
f
(
m
)
+
f
(
n
)
)
=
f
(
f
(
m
)
)
+
f
(
n
)
f(f(m) + f(n)) = f(f(m)) + f(n)
f
(
f
(
m
)
+
f
(
n
))
=
f
(
f
(
m
))
+
f
(
n
)
for all
m
,
n
∈
N
m, n \in \mathbb{N}
m
,
n
∈
N
. Prove that
f
(
n
)
=
n
+
2
f(n) = n + 2
f
(
n
)
=
n
+
2
for all
n
≥
2
n \ge 2
n
≥
2
.
3
3
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Deciman digits
Let
n
≥
2
n \ge 2
n
≥
2
be a positive integer, and write in a digit form
1
n
=
0.
a
1
a
2
…
.
\frac{1}{n}=0.a_1a_2\dots.
n
1
=
0.
a
1
a
2
…
.
Suppose that
n
=
a
1
+
a
2
+
⋯
n = a_1 + a_2 + \cdots
n
=
a
1
+
a
2
+
⋯
. Determine all possible values of
n
n
n
.
Prove that line PQ bisects diagonal BD
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram. A line
ℓ
\ell
ℓ
intersects lines
A
B
,
B
C
,
C
D
,
D
A
AB,~ BC,~ CD, ~DA
A
B
,
BC
,
C
D
,
D
A
at four different points
E
,
F
,
G
,
H
,
E,~ F,~ G,~ H,
E
,
F
,
G
,
H
,
respectively. The circumcircles of triangles
A
E
F
AEF
A
EF
and
A
G
H
AGH
A
G
H
intersect again at
P
P
P
. The circumcircles of triangles
C
E
F
CEF
CEF
and
C
G
H
CGH
CG
H
intersect again at
Q
Q
Q
. Prove that the line
P
Q
P Q
PQ
bisects the diagonal
B
D
BD
B
D
.
Complicated expressions with inequalties
Let
a
,
b
a, b
a
,
b
be two nonnegative real numbers and
n
n
n
a positive integer. Prove that
(
1
−
2
−
n
)
∣
a
2
n
−
b
2
n
∣
≥
a
b
∣
a
2
n
−
1
−
b
2
n
−
1
∣
.
\left(1-2^{-n}\right)\left|a^{2^n}-b^{2^n}\right|\ge\sqrt{ab}\left|a^{2^n-1}-b^{2^n-1}\right|.
(
1
−
2
−
n
)
a
2
n
−
b
2
n
≥
ab
a
2
n
−
1
−
b
2
n
−
1
.
2
3
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Difference between cubes is 91
Prove that among any
16
16
16
perfect cubes we can always find two cubes whose difference is divisible by
91
91
91
.
Find the minimum number of elements in S
Let
N
\mathbb{N}
N
denote the set of positive integers, and let
S
S
S
be a set. There exists a function
f
:
N
→
S
f :\mathbb{N} \rightarrow S
f
:
N
→
S
such that if
x
x
x
and
y
y
y
are a pair of positive integers with their difference being a prime number, then
f
(
x
)
≠
f
(
y
)
f(x) \neq f(y)
f
(
x
)
=
f
(
y
)
. Determine the minimum number of elements in
S
S
S
.
prove that Q,X,Y,Z, are collinear
Circles
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
meet at
P
P
P
and
Q
Q
Q
. Segments
A
C
AC
A
C
and
B
D
BD
B
D
are chords of
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
respectively, such that segment
A
B
AB
A
B
and ray
C
D
CD
C
D
meet at
P
P
P
. Ray
B
D
BD
B
D
and segment
A
C
AC
A
C
meet at
X
X
X
. Point
Y
Y
Y
lies on
ω
1
\omega_1
ω
1
such that
P
Y
∥
B
D
P Y \parallel BD
P
Y
∥
B
D
. Point
Z
Z
Z
lies on
ω
2
\omega_2
ω
2
such that
P
Z
∥
A
C
P Z \parallel AC
PZ
∥
A
C
. Prove that points
Q
,
X
,
Y
,
Z
Q,~ X,~ Y,~ Z
Q
,
X
,
Y
,
Z
are collinear.
1
3
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Proper divisor function
A positive proper divisor is a positive divisor of a number, excluding itself. For positive integers
n
≥
2
n \ge 2
n
≥
2
, let
f
(
n
)
f(n)
f
(
n
)
denote the number that is one more than the largest proper divisor of
n
n
n
. Determine all positive integers
n
n
n
such that
f
(
f
(
n
)
)
=
2
f(f(n)) = 2
f
(
f
(
n
))
=
2
.
Minimum of summation
Find the minimum of
∑
k
=
0
40
(
x
+
k
2
)
2
\sum\limits_{k=0}^{40} \left(x+\frac{k}{2}\right)^2
k
=
0
∑
40
(
x
+
2
k
)
2
where
x
x
x
is a real numbers
Find all functions f:N--> (0,∞)
Find all functions
f
:
N
→
(
0
,
∞
)
f:\mathbb{N}\rightarrow(0,\infty)
f
:
N
→
(
0
,
∞
)
such that
f
(
4
)
=
4
f(4)=4
f
(
4
)
=
4
and
1
f
(
1
)
f
(
2
)
+
1
f
(
2
)
f
(
3
)
+
⋯
+
1
f
(
n
)
f
(
n
+
1
)
=
f
(
n
)
f
(
n
+
1
)
,
∀
n
∈
N
,
\frac{1}{f(1)f(2)}+\frac{1}{f(2)f(3)}+\cdots+\frac{1}{f(n)f(n+1)}=\frac{f(n)}{f(n+1)},~\forall n\in\mathbb{N},
f
(
1
)
f
(
2
)
1
+
f
(
2
)
f
(
3
)
1
+
⋯
+
f
(
n
)
f
(
n
+
1
)
1
=
f
(
n
+
1
)
f
(
n
)
,
∀
n
∈
N
,
where
N
=
{
1
,
2
,
…
}
\mathbb{N}=\{1,2,\dots\}
N
=
{
1
,
2
,
…
}
is the set of positive integers.