MathDB
Problems
Contests
National and Regional Contests
Saudi Arabia Contests
Saudi Arabia BMO TST
2015 Saudi Arabia BMO TST
2015 Saudi Arabia BMO TST
Part of
Saudi Arabia BMO TST
Subcontests
(4)
1
2
Hide problems
x_{i+1} - 8x_i^3 -4x_i + 3x_{i-1} + 1 = 0
Prove that for any integer
n
≥
2
n \ge 2
n
≥
2
, there exists a unique finite sequence
x
0
,
x
1
,
.
.
.
,
x
n
x_0, x_1,..., x_n
x
0
,
x
1
,
...
,
x
n
of real numbers which satisfies
x
0
=
x
n
=
0
x_0 = x_n = 0
x
0
=
x
n
=
0
and
x
i
+
1
−
8
x
i
3
−
4
x
i
+
3
x
i
−
1
+
1
=
0
x_{i+1} - 8x_i^3 -4x_i + 3x_{i-1} + 1 = 0
x
i
+
1
−
8
x
i
3
−
4
x
i
+
3
x
i
−
1
+
1
=
0
for all
i
=
1
,
2
,
.
.
.
,
n
−
1
i = 1,2,...,n - 1
i
=
1
,
2
,
...
,
n
−
1
. Prove moreover that
∣
x
i
∣
≤
1
2
|x_i| \le \frac12
∣
x
i
∣
≤
2
1
for all
i
=
1
,
2
,
.
.
.
,
n
−
1
i = 1,2,...,n - 1
i
=
1
,
2
,
...
,
n
−
1
.Nguyễn Duy Thái Sơn
for any m, n in Z there exists a k in Z such that f(k) = f(m) - f(n).
Find all strictly increasing functions
f
:
Z
→
R
f : Z \to R
f
:
Z
→
R
such that for any
m
,
n
∈
Z
m, n \in Z
m
,
n
∈
Z
there exists a
k
∈
Z
k \in Z
k
∈
Z
such that
f
(
k
)
=
f
(
m
)
−
f
(
n
)
f(k) = f(m) - f(n)
f
(
k
)
=
f
(
m
)
−
f
(
n
)
.Nguyễn Duy Thái Sơn
4
2
Hide problems
gcd (k + p_1p_2...p_i, p_1p_2...p_n) = 1
Let
n
≥
2
n \ge 2
n
≥
2
be an integer and
p
1
<
p
2
<
.
.
.
<
p
n
p_1 < p_2 < ... < p_n
p
1
<
p
2
<
...
<
p
n
prime numbers. Prove that there exists an integer
k
k
k
relatively prime with
p
1
p
2
.
.
.
p
n
p_1p_2... p_n
p
1
p
2
...
p
n
and such that
g
c
d
(
k
+
p
1
p
2
.
.
.
p
i
,
p
1
p
2
.
.
.
p
n
)
=
1
gcd (k + p_1p_2...p_i, p_1p_2...p_n) = 1
g
c
d
(
k
+
p
1
p
2
...
p
i
,
p
1
p
2
...
p
n
)
=
1
for all
i
=
1
,
2
,
.
.
.
,
n
−
1
i = 1, 2,..., n - 1
i
=
1
,
2
,
...
,
n
−
1
.Malik Talbi
7^{n-1} - 3^{n-1} is divisible by n, non prime n
Prove that there exist infinitely many non prime positive integers
n
n
n
such that
7
n
−
1
−
3
n
−
1
7^{n-1} - 3^{n-1}
7
n
−
1
−
3
n
−
1
is divisible by
n
n
n
.Lê Anh Vinh
2
2
Hide problems
2015 subsets of the set {1, 2,..., 1000}
Given
2015
2015
2015
subsets
A
1
,
A
2
,
.
.
.
,
A
2015
A_1, A_2,...,A_{2015}
A
1
,
A
2
,
...
,
A
2015
of the set
{
1
,
2
,
.
.
.
,
1000
}
\{1, 2,..., 1000\}
{
1
,
2
,
...
,
1000
}
such that
∣
A
i
∣
≥
2
|A_i| \ge 2
∣
A
i
∣
≥
2
for every
i
≥
1
i \ge 1
i
≥
1
and
∣
A
i
∩
A
j
∣
≥
1
|A_i \cap A_j| \ge 1
∣
A
i
∩
A
j
∣
≥
1
for every
1
≤
i
<
j
≤
2015
1 \le i < j \le 2015
1
≤
i
<
j
≤
2015
. Prove that
k
=
3
k = 3
k
=
3
is the smallest number of colors such that we can always color the elements of the set
{
1
,
2
,
.
.
.
,
1000
}
\{1, 2,..., 1000\}
{
1
,
2
,
...
,
1000
}
by
k
k
k
colors with the property that the subset
A
i
A_i
A
i
has at least two elements of different colors for every
i
≥
1
i \ge 1
i
≥
1
.Lê Anh Vinh
a_1 + a_2 + a_3 + a_4 + a_5 + a_6 = 30 , on sides of hexagon
Find the number of
6
6
6
-tuples
(
a
1
,
a
2
,
a
3
,
a
4
,
a
5
,
a
6
)
(a_1,a_2, a_3,a_4, a_5,a_6)
(
a
1
,
a
2
,
a
3
,
a
4
,
a
5
,
a
6
)
of distinct positive integers satisfying the following two conditions: (a)
a
1
+
a
2
+
a
3
+
a
4
+
a
5
+
a
6
=
30
a_1 + a_2 + a_3 + a_4 + a_5 + a_6 = 30
a
1
+
a
2
+
a
3
+
a
4
+
a
5
+
a
6
=
30
(b) We can write
a
1
,
a
2
,
a
3
,
a
4
,
a
5
,
a
6
a_1,a_2, a_3,a_4, a_5,a_6
a
1
,
a
2
,
a
3
,
a
4
,
a
5
,
a
6
on sides of a hexagon such that after a finite number of time choosing a vertex of the hexagon and adding
1
1
1
to the two numbers written on two sides adjacent to the vertex, we obtain a hexagon with equal numbers on its sides. Lê Anh Vinh
3
2
Hide problems
tangent circles wanted, circumcircle, incenter related
Let
A
B
C
ABC
A
BC
be a triangle,
Γ
\Gamma
Γ
its circumcircle,
I
I
I
its incenter, and
ω
\omega
ω
a tangent circle to the line
A
I
AI
A
I
at
I
I
I
and to the side
B
C
BC
BC
. Prove that the circles
Γ
\Gamma
Γ
and
ω
\omega
ω
are tangent.Malik Talbi
collinear intersectionds of circumcircles, incenter, altitudes related
Let
A
B
C
ABC
A
BC
be a triangle,
H
a
,
H
b
H_a, H_b
H
a
,
H
b
and
H
c
H_c
H
c
the feet of its altitudes from
A
,
B
A, B
A
,
B
and
C
C
C
, respectively,
T
a
,
T
b
,
T
c
T_a, T_b, T_c
T
a
,
T
b
,
T
c
its touchpoints of the incircle with the sides
B
C
,
C
A
BC, CA
BC
,
C
A
and
A
B
AB
A
B
, respectively. The circumcircles of triangles
A
H
b
H
c
AH_bH_c
A
H
b
H
c
and
A
T
b
T
c
AT_bT_c
A
T
b
T
c
intersect again at
A
′
A'
A
′
. The circumcircles of triangles
B
H
c
H
a
BH_cH_a
B
H
c
H
a
and
B
T
c
T
a
BT_cT_a
B
T
c
T
a
intersect again at
B
′
B'
B
′
. The circumcircles of triangles
C
H
a
H
b
CH_aH_b
C
H
a
H
b
and
C
T
a
T
b
CT_aT_b
C
T
a
T
b
intersect again at
C
′
C'
C
′
. Prove that the points
A
′
,
B
′
,
C
′
A',B',C'
A
′
,
B
′
,
C
′
are collinear.Malik Talbi